Engineering Calculators (Mechanical, Civil, Fluids & HVAC)

Compute Re from density, velocity, pipe diameter and viscosity — with instant laminar/turbulent classification.

Classify hydraulic-oil or process-fluid flow regime from kinematic viscosity, line velocity and tube bore.

Solve downstream pressure between two points from velocities, elevations and fluid density (ideal, lossless).

Q = A·V — volumetric flow from pipe diameter and velocity, in L/s, m³/h and US gpm.

Back out water velocity in a pipe from flow rate and bore, checked against recommended design limits.

Convert between dynamic (cP, Pa·s) and kinematic (cSt, m²/s) viscosity through fluid density.

Convert volumetric flow to mass flow (kg/s, kg/h, t/day) for process and metering work.

Flow through a sharp-edged orifice from differential pressure, bore and discharge coefficient.

Flow rate through a venturi from inlet/throat diameters and measured pressure drop.

Check line velocity and dynamic pressure for a given duty against erosion and noise limits.

Reynolds number and flow regime for air (dry air at 20 °C, 1 atm) from velocity and pipe bore.

Reynolds number and flow regime for seawater (standard seawater, 3.5% salinity, 20 °C) from velocity and pipe bore.

Reynolds number and flow regime for hot water (80 °c) (water at 80 °C) from velocity and pipe bore.

Reynolds number and flow regime for chilled water (5 °c) (water at 5 °C) from velocity and pipe bore.

Reynolds number and flow regime for sae 30 engine oil (SAE 30 at ~20 °C) from velocity and pipe bore.

Reynolds number and flow regime for iso vg46 hydraulic oil (ISO VG46 at 40 °C (ν ≈ 46 cSt)) from velocity and pipe bore.

Reynolds number and flow regime for diesel fuel (No. 2 diesel at 20 °C) from velocity and pipe bore.

Reynolds number and flow regime for petrol / gasoline (gasoline at 20 °C) from velocity and pipe bore.

Reynolds number and flow regime for kerosene / jet-a (kerosene at 20 °C) from velocity and pipe bore.

Reynolds number and flow regime for 50% ethylene glycol (50/50 EG–water at 20 °C) from velocity and pipe bore.

Reynolds number and flow regime for milk (whole milk at 20 °C) from velocity and pipe bore.

Reynolds number and flow regime for crude oil (light crude (≈35° API) at 20 °C; heavy crudes run far higher) from velocity and pipe bore.

Reynolds number and flow regime for glycerine (pure glycerine at 20 °C) from velocity and pipe bore.

Reynolds number and flow regime for natural gas (methane-rich gas at 1 atm, 20 °C — use line density at pressure) from velocity and pipe bore.

How many GPM / L/s a ½" schedule-40 steel pipe carries at your chosen design velocity.

How many GPM / L/s a ¾" schedule-40 steel pipe carries at your chosen design velocity.

How many GPM / L/s a 1" schedule-40 steel pipe carries at your chosen design velocity.

How many GPM / L/s a 1½" schedule-40 steel pipe carries at your chosen design velocity.

How many GPM / L/s a 2" schedule-40 steel pipe carries at your chosen design velocity.

How many GPM / L/s a 3" schedule-40 steel pipe carries at your chosen design velocity.

How many GPM / L/s a 4" schedule-40 steel pipe carries at your chosen design velocity.

How many GPM / L/s a 6" schedule-40 steel pipe carries at your chosen design velocity.

How many GPM / L/s a 8" schedule-40 steel pipe carries at your chosen design velocity.

How many GPM / L/s a 10" schedule-40 steel pipe carries at your chosen design velocity.

How many GPM / L/s a 12" schedule-40 steel pipe carries at your chosen design velocity.

Check pump suction line velocity against the recommended 0.6–1.2 m/s design window for the duty.

Check pump discharge line velocity against the recommended 1.5–3 m/s design window for the duty.

Check chilled water line velocity against the recommended 1.2–2.4 m/s design window for the duty.

Check condenser water line velocity against the recommended 1.5–3 m/s design window for the duty.

Check fire sprinkler main velocity against the recommended 3–6 m/s design window for the duty.

Check irrigation mainline velocity against the recommended 0.9–1.5 m/s design window for the duty.

Check compressed air main velocity against the recommended 6–9 m/s design window for the duty.

Check natural gas line velocity against the recommended 10–20 m/s design window for the duty.

Check hot water recirculation line velocity against the recommended 0.6–1.2 m/s design window for the duty.

Time to empty a vertical tank through a bottom orifice (Torricelli, integrated).

Outflow velocity and flow from an opening h metres below the liquid surface.

Fluid velocity from pitot–static differential pressure for liquids or gases.

Open-channel discharge through a Parshall flume from throat width and head.

Discharge over a trapezoidal (Cipolletti) weir — irrigation's favourite, no end-contraction math.

Velocity and discharge of a rectangular open channel via Manning's equation.

Gravity-flow capacity of a circular pipe flowing full, from diameter, slope and roughness.

Froude number and sub/supercritical classification for open-channel flow.

Surge pressure from sudden valve closure via the Joukowsky equation.

Ratio of inertia to surface tension — governs droplet breakup and atomization.

Convert pressure to metres or feet of liquid column for any specific gravity.

Convert a liquid column height into static pressure for any specific gravity.

Absolute and gauge pressure at depth in fresh or sea water — for tanks, dams and diving.

Convert velocity to velocity head V²/2g — the building block of every loss calculation.

Head loss per unit length between two gauges — slope of the hydraulic grade line.

Convert between specific gravity, density and unit weight for any liquid.

Convert between API gravity and specific gravity / density for crude and products.

Convert Saybolt Universal Seconds to centistokes (and back) for lube-oil specs.

Hydraulic diameter of a rectangular duct for Re and friction calculations.

Hydraulic diameter of an annular gap (pipe-in-pipe, heat-exchanger shell side).

One flow, every unit: L/s, m³/h, US gpm, UK gpm, cfm, MGD side by side.

Convert standard to actual air flow at your line pressure and temperature.

Convert gauge to absolute pressure (and vacuum) at any barometric pressure.

Gas density at any pressure and temperature from molar mass via the ideal-gas law.

Pressure drop in a compressed-air line via the empirical Harris formula.

Terminal settling speed of a small particle in liquid or gas (Stokes' law).

Partial-fill volume of a horizontal cylinder tank from dip level — fuel, water, oil.

