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Shop Math, Finally Simple

The stuff the handbook makes you squint at — trig, tapers, bolt circles, thread wires — explained like a human and backed by calculators that do the work for you.

Shop Trig Right Triangle

Almost every angle problem on a print is secretly a right triangle. If you can find the right triangle hiding in the part, three buttons on a calculator solve it. You only ever need two pieces of info: either one angle + one side, or two sides.

Where you use it: finding an angle between two features, how far to move for a given slope, the reach of a chamfer, laying out an offset.

Aadj (b)opp (a)hyp (c)
Opposite is across from angle A · adjacent is beside it · hypotenuse is the slant
Right angle at the corner. opp = across from the angle · adj = next to