General Math / Conversions
Unit Conversions
Common engineering conversions across mechanical, electrical, and thermal domains.
in ↔ mm: mm = in × 25.4
ft ↔ m: m = ft × 0.3048
lbf ↔ N: N = lbf × 4.448
psi ↔ bar: bar = psi × 0.06895
°F ↔ °C: °C = (°F − 32) × 5/9
GPM ↔ L/min: L/min = GPM × 3.785
HP ↔ W: W = HP × 745.7
oz·in ↔ N·m: N·m = oz·in × 0.00706
kg ↔ lbs: lbs = kg × 2.205
Circle Area & Circumference
Fundamental geometry for gears, pulleys, pistons, pipe cross-sections, and any round component. Enter diameter.
C = circumference (unit)
r = radius d = diameter
Motion / Drive Systems
Motor RPM → Wheel / Roller Surface Speed
Given motor RPM and wheel diameter, compute surface speed. Useful for drive wheels, rollers, conveyor drums — any case where rotational speed maps to linear velocity.
d = wheel diameter (inches)
1056 = 63,360 in/mi ÷ 60 min/hr
Gear Ratio
Gear ratio maps input RPM and torque to output shaft. Higher ratio = more torque, less speed. Enter tooth counts or diameter values.
RPMout = RPMin / ratio
Torqueout = Torquein × ratio × η
Belt / Pulley Speed Ratio
Belt speed is the same on both pulleys — so RPM scales inversely with diameter. A smaller driven pulley spins faster; a larger one slower with more torque.
Belt surface speed is equal on both pulleys
Speed ratio = Dout / Din
Lead Screw: Motor Rotations → Linear Travel
Lead screws convert rotation to linear motion. Lead is the distance traveled per full motor rotation. Critical for CNC axes and linear actuators.
rotations = steps / steps_per_rev
steps/rev: 200 (1.8° motor, no microstepping)
400 (0.9° motor)
1600 (1.8° + 1/8 microstep)
Torque Required to Move a Load
Estimates minimum motor torque needed to overcome friction and gravity. Add a safety factor (1.5–2×) for acceleration and real-world losses.
μ = friction coefficient (0.1 smooth, 0.3 rubber)
θ = incline angle (0° = flat)
r = drive wheel/pulley radius (mm)
Electrical
Ohm's Law
Fill any two fields — the third is solved. The foundation of all DC circuit analysis.
I = current (amperes)
R = resistance (ohms, Ω)
I = V / R
R = V / I
Fill any two — third is solved.
Electrical Power
Three equivalent power formulas — all three are shown for whichever pair of values you provide.
V = voltage (V) I = current (A)
R = resistance (Ω)
Fill any two fields.
Voltage Divider
Two resistors in series divide a supply voltage. Essential for ADC input scaling, level shifting, and transistor biasing.
R2 = bottom resistor (Vout to GND)
I = Vin / (R1 + R2) — quiescent current
RC Time Constant
At t = τ a charging capacitor reaches 63.2% of supply. At 5τ it's essentially full (99.3%). Used for filter design, debouncing, and delay circuits.
R = resistance (Ω)
C = capacitance (µF)
Vs = supply voltage
Battery Runtime
Theoretical runtime from capacity divided by load. Real-world runtime is typically 75–85% of theoretical due to Peukert effect and cutoff voltage.
C = capacity (mAh)
I = load current (mA)
Energy = C × V / 1000 (Wh)
1000 mAh @ 3.7V = 3.7 Wh
Wire Gauge Voltage Drop
Resistance in a wire run drops voltage proportional to current. Long runs or high current require heavier gauge. Keep drop below 3% of supply for sensitive loads.
I = current (A)
Rft = resistance per foot (Ω/ft)
×2 accounts for round-trip (hot + return)
Copper Ω/ft: 8AWG 0.00199 10AWG 0.00316
12AWG 0.00501 14AWG 0.00797
16AWG 0.01267 18AWG 0.02014
Electronics / Control
PWM Duty Cycle → Average Voltage
PWM switches a signal on/off rapidly. The average voltage seen by a motor or low-pass filter is proportional to duty cycle. 50% duty = half voltage.
Vin = supply voltage
Motor speed is approximately
proportional to Vavg
LED Current Limiting Resistor
Every LED needs a series resistor. Forward voltage by color: red ~2.0V, green/yellow ~2.1V, blue/white ~3.2V. Typical current: 20 mA for standard LEDs.