Capacity and current contents of a vertical cylinder tank from level.

Volume of rectangular tanks and sumps in litres, m³ and US gallons.

Total capacity of a vertical tank with a conical bottom section.

How long a pump or supply line takes to fill a tank of known capacity.

Flow through a siphon hose from elevation drop, hose bore and length (with friction).

Jet reaction force on a fire or wash-down nozzle from flow and tip diameter.

Force of a liquid jet striking a flat plate or vane (momentum flux).

Flow from any K-factor nozzle (sprinkler, spray, wash-down) at a given pressure.

Development length before flow becomes fully developed — laminar and turbulent.

Cavitation index σ for valves, propellers and venturis — margin above vapour pressure.

Kármán vortex frequency behind a cylinder — chimneys, thermowells, cables.

Laminar/turbulent boundary-layer growth along a flat plate at distance x.

Terminal fall speed of a sphere in air or liquid, iterating drag coefficient.

Height a liquid climbs in a fine tube from surface tension, contact angle and bore.

Flow through an ISA-1932 nozzle from differential pressure — high-velocity steam/water metering.

Air loss through a leak hole at line pressure (choked-orifice model).

Annual energy cost of compressed-air leaks from leak rate, specific power and tariff.

Realistic garden-hose flow from bore, length and supply pressure (Hazen-Williams).

Discharge from a smooth-bore fire nozzle: Q = 29.7·d²·√P (NFPA form).

Re for air in rectangular ducting using hydraulic diameter.

The velocity at which pipe flow leaves the laminar regime, for any fluid and bore.

Skip velocity — get Re and regime straight from flow rate, bore and fluid.

Peak stormwater runoff from catchment area, runoff coefficient and rain intensity.

Minimum slope for self-cleansing velocity in a drain of given size and flow.

Design capacity of a gravity sewer at the standard d/D = 0.8 fill ratio.

Flow capacity of a circular culvert under head, outlet-control energy balance.

Critical depth, velocity and minimum specific energy for a rectangular channel.

Open-channel velocity by the Chézy formula with Bazin or direct C.

Dynamic and kinematic viscosity of water at any temperature 0–100 °C (Vogel equation).

Head loss in a pipe with the friction factor solved automatically (Swamee–Jain) from roughness and Reynolds number.

Add static lift, discharge head, and suction/discharge friction into the total head a pump must develop.

Net positive suction head available at the pump eye, from atmospheric pressure, lift, friction and vapour pressure.

Minimum pipe diameter to carry a flow at your chosen design velocity, with the next standard size suggested.

Friction loss for water pipes by the Hazen–Williams formula with your C coefficient.

Direct head-loss computation when you already know the friction factor.

Hydraulic, shaft and electrical power for a pumping duty with pump and motor efficiency.

Head loss through elbows, tees and valves from the sum of K-values.

New flow, head and power when pump speed changes — the VFD designer's first tool.

Flow, head and power after trimming an impeller diameter (affinity approximation).

Darcy–Weisbach head loss for water in commercial steel pipe (ε = 0.045 mm) with auto friction factor.

Darcy–Weisbach head loss for water in pvc / upvc pipe (ε = 0.0015 mm) with auto friction factor.

Darcy–Weisbach head loss for water in copper tube pipe (ε = 0.0015 mm) with auto friction factor.

Darcy–Weisbach head loss for water in hdpe pipe (ε = 0.007 mm) with auto friction factor.

Darcy–Weisbach head loss for water in cast iron pipe (ε = 0.26 mm) with auto friction factor.

Darcy–Weisbach head loss for water in galvanized steel pipe (ε = 0.15 mm) with auto friction factor.

Darcy–Weisbach head loss for water in stainless steel pipe (ε = 0.015 mm) with auto friction factor.

Darcy–Weisbach head loss for water in concrete pipe (ε = 0.5 mm) with auto friction factor.

Water friction loss in pvc pipe with the standard C = 150 baked in.

Water friction loss in copper pipe with the standard C = 135 baked in.

Water friction loss in new steel pipe with the standard C = 120 baked in.

Water friction loss in old steel (20 yr) pipe with the standard C = 95 baked in.

Water friction loss in cast iron (new) pipe with the standard C = 130 baked in.

Water friction loss in cement-lined ductile iron pipe with the standard C = 140 baked in.

Water friction loss in hdpe pipe with the standard C = 150 baked in.

Water friction loss in concrete pressure pipe pipe with the standard C = 130 baked in.

Diameter needed for a pump suction line at the duty's recommended 1 m/s design velocity.

Diameter needed for a pump discharge line at the duty's recommended 2.5 m/s design velocity.

Diameter needed for a gravity drain line at the duty's recommended 1 m/s design velocity.

Diameter needed for a cooling water line at the duty's recommended 2 m/s design velocity.

Diameter needed for a irrigation mainline at the duty's recommended 1.5 m/s design velocity.

Diameter needed for a fire protection main at the duty's recommended 4.5 m/s design velocity.

Darcy head loss and pressure drop for 30% Propylene Glycol (0 °C) in steel pipe, laminar or turbulent handled automatically.

Darcy head loss and pressure drop for 50% Ethylene Glycol (−10 °C) in steel pipe, laminar or turbulent handled automatically.

Darcy head loss and pressure drop for Seawater (20 °C) in steel pipe, laminar or turbulent handled automatically.

Darcy head loss and pressure drop for ISO VG46 Oil (40 °C) in steel pipe, laminar or turbulent handled automatically.

NPSH available when the pump draws water up from a well, sump or river below it.

NPSH available with the supply tank above the pump (flooded suction).

How temperature destroys NPSH margin — vapour pressure computed from the steam tables fit.

Estimate NPSH required at a new speed — NPSHr scales roughly with N².

Required sump-pump flow and head from inflow rate, lift and discharge run.

Head, power and stage estimate for a borewell submersible from depth, tank height and flow.

Boost pressure a building set must add to serve the top floor at proper pressure.

Flow needed to turn a pool over in your chosen period, with pipe-velocity check.

Total zone demand from head count and nozzle flow — checks it against supply capacity.

Water a ram pump can lift — no electricity, just fall and water hammer.

Usable water per pump cycle from a bladder tank via Boyle's law, plus motor cycling check.

Annual kWh and cost of running a pump from shaft power, motor efficiency and schedule.

Energy saved by slowing a pump with a VFD instead of throttling, via affinity laws.

