Skilled Trades
Frame walls, run wire, and slope pipe with rock-solid math.
Build the base
foundationReading a Tape Measure (Fractional ↔ Decimal Algebra)
The lead carpenter calls out '14 and eleven-sixteenths.' You must place the mark, add and subtract those readings on the fly, and round to the nearest 1/16 — no calculator. Drag the indicator and watch the fraction snap. Every framing cut, every door jamb, every cabinet reveal depends on this becoming muscle memory.
foundationMetric ↔ Imperial Conversion & Tolerance Stacking
Blueprint specifies a 38 mm hole through 3/4" plywood. Your drill index is fractional, your CNC is metric, and the engineer specifies ±0.5 mm tolerance. Convert, round to the nearest drill bit, and check whether the slop is within spec. Get this wrong and the dowel won't seat.
Daily-job math
coreThe 3-4-5 Rule & Pythagorean Layout
You framed a 14×20 ft deck and need to prove it's square before nailing joists. A speed square checks 90° at a 6" scale — useless over 20 ft. Use the Pythagorean theorem at full deck scale: pick a 3-4-5 multiple that fits, then measure the diagonal. If it's off by even 1/2", the far corner is over an inch out of square.
coreStair Rise, Run & IRC Code Compliance
Build a stair from a 48" deck to a patio. IRC R311.7.5 caps riser ≤ 7 3/4", requires tread ≥ 10", and limits variation between any two risers to 3/8". The classic 'rule of 25' (2R + T ≈ 25) gives comfort. Compute step count, exact riser, total run, and verify code in one pass.
coreDrainage Slope, Velocity & IPC Sizing
60-ft of 3" sanitary drain under floor joists with only 10" of vertical headroom. IPC 704.1 mandates 1/4" per ft slope for ≤3" pipe (1/8"/ft allowed for 4"+). Manning's equation governs flow velocity — too slow and solids settle, too fast (>10 fps) and the slug runs away. Compute total drop, percent grade, and verify the line fits.
Master-class problems
masterOhm's Law, Power & NEC 80% Continuous-Load Rule
A 1500 W space heater on a 20 A / 120 V circuit. NEC 210.20(A) limits continuous loads (≥3 hr) to 80% of breaker rating. Compute draw, verify safety margin, then check what happens if the homeowner adds a 600 W toaster oven mid-shift. Wire ampacity, voltage drop on long runs, and resistive heating all interlock.
masterEMT Conduit Offsets: Multipliers, Shrink & Saddle Bends
Your 1/2" EMT run hits a junction box that's 6" higher than the conduit. Pick a bend angle, compute the distance between bend marks, then add the shrink to the cut length so the conduit lands exactly in the box. The math comes from the right triangle — multiplier = 1/sin(θ), shrink ≈ cot(θ/2) − cot(θ), tabulated for speed.