Vf = LED forward voltage
If = desired current (amps)
PR = If² × R (resistor power)
Op-Amp Gain
Two fundamental op-amp configurations. Inverting flips signal polarity; non-inverting preserves it with high input impedance. Gain set entirely by resistor ratio.
Rin = input resistor
Vout = G × Vin
dB = 20 × log10(|G|)
Structural / Forces
Torque (Force × Moment Arm)
Torque is the rotational effect of a force applied at a distance from a pivot. Fundamental to levers, wrenches, motor shafts, and structural connections.
F = force (N or lbf)
d = moment arm length
θ = angle between force and arm
θ = 90° gives maximum torque
Mechanical Advantage — Levers & Pulleys
Trade force for distance. For a lever: arm lengths. For a pulley system: count the rope segments supporting the load. Work in always equals work out.
Pulleys: MA = number of rope segments
supporting the load
Work in = Work out (no free energy)
Beam Deflection — Simply Supported, Center Load
Maximum deflection of a beam supported at both ends with a point load at center. Common for shelf loading, horizontal spans, and structural members.
P = point load (N)
L = span length (m)
E = Young's modulus (Pa)
I = second moment of area (m&sup4;)
Rect section: I = b×h³ / 12
Circ section: I = π×d&sup4; / 64
Pressure (Force ÷ Area)
Pressure is force distributed over area. Used in hydraulics, pneumatics, contact stress, and structural bearing calculations.
F = force (N or lbf)
A = area (m², cm², in², mm²)
1 bar = 100,000 Pa = 14.504 psi
1 psi = 6,894.76 Pa = 0.0689 bar
Safety Factor
Margin between working load and failure. SF = 1 means operating at the limit. Static structures typically require SF ≥ 3–4; lifting equipment SF ≥ 5 (regulatory).
Static structures: SF ≥ 3–4
Dynamic / impact: SF ≥ 5–8
Lifting equipment: SF ≥ 5 (regulatory)
Consumer products: SF ≥ 2–3
Units must match (both N, both lbf, etc.)
Fluid / Pneumatic
Flow Rate → Pipe Velocity
Velocity of fluid in a pipe from volumetric flow rate and cross-section. Higher velocity means more friction losses. Recommended: water 1–3 m/s, air 5–15 m/s.
Q = volumetric flow rate (m³/s)
A = pipe cross-sectional area (m²)
d = inner diameter
Pressure Drop (Darcy-Weisbach)
Pressure loss due to friction in a pipe run. Use f ≈ 0.02 for turbulent flow in smooth pipes as a starting estimate. Water: ρ = 1000, air at 20°C: ρ = 1.2 kg/m³.
L = pipe length (m)
D = inner diameter (m)
ρ = fluid density (kg/m³)
v = flow velocity (m/s)
Pneumatic / Hydraulic Cylinder Force
Extend and retract forces differ because the rod reduces the effective piston area on the rod side. Rod diameter is only needed for retract force.
dbore = cylinder bore diameter
drod = rod diameter
Tank Hold Time at Flow Rate
How long a pressurized reservoir supplies flow at a given rate. Enter pressures as gauge (not absolute) — the calculation converts internally.
P0 = initial pressure (bar absolute)
Pmin = min usable pressure (bar abs)
Q = free-air flow rate (L/min)
Patm = 1.013 bar
Thermal
Heat Dissipation & Temperature Rise
Power dissipated as heat causes a junction temperature rise that depends on thermal resistance. Critical for heatsink sizing on MOSFETs, LDOs, and power resistors.
Rθ = thermal resistance (°C/W)
Tj = junction temperature (°C)
Typical RθJA:
TO-220 + heatsink: ~5–15 °C/W
TO-220 no heatsink: ~50–60 °C/W
SOT-23 no heatsink: ~150–200 °C/W
Brewing Calculators
Gravity / Fermentation
ABV from Original & Final Gravity
Alcohol by volume from pre- and post-fermentation gravity readings. The simple formula is accurate within ±0.1% ABV for typical beers; the Balling formula is more precise for high-gravity brews.
ABV = 76.08(OG−FG)/(1.775−OG) × FG/0.794 Balling formula — better for high-gravity
OG = original gravity (e.g. 1.055)
FG = final gravity (e.g. 1.010)
Apparent Attenuation
How much of the fermentable sugar the yeast consumed, expressed as a percentage. Most ale yeasts attenuate 72–80%. Low attenuation can indicate stuck fermentation or poor yeast health.