Energy per cubic metre pumped — the KPI water utilities benchmark on.

Overall pumping-plant efficiency from measured flow, head and electrical power.

Compare two pumps on purchase + energy + maintenance over the ownership period.

Expected running amps of a three-phase pump motor — sanity-check against nameplate.

Flow through a valve of known Cv at a given pressure drop (liquid service).

Pressure a valve of known Cv eats at your flow — find out what that ball valve costs you.

Convert between metric Kv (m³/h @ 1 bar) and US Cv (gpm @ 1 psi) plus equivalent K-factor.

Build H = H_static + k·Q² from one known duty point; outputs head at 25/50/75/125% flow.

Combined head of identical pumps in series at a common flow.

Combined flow of identical pumps in parallel — with the honest warning about system curves.

How flow divides between two parallel pipes of different size and length.

Convert a fitting K-value into metres of straight pipe (and back) for quick sizing.

Actual developed head from suction & discharge gauges, including gauge elevation offset.

Maximum possible discharge pressure (dead-headed pump) for relief-valve and flange rating checks.

Thermal minimum flow from allowed temperature rise — protects pumps at low demand.

Liquid temperature rise across a running pump from power, flow and efficiency.

Hydraulic Institute B-parameter — does your viscous fluid de-rate the centrifugal pump?

How motor load changes when the same pump moves a heavier or lighter liquid.

Dimensionless N_s identifies the impeller type your duty wants — radial, mixed or axial.

N_ss — the cavitation-aggressiveness number reliability engineers screen pumps with.

VFD speed needed to hit a reduced target head or flow, via affinity laws.

Fit H = a − b·Q² through three catalogue points and read head at any flow.

Pump starts per hour for a lift-station wet well — worst case checked automatically.

Specific capacity (flow per metre of drawdown) — the health metric of any production well.

Dosing-pump flow to hit a target ppm in a process stream from stock-solution strength.

Allowable internal pressure of a pipe from wall thickness and material stress.

Minimum wall required for a design pressure, plus corrosion and mill-tolerance allowances.

Working pressure of HDPE pipe from its SDR and material grade (PE80/PE100).

Code test pressure for piping (1.5×) with the temperature-stress ratio correction.

FL = C·Q²·L — the fireground friction formula with standard hose coefficients.

Number of stages a multistage/borewell pump needs for your total head.

Flow that elevation alone drives through a connecting pipe — no pump.

Water content of a pipeline — for flushing, dosing, draining and freeze-protection planning.

Emitter count, zone flow and run time to deliver a target water depth.

Water needed to prime a suction line and pump casing before first start.

Peripheral speed of an impeller — wear, NPSH and emulsification screening.

Stroke % to set on a metering pump for a target dose rate.

Verifies your suction line against the one-size-up rule and the 1.2 m/s velocity ceiling.

Where your duty sits relative to best-efficiency point — HI preferred/allowable bands.

Minimum liquid depth over a suction bell to prevent air-entraining vortices.

Estimate flow from a horizontal open pipe by measuring how far the jet lands (Purdue method).

Flow and head for a condensate receiver pump from boiler load, with the 3× intermittent factor.

Resultant thrust at a pipe bend that a concrete block (or restrained joints) must resist.

Flow, time and water volume to flush a main at scouring velocity for the required pipe volumes.

Probable peak flow from total Water Supply Fixture Units via a Hunter-curve fit.

Head loss per 100 m of ½" Sch 40 steel pipe at your flow — the friction-table replacement.

Head loss per 100 m of 1" Sch 40 steel pipe at your flow — the friction-table replacement.

Head loss per 100 m of 2" Sch 40 steel pipe at your flow — the friction-table replacement.

Head loss per 100 m of 3" Sch 40 steel pipe at your flow — the friction-table replacement.

Head loss per 100 m of 4" Sch 40 steel pipe at your flow — the friction-table replacement.

Head loss per 100 m of 6" Sch 40 steel pipe at your flow — the friction-table replacement.

Round duct diameter for an airflow at your target velocity, with rectangular equivalent.

Size a duct to a constant friction rate (Pa/m) — the standard low-rise design method.

Dew point, humidity ratio, enthalpy, wet-bulb and vapour pressure from dry-bulb & RH.

Cooling tonnage for a room from area, occupancy, glazing and equipment — better than sq-ft rules.

Airflow a room needs for a target air-change rate from its dimensions.

Total external static pressure of a duct run: straight friction + fittings + components.

Will a cold surface sweat? Compares surface temperature against indoor dew point.

Moisture content of air (g/kg dry air) — the conserved quantity in HVAC mixing math.

Air changes per hour your current fan/AHU actually achieves in a space.

Room-by-room sensible load from walls, roof, glass, people and equipment with U-values.

Ventilation airflow for a bedroom at the recommended 5 air changes per hour.

Ventilation airflow for a kitchen at the recommended 15 air changes per hour.

Ventilation airflow for a bathroom at the recommended 8 air changes per hour.

Ventilation airflow for a office room at the recommended 6 air changes per hour.

Ventilation airflow for a classroom at the recommended 6 air changes per hour.

Ventilation airflow for a home gym / fitness room at the recommended 10 air changes per hour.

Ventilation airflow for a restaurant dining at the recommended 8 air changes per hour.

Ventilation airflow for a server room at the recommended 20 air changes per hour.

Ventilation airflow for a workshop / garage at the recommended 8 air changes per hour.

Ventilation airflow for a basement at the recommended 4 air changes per hour.

Ventilation airflow for a grow room at the recommended 30 air changes per hour.

Ventilation airflow for a warehouse at the recommended 2 air changes per hour.

Tonnage estimate tuned to a hot-humid climate (~14 m²/ton) with occupant & sun corrections.

Tonnage estimate tuned to a hot-dry climate (~16 m²/ton) with occupant & sun corrections.

Tonnage estimate tuned to a temperate climate (~20 m²/ton) with occupant & sun corrections.

Tonnage estimate tuned to a composite climate (north india) (~15 m²/ton) with occupant & sun corrections.

BTU class (9/12/18/24k) for a ductless mini-split, with insulation quality factor.

ENERGY STAR-style BTU sizing for window units with the standard room corrections.

Cooling for an IT room: every watt of IT load is a watt of heat, plus room gains.

Translates marketing BTU to DOE SACC and checks it against your room.

kW of electric heating for a room from volume, insulation and design temperature lift.

Input & output BTU for a furnace from the design heat loss and AFUE efficiency.

Capacity check at the winter design point using the capacity-derate curve.

Real heat output of a radiator at your actual water/room ΔT versus the ΔT50 catalogue rating.

Water flow a heating circuit needs to deliver its kW at the chosen ΔT.

W/m² a wet UFH floor can emit from floor surface and room temperatures.

Wet-bulb from dry-bulb and RH (Stull 2011) — cooling towers, evap coolers, heat stress.

What heating (or cooling) air does to its relative humidity at constant moisture.

Mixed-air temperature and moisture when return and outdoor air streams combine.

Air density from temperature, RH and barometric pressure — humid air is LIGHTER.

Total energy content of moist air — the currency of coil and energy-recovery calculations.

Suction superheat from measured line temperature and saturation temperature off the gauge.

Liquid-line subcooling — the TXV-system charging metric.

Translate between the three cooling-efficiency ratings (with the SEER↔EER field approximation).

Chiller plant efficiency from measured load and power — the operator's daily KPI.

Absolute compression ratio of a refrigeration compressor and what it means for its life.

Live cooling duty from chilled-water flow and ΔT — the BTU-meter formula.

Convert between R and U in both SI and US unit systems — without the 5.68× trap.

Total R and U of a wall assembly from up to four layers plus air films.

Conduction heat flow through any envelope element: Q = U·A·ΔT.

Solar + conduction gain through glazing from SHGC, U-value and incident solar.

Energy carried away by air leakage from ACH, volume and temperature difference.

Sensible kW an airflow carries — or the CFM needed to move a sensible load.

Latent (moisture) load an airflow carries from the humidity-ratio difference.

Total (sensible + latent) duty from airflow and enthalpy difference — coil selection math.

New CFM, static pressure and power when fan speed changes — the air-side affinity laws.

Shaft and electrical power to move an airflow against a static pressure.

Convert between air velocity and velocity pressure for pitot traverses.

Pa/m friction of a round galvanized duct at your airflow — the friction chart in a box.

Width of rectangular duct (at your height limit) matching a round duct's friction.

Face velocity and noise expectation for a supply/return grille at your airflow.

Actual airflow from an averaged pitot/anemometer traverse of a duct.

How much airflow compressed flexible duct really delivers versus rigid metal.

Exhaust for a commercial wall-canopy hood from length and cooking duty.

HVI-method fan size from floor area and fixtures, with duct-length reality check.

Fan size and attic outlet area for night-flush cooling of an entire home.

Net free vent area for an attic, split between soffit intake and ridge exhaust.

Makeup air a big range hood or exhaust fan demands before it depressurizes the house.

Cross-draft booth airflow from face area at the 100 fpm capture velocity.

Chilled/condenser water flow per ton at any design ΔT — and the classic 2.4/3.0 anchors.

Range, approach and effectiveness of a cooling tower from its three temperatures.

Condenser-side flow from load and ΔT, including the compressor heat of rejection.

Bladder expansion tank volume for a closed hydronic loop from system volume and temperatures.

Heating/cooling water pipe diameter from kW and ΔT at a sane velocity.

Season heating energy and cost from your design heat loss and local degree-days.

Monthly electricity cost of an air conditioner from capacity, efficiency and duty cycle.

Cost per kWh of delivered heat for any two fuels, efficiencies included.

Moisture load (kg/h) to hold an indoor RH target against ventilation in winter.

Pints/litres per day for a damp space, AHAM-2019 style, from area and dampness class.

One airflow, every unit used across HVAC catalogues worldwide.

Pascal ↔ inches & mm of water gauge — manometer language across continents.

Tons of refrigeration, kilowatts, BTU/h and kcal/h in one card.

Outdoor air per person to hold a CO₂ target, from the steady-state mass balance.

Fan sweep (mm) and count for a room, plus the airflow you should expect.

Swamp-cooler CFM for a space (30–40 ACH) plus the exit-air temperature it can reach.

System leakage from fan flow versus the sum of grille airflows.

Heating/cooling energy saved by night or away setbacks (% per degree rule).

Sanity-check a load calc against benchmark W/m² for the building type.

Required supply temperature (or airflow) to absorb a room's sensible load.

Heat loss per metre of insulated pipe via radial conduction — bare vs lagged.

Heater kW to raise pool temperature in a chosen time, plus surface-loss top-up.

Outlet temperature (or ratio) when hot tank water blends with cold supply.

How long a geyser/tank heater takes to reheat after a draw.

Max indoor RH before windows start sweating, from outdoor temperature and glazing type.

Evaporator temperature split — the 60-second airflow & charge health check.

Sensible effectiveness and recovered kW of a heat-recovery ventilator from three temperatures.

OA fraction of an AHU from the classic three-temperature field formula.

Airflow per ton of cooling against the 350–450 CFM/ton commissioning window.

Free area and nominal size a return-air grille needs to stay quiet.

Deflection, max moment and reactions for a point load at midspan.

Deflection, moment and reactions under a uniformly distributed load.

Tip deflection and fixed-end moment for a load at the cantilever tip.

Tip deflection and support moment for a uniformly loaded cantilever (canopies, balconies).

Deflection and end/midspan moments for a built-in beam under UDL.

Built-in beam with central point load: deflection and end moments.

One end fixed, one propped: reactions, max moment and deflection.

Reactions, moment under the load and deflection for a point load anywhere on a simple span.

Simple beam under linearly varying (triangular) load — water pressure, soil, snow drift.

Support and span moments for equal two-span continuous beam under UDL.

Superpose a UDL and a midspan point load on a simple beam — the real-world loading.

Check computed deflection against the code serviceability limits L/240, L/360.

First bending frequency of a simply supported beam — resonance and vibration screening.

Estimate floor natural frequency straight from its dead-load deflection.

Dynamic amplification when a weight drops onto a structure from height h.

Elastic section modulus, inertia and area of a rectangular section.

I, S, J and area for solid round bars and shafts.

Section properties of CHS / pipe — the most weight-efficient bending shape.

I, S and area of a doubly-symmetric I-section from its plate dimensions.

Centroid, I and both section moduli for a tee — the asymmetric workhorse.

Shift any shape's I to a new axis: I = I₀ + A·d².

Beam + top/bottom cover plates: combined I and S — strengthening math.

r = √(I/A) — the column designer's favourite section property.