English ales: 72–76%
American ales / lagers: 75–80%
Belgian / saison: 78–85%+
Stouts: 70–76%
Calories per 12 oz
Uses the ASBC standard method: real extract derived from original and final gravity, then calories calculated from alcohol and residual carbohydrate contributions.
RE = real extract (°Plato)
RE = 0.1808 × OGP + 0.8192 × FGP
°Plato = 259 × (1 − 1/SG)
Alcohol contributes ~7 cal/g
Carbs contribute ~4 cal/g
Mash & Grain
Strike Water Temperature
The temperature water needs to be when added to grain to hit your target mash temperature, accounting for heat absorbed by the grain and equipment. Assumes ambient grain temperature.
Tmash = target mash temp (°F)
Tgrain = grain temp (usually room temp)
Typical single infusion mash: 148–158°F
Lower temp = more fermentable (drier beer)
Higher temp = more body (fuller beer)
Mash Thickness
Water-to-grain ratio affects enzyme activity, conversion efficiency, and body. Thicker mashes (lower ratio) tend to produce fuller-bodied, less fermentable wort. Standard range: 1.25–1.5 qt/lb.
Thick mash (<1.25 qt/lb):
Higher enzyme concentration
More dextrinous, fuller body
Thin mash (>1.5 qt/lb):
Better enzyme activity range
More fermentable, drier finish
Extract Efficiency
How much sugar you extracted from the grain versus the theoretical maximum. Affected by crush quality, mash temperature, pH, and sparge technique. Typical homebrewing range: 65–80%.
potential points = grain lb × PPG
PPG (points per pound per gallon):
2-row / pale malt: 37
Munich malt: 35
Crystal / caramel: 33–35
Roasted barley: 25
Flaked oats/wheat: 32–36
Expected OG from Grain Bill
Predict your original gravity before brew day given grain weight, average PPG of your grain bill, expected efficiency, and batch volume.
eff% = mash + lauter efficiency
To hit a target OG, rearrange:
lb needed = (target points × gal) / (PPG × eff%)
Hops & Bitterness
IBU (Tinseth Formula)
Calculates bitterness units for a single hop addition. Run it multiple times and sum the results for multi-addition recipes. Higher OG wort extracts less alpha acids — the Tinseth bigness factor accounts for this.
bigness = 1.65 × 0.000125(OG−1)
time_factor = (1 − e−0.04t) / 4.15
AA = alpha acid %
vol = wort volume (gallons)
t = boil time (minutes)
BU:GU Ratio (Bitterness Balance)
Relates bitterness units to gravity units to express perceived bitterness balance. A ratio of ~0.5 is balanced; West Coast IPAs typically run 0.7–1.0+.
Style ranges:
Balanced (pale ale, amber): ~0.5
Malt-forward (stout, bock): 0.3–0.4
IPA / DIPA: 0.6–1.0+
Imperial stout: 0.3–0.5
Water & Volumes
Priming Sugar for Bottle Carbonation
Sugar weight needed to hit a target CO₂ volume in the bottle. Residual CO₂ from fermentation is subtracted — beer fermented warmer retains less CO₂ and needs more priming sugar.
T = highest fermentation temp (°F)
Factors (oz/gal/vol):
Corn sugar (dextrose): 0.91
Table sugar (sucrose): 0.84
DME: 0.74
Target CO2 volumes by style:
Ales: 1.5–2.5 Lagers: 2.4–2.6
Wheat beer: 3.0–4.5 Stout: 1.7–2.3
Boil-Off & Pre-Boil Volume
Work backwards from your target post-boil volume to determine how much wort you need going into the kettle. Accounts for boil-off, trub loss, and grain absorption.
trub loss: typically 0.5 gal
grain absorption: ~0.125 qt/lb
(0.5 qt/lb for all-grain sparge losses)
Typical boil-off rates:
Homebrewing (propane): 1.0–1.5 gal/hr
Electric: 0.75–1.0 gal/hr
Hydrometer Temperature Correction
Hydrometers are calibrated at a specific temperature (usually 60°F). Taking a reading at a different temperature requires a correction — hot wort reads artificially low, cold wort reads high.
+ 2.041×10−6T²
− 2.328×10−9T³
T = temperature (°F)
Tcal = calibration temp (usually 60°F)
Rule of thumb: add ~0.001 for every
10°F above calibration temperature