Z, shape factor and plastic moment for a rectangle — limit-state bending capacity.

Weight of mild-steel plate from size and thickness — per piece and per m².

Steel round bar weight by the d²/162 rule, exact density math shown alongside.

Solid square steel bar weight per metre and per length (s²/127 rule).

MS flat (patti) weight from width × thickness — fabrication estimating staple.

Pipe weight from OD and wall — the universal (OD−t)×t×0.0246 formula.

Rectangular & square hollow section weight with corner-radius correction.

Equal/unequal MS angle weight from legs and thickness.

ISMC-style channel weight from web, flanges and thicknesses.

Weight of an I-beam from its plate dimensions, with the ISMB quick table in the note.

Floor-plate weight including the pattern allowance over plain plate.

Critical buckling load of a column with any end condition (K factor).

KL/r with the IS 800 limit verdicts — the first column check.

Rankine-Gordon capacity bridging crushing and buckling for intermediate columns.

Effective lengths for all six classical end-condition cases at once.

P/A against the slenderness-reduced permissible stress (IS 800 WSD-style curve).

Slab + finishes + services dead load in kN/m² from layer thicknesses.

Total live load on an area with the standard occupancy intensities in the help text.

Line load on a beam and point load on a column from slab tributary geometry.

Factored design load under IS/ASCE combinations — the governing case highlighted.

Design wind pressure from basic wind speed with terrain/height factor.

Design base shear V = A_h·W from zone, importance, response reduction and Sa/g.

Flat-roof snow load from ground snow with exposure, thermal and importance factors.

Compare planned equipment/storage loading against the floor's design rating.

Ultimate moment of a singly-reinforced section from steel area, grade and depth.

A_st needed for a factored moment — the daily design equation inverted.

Code minimum and maximum reinforcement for beams, slabs and columns.

Bar spacing for a required steel area per metre — slab detailing in one step.

Anchorage length L_d for deformed bars in tension by grade combination.

Splice lap for tension and compression bars, with the 50Ø site rule context.

Deflection control by L/d limits for slabs and beams (IS 456 basic values).

Vertical stirrup spacing for the shear the concrete can't carry (IS 456).

Squash capacity of a tied column: concrete plus longitudinal steel.

ly/lx ratio decides the load path — and roughly how thick the slab wants to be.

Design shear strength of an ordinary bolt — single or double shear.

Plate bearing strength at a bolt hole — often governs over bolt shear.

Design tensile strength of a bolt — hangers, end plates, uplift anchors.

Slip capacity of a friction-grip bolt from preload, friction and surfaces.

ACI-style concrete cone breakout for a single anchor far from edges.

Required section modulus and a suggested rectangular size for a timber beam.

Deflection-governed max span of rectangular timber joists at your spacing.

Axial capacity of a rectangular timber post with slenderness reduction.

σ = M/S with utilisation against the material's allowable stress.

Peak web shear stress: 1.5V/A for rectangles, V/A_web for I-beams.

P/A ± M/S — stress at both extreme fibres of an eccentrically loaded member.

Is the resultant inside b/6? No-tension check for footings, dams and masonry.

Required section modulus from load & span, with the lightest ISMB suggested.

Load and size for an RC lintel over a door/window using the masonry triangle rule.

Accumulated column load floor-by-floor from tributary area and storey count.

Plate area so a beam reaction doesn't crush masonry or concrete below.

Moment and required section modulus for a railing post under code loads.

Load per standard (leg) from platform loading class, bays and lifts.

Tension in a wire/cable from its span, sag and loading (parabolic).

Tension per sling leg versus angle — the rigger's load chart.

Net uplift on a roof area and the hold-down each fastener row must resist.

Design pressure on cladding panels and load per fastener.

Inter-storey drift ratio against the 0.004h seismic limit.

Notional restraint force a brace must carry to stabilise a compression member.

Load, moment and required Z for a roof purlin from sheeting load and spacing.

Quick kg of rebar in a slab by area, thickness and the standard percentage.

Rebar tonnage in beams from concrete volume at standard percentages.

Allowable line load on a masonry wall with slenderness reduction (IS 1905 style).

Line load (kN/m) a wall puts on the beam or slab below — brick, block or AAC.

Isolated footing plan size from column load and safe bearing capacity.

Starting beam depth & width from span — the experienced engineer's first guess.

Forces in a simple two-bar bracket/truss supporting a point load.

Max safe UDL a given beam section can carry over a span — strength and deflection both.

Bending in a lifting spreader and the section it needs.

Is your roof slab happy under that new water tank? Pressure and slab-rating check.

Added dead load + wind uplift from a rooftop PV array.

Embedment depth for an unconstrained post by the IBC pole formula.

Wind force and base moment on a sign or hoarding post.

Line load on a deck ledger and the lag/bolt spacing it demands.

Load per stud and a crushing/buckling sanity check for light-frame walls.

Line load on a structural ridge from rafter span and roof loading.

Effective spread of a concentrated load through a slab and the local moment it adds.

Beam UDL and column loads for a storage mezzanine layout.

Axial capacity of a steel pipe prop/column — site props, shoring, stub columns.

Engineering stress, strain and elongation of a bar under axial load.

E from a tension test: load, area, gauge length and measured extension.

Brinell to Vickers and an estimated tensile strength for steels.

FoS against yield and ultimate, with typical industry target bands.

Clamp force a given torque produces — the K-factor (nut factor) equation.

ν from measured axial and lateral strains, with the volume-change insight.

Cycles to failure from stress amplitude via the high-cycle power law.

Density of a two-component alloy or composite from weight fractions.

G from E and ν — the isotropic elastic-constant relations in one card.

Expansion of aluminium (α = 23.1 µm/m·K) over your temperature range.

Expansion of stainless steel (304) (α = 17.3 µm/m·K) over your temperature range.

Expansion of copper (α = 16.5 µm/m·K) over your temperature range.

Expansion of brass (α = 19 µm/m·K) over your temperature range.

Expansion of concrete (α = 12 µm/m·K) over your temperature range.

Expansion of glass (soda-lime) (α = 8.9 µm/m·K) over your temperature range.

Expansion of titanium (α = 8.6 µm/m·K) over your temperature range.

Expansion of invar 36 (α = 1.2 µm/m·K) over your temperature range.

Thermal movement of a steel pipe run and the expansion-loop size it needs.

Thermal movement of a copper pipe run and the expansion-loop size it needs.

Thermal movement of a pvc pipe run and the expansion-loop size it needs.

Thermal movement of a cpvc pipe run and the expansion-loop size it needs.

Thermal movement of a pex pipe run and the expansion-loop size it needs.

Stress when expansion is prevented — the hidden force in rigid assemblies.

P/A with utilisation against yield — the fastest of all checks.

Average shear on pins, rivets and glue lines — single or double shear.

Contact crushing stress between pin and hole, or plate and support.

Equivalent stress from σx, σy and τxy — the ductile-failure yardstick.

σ1, σ2, max shear and principal angle — Mohr's circle without the drawing.

Engineering ↔ true values — essential beyond a few percent strain.

Elastic energy stored in a loaded bar — and what it means when it lets go.

Elastic energy a material absorbs per unit volume before yielding.

Press tonnage to punch a hole — perimeter × thickness × shear strength.

Kt for a circular hole in a finite-width plate under tension (Howland).

K from a pressure-volume measurement — liquid or solid compressibility.

Measured E and G give you Poisson's ratio and bulk modulus instantly.

Elastic modulus from longitudinal sound velocity and density — NDT in action.

E from the natural 'ring' frequency of a rectangular bar (impulse excitation).

HV → HB with the validity window every QA inspector should know.

UTS and approximate yield from hardness — by material family.

Mean-stress fatigue check: is your (σm, σa) point inside the Goodman line?

The yield-conservative cousin of Goodman for no-permanent-set designs.

Cumulative fatigue damage from up to four load blocks.

First-pass S_e for steel from UTS, with the Marin correction factors listed.

R-ratio, amplitude and mean from the max/min stress of a fatigue cycle.

How long a crack becomes unstable, from fracture toughness and stress.

Cycles for a crack to grow between two sizes under constant ΔK loading.

Trade temperature against time for creep life — the boiler-tube equation.

Weight of aluminium sheet from area and gauge (ρ = 2700 kg/m³).

Weight of copper sheet from area and gauge (ρ = 8960 kg/m³).

Weight of brass sheet from area and gauge (ρ = 8500 kg/m³).

Weight of stainless steel sheet from area and gauge (ρ = 7930 kg/m³).

kg per km of copper conductor from cross-section — cable estimating basic.

Strength-to-weight and stiffness-to-weight — the Ashby selection numbers.

Longitudinal and transverse moduli of a fiber composite from constituents.

V_f from weight fractions and densities — the layup-shop conversion.

Failure probability of a brittle material at a given stress — ceramics & glass honesty.

Tensile stress area of metric threads — the area that actually carries load.

Proof load and recommended preload for metric property classes.

Elongation at target preload — the gauge-and-micrometer tightening method.

Decode 8.8 / 10.9 / 12.9 markings into tensile and yield strength.

Ductility from gauge marks before and after a tensile test.

The other ductility number — area shrink at the fracture neck.

Corrosion rate from coupon weight loss — the plant-inspection standard.

Potential difference between coupled metals and the severity verdict.

Flat-pattern length through a bend from K-factor, radius and angle.

Tightest bend a sheet tolerates before cracking, from its ductility.

Peak pressure where a ball meets a flat — bearings, balls, cam followers.

Sliding-wear volume from load, distance and the wear coefficient.

µ from the angle at which an object starts to slide — the incline test.

Constructional plus elastic stretch of new wire rope under load.

Strain from resistance change via gauge factor — and what stress that means.

Galvanizing: g/m² ↔ µm ↔ service-life estimate.

Deposit thickness from current density and time via Faraday's law.

Dry film from wet film and volume solids — the coating inspector's triangle.

Litres of coating for an area at target DFT, with loss factor.

Anodic film thickness from current density and time (the 720 rule).

kWh to melt a charge — sensible heat plus latent heat of fusion.

Pattern oversize for solid shrinkage — by alloy.

Grain diameter and grains-per-area from the ASTM grain size number.

Yield strength from grain size — why fine grain wins.

Effective case depth from time and temperature (Harris equation).

Hold time at temperature by section thickness — the hour-per-inch family of rules.

Cavity size for a target part dimension, by polymer shrink rate.

Max sudden ΔT a brittle material survives — the quench test number.

Tightening torque for an M6 bolt by property class and lubrication.

Tightening torque for an M8 bolt by property class and lubrication.

Tightening torque for an M10 bolt by property class and lubrication.

Tightening torque for an M12 bolt by property class and lubrication.

Tightening torque for an M16 bolt by property class and lubrication.

Tightening torque for an M20 bolt by property class and lubrication.

Same part, different metal: weight, stiffness and deflection ratios.

Two candidate materials priced per finished part, density included.

Heat stored in a mass over a temperature swing — buffers, beds, buildings.

Copper-industry %IACS ↔ MS/m ↔ resistivity.

W/m·K ↔ BTU and kcal forms — datasheet languages unified.

Ultimate and safe bearing capacity of a strip footing from c, φ and depth.

Allowable bearing for footings on sand straight from corrected SPT N-value.

The three classical factors for any friction angle — exam and design staple.

e, n and their interconversion — the soil phase diagram's first pair.

Water content from oven-dry weights — the most-run lab test on earth.

How full the voids are — links e, w and G_s in one identity.

γ_d from field bulk density and moisture — the compaction-control conversion.

Field density as % of Proctor maximum — the pass/fail of earthworks.

Theoretical max dry density at a moisture content — the line no test may cross.

Total, pore and effective stress at depth with a water table.

Pressure felt at depth under a footing by the 2:1 spread approximation.

Primary settlement of a clay layer from Cc, e₀ and the stress change.

How long settlement takes: t₅₀/t₉₀ from c_v and drainage path.

Active, passive and at-rest coefficients from friction angle.

Active thrust on a wall — soil, surcharge and water components separated.

Factor of safety of a long natural slope — dry or with seepage.

Groundwater flow rate through soil from k, gradient and area.

Upward gradient that floats the soil — boiling/piping check for excavations.

k of fine soils from the falling-head lab test readings.

Plasticity, liquidity and consistency indices — plus the A-line classification.

Bank → loose → compacted volume conversions for earthmoving.

Resilient modulus and k-value estimates from the CBR test.

End bearing + skin friction capacity of a single pile.

Converse-Labarre group efficiency for a pile block.

Vertical stress at any depth/offset from a surface point load.

Extrapolate footing settlement from a plate load test on sand — the Terzaghi & Peck size correction.

Correct a field SPT blow count for hammer energy and overburden — the N60 and (N1)60 every correlation needs.

Cu and Cc from D10, D30 and D60 — with the USCS well-graded / poorly-graded verdict.

Coefficient of permeability k from the constant-head lab test — the standard for sands and gravels.

qu and undrained shear strength from an unconfined compression test, with the area correction and consistency verdict.

Cubic metres / yards of concrete for a slab, with 5% waste included.

How many premix bags (50 lb / 60 lb / 80 lb / 25 kg) your pour needs.

Cement bags, sand and aggregate for any nominal mix (1:2:4, 1:1.5:3 …).

The 1.54 (concrete) / 1.33 (mortar) factor explained and applied.

Concrete for isolated footings — straight pad or stepped/trapezoidal.

Concrete for rectangular or circular columns, full height, any count.

Concrete in beams from total running length and section.

Waist slab + steps concrete for a straight flight.

Yardage, cost band and rebar/mesh for a residential driveway.

Material + labour + pumping for a pour, line-item style.

Bricks and mortar for a wall — standard modular sizes, openings deducted.

8×8×16 blocks (or AAC) plus mortar for a wall.

Cement bags and sand for any mortar ratio and volume.

Cement and sand for wall/ceiling plaster by area, thickness and mix.

Tiles, boxes and wastage for a floor or wall — any tile size.

20 kg adhesive bags by trowel size and area.

Grout kilograms from tile size, joint width and depth.

Litres for walls & ceiling by coats and coverage class.

Putty kilograms for two-coat finishing before paint.

Tonnes/tons of gravel or crushed stone for an area at depth.

Sand tonnage for filling/bedding — with the Indian 'brass' unit included.

Hot-mix tonnes for a paving area at compacted thickness.

Pavers, bedding sand and base depth for a patio or driveway.

Bags per post hole and total for the fence run.

Grout volume to fill CMU cores — full or spaced fills.

Water per batch and the strength meaning of your w/c number.

Expected strength at any age from the 28-day grade (ACI time curve).

Translate between IS/BS cube and ACI cylinder test results.

M-grades, MPa, psi and typical-use guide in one card.

Lean-mix levelling course quantities under footings and floors.

Cement-sand screed (1:4) quantities for floor levelling.

Segmental (SRW) blocks, cap units and gravel backfill for a garden wall.

How much concrete weighs — slab, cylinder or volume, wet and cured.

Cubic metres/yards and tonnes of topsoil for beds and lawns.

Weight of 8 mm TMT/rebar per metre, per 12 m rod and per bundle.

Weight of 10 mm TMT/rebar per metre, per 12 m rod and per bundle.

Weight of 12 mm TMT/rebar per metre, per 12 m rod and per bundle.

Weight of 16 mm TMT/rebar per metre, per 12 m rod and per bundle.

Weight of 20 mm TMT/rebar per metre, per 12 m rod and per bundle.

Weight of 25 mm TMT/rebar per metre, per 12 m rod and per bundle.

Total bars and steel weight for a two-way mesh from spacing.

Dig volume with side slopes, truck trips and the soil-bulking surprise.

Board feet of lumber from thickness, width, length and piece count — the unit sawmills and lumberyards price by.

Sheets, screws, joint compound and tape for a wall or ceiling area — with waste built in.

Studs and top/bottom plate lumber for a framed wall at any on-center spacing.

Rolls needed from wall width, height and the pattern repeat — the drop-matching math done right.

Linear metres of 3.66 m broadloom and the real order area for a room — seams and nap direction included.

Boxes of laminate and underlayment rolls for a floor area, with cutting waste.

Cartons of luxury vinyl plank for a room — LVP cartons run smaller than laminate, so the count surprises people.

Bundles of solid or engineered hardwood with the higher 12% grading-and-cutting waste real wood needs.

Decking boards and screws for a deck from board size and gap — the row-by-row math, not just area.

Roofing squares, shingle bundles, felt and nails from the house footprint and pitch.

Rib panels, total coverage and screws for a metal roof from eave length and slope length.

Siding squares and cartons for exterior walls, netting out doors, windows and gables.

Concrete volume and premix bags for cylindrical form-tube piers — deck footings, post bases, fence terminals.

Tons of crushed stone / crusher run for a base or driveway lift from area and depth.

Cubic yards/metres and 2 ft³ bags of mulch for beds at the depth that actually suppresses weeds.

Tonnage of decorative river rock by stone size — bigger stones need a deeper layer to hide the ground.

Litres and gallons of exterior paint — rough render and fibre-cement drink far more than the can claims.

Litres of primer/sealer for new drywall, patches or colour changes — one coat at primer spreading rates.

4×8 sheets for subfloor, sheathing or roof decking from area, with the standard 10% cut waste.

Depth and bags of blown-in cellulose to reach a target R-value over what's already there.

Wall tiles and boxes for showers, backsplashes and full-height bathroom walls.

Linear metres and 8 ft sticks of baseboard from room perimeter, minus the door openings.

Posts, rails and pickets for a wood fence run — solid privacy or spaced picket, either way.

Steps, exact riser height, total run and the stringer board length from a total rise.

Baluster count and the exact even gap that passes the 4-inch-sphere code rule on any railing run.

Compacted gravel base and bedding sand quantities under a paver patio, path or driveway.

Litres of epoxy from area, film thickness and solids content — the wet-film math coating reps use.

Tension lap and development length for any bar diameter using the multiplier practice (50d etc.).

Shuttering contact area for rectangular RCC columns — the m² number quantity surveyors bill.

Kilograms of 18-gauge binding wire from the tonnage of reinforcement steel being tied.

Litres and gallons a round tank holds from its diameter and height — plus what it weighs full.

Litres in a rectangular tank or sump from inside dimensions — and the size you need for a target capacity.

FM of fine aggregate from cumulative %-retained on the six standard sieves — with the coarseness verdict.

Area of an irregular four-sided plot from its sides and one diagonal (two Herons).

Triangular plot area from three sides — corner plots and leftovers.

Exact area of a 5-corner plot from X-Y coordinates — total-station math.

Plot dimension to sqft / sqm / gaj / cents with the common-site cheat sheet.

Convert bigha to square feet, square metres, gaj and acres.

Convert marla to square feet, square metres, gaj and acres.

Convert kanal to square feet, square metres, gaj and acres.

Convert gunta to square feet, square metres, gaj and acres.

Convert cent to square feet, square metres, gaj and acres.

Convert ground to square feet, square metres, gaj and acres.

Convert katha to square feet, square metres, gaj and acres.

Convert biswa to square feet, square metres, gaj and acres.

Convert decimal to square feet, square metres, gaj and acres.

Convert ankanam to square feet, square metres, gaj and acres.

Sqft ↔ gaj (square yard) — the North-Indian property unit.

Buildable area from plot size and the sanctioned Floor Space Index.

Max footprint after setbacks and the coverage percentage check.

Convert between rise/run, percent, ratio (1 in N) and degrees.

Earthwork volume between two cross-sections by the average-end-area method.

Trench dig volume — straight or battered sides — plus bedding and backfill.

Earth volume from a 4-square grid of depth readings.

Temperature, pull and sag corrections for precise taping in one pass.

RLs from a level run: backsight, foresights and the height of instrument.

Earth-curvature and refraction corrections for long level sights.

WCB (0–360°) to N/S-E/W reduced bearings and back.

Fore bearing ↔ back bearing, with the local-attraction check explained.

Inverse computation: bearing and distance between two E,N coordinate pairs.

New point coordinates from a known station + bearing + distance.

Misclosure and relative precision (1 : N) of a closed traverse.

Distance and elevation from stadia readings and vertical angle.

SSD from speed, reaction time and friction — IRC/AASHTO style.

Banking required for a curve at design speed (IRC 75% rule included).

Arc-definition degree of curve to radius, both unit systems.

Riser/tread layout for a floor height — with the comfort-formula verdict.

Pitch ↔ angle ↔ rafter length from span and rise.

Check a layout corner for square using diagonal measurement.

Crest curve length for sight distance from grades and SSD (K-value method).

Hectares ↔ acres, square metres and square feet — the two big land units, converted exactly.

Acres to square feet, square metres, gaj, cents and bigha — every unit a land deed might quote.

Gunter's chains and links to feet and metres — the unit old revenue maps and legal descriptions still use.

The diagonal a rectangular plot or slab SHOULD measure — and how far out of square yours is.

Buildable footprint and ground coverage left on a plot after front, rear and side setbacks.

kJ/mm from volts, amps and travel speed — with process efficiency.

The speed your welder must hold to stay inside a heat-input window.

Design capacity of a fillet from leg size, length and electrode strength.

Leg size needed for a load — and the min/max size rules checked.

Theoretical and effective throat from leg sizes — unequal legs handled.

Full- and partial-penetration groove weld capacity.

Code minimum and maximum fillet for your plate combination.

Nugget size and shear strength for resistance spot welds in sheet.

Deposited kilograms for fillet runs — the consumables takeoff.

Deposited weight in a single-V joint from thickness, angle and gap.

Rods needed from deposited weight — stub loss and efficiency included.

kg/hr from wire size, feed speed (MIG/FCAW) — or amps for stick.

Welding hours for a job from weld weight and deposition rate.

Full cost build-up: consumables, gas, power and labour per metre of weld.

Gas litres for a job and how long a cylinder lasts.

Weldability index from chemistry — predicts preheat requirements.

The modern low-alloy cracking index (Ito-Bessyo) for low-carbon steels.

Preheat estimate from CE and combined thickness (EN 1011 style).

Transverse shrinkage of butt joints — set your pre-allowance.

Current window for an electrode size — the 35–40 A/mm rule plus position trim.

Wire feed speed from target current — by wire size (burn-off relation).

kWh and electricity cost of welding work.

Can your machine hold this current? Duty-cycle math in both directions.

Electrodes per pipe joint from diameter and wall — the pipeline estimator's number.

Electrode size and gas-cup match for your TIG current range.

Starting amperage and wire feed speed for short-circuit MIG on mild steel from material thickness.

Starting TIG amps for steel and stainless from thickness — the one-amp-per-thou bench rule, with the aluminium correction.

Rate of a compression spring from wire, coil diameter and active turns.

Corrected shear stress in a loaded coil spring (Wahl factor).

All key gear dimensions from module and tooth count.

Output speed/torque through one or two reduction stages.

Shaft spacing for a meshing gear pair (and the ratio it gives).

Driven speed from pulley diameters — the workshop's oldest calculation.

Belt length for a two-pulley drive from diameters and centre distance.

Tight/slack tensions and transmittable power from wrap and friction.

Chain length in pitches and links, plus output speed.

Million-revolution and hour life of a rolling bearing from load ratio.

BPFO, BPFI, BSF and FTF for vibration analysis.

The P = 2πNT/60 triangle solved for torque.

Shear stress and twist angle of a solid shaft under torque.

First whirling speed from static deflection (Rayleigh).

Shear and crushing checks on a parallel key.

Engagement so the bolt breaks before threads strip — by material pair.

Hoop and longitudinal stress in cylinders — tanks, pipes, air receivers.

Bore stress of hydraulic cylinders and HP tubing where thin-wall lies.

Push and pull force from bore, rod and pressure.

Extend/retract speed and time from pump flow.

kW in a hydraulic circuit — the P·Q/600 rule.

Air-cylinder force at line pressure with the load-ratio sizing rule.

Free-air usage of a cycling cylinder — compressor sizing input.

Effort to lift with block-and-tackle, friction included.

Effort, load or arm lengths for all three lever classes.

Torque to raise a load on a power screw, with the self-locking check.

Pull to drag or winch a load up a slope, with rolling/sliding friction.

Motor torque to spin a load up to speed in a target time.

Spindle RPM from surface speed and diameter — lathe, mill and drill.

Drill size for any metric or unified thread at chosen % engagement.

Aerodynamic lift from CL, wing area and airspeed.

Power in the wind and what a rotor can really capture (Betz).

Outward force on a rotating mass — balancing and guard design.

f_n of a mass on a spring or mount — resonance avoidance 101.

Transmissibility and isolation % from frequency ratio.

Torque capacity of a plate clutch from springs, friction and geometry.

Braking torque from pad force, friction and effective radius.

Interference pressure and press-in force for a shaft-hub fit.

Radial shaft load from a sprocket/pulley vs the gearbox rating.

Cycle time for a turning pass from feed, speed and length.

Surface/peripheral speed of anything that spins — wheels, rolls, blades.

Relative centrifugal force from rotor radius and speed.

Static compression ratio from swept volume and combustion-chamber volume — with gasket and deck volume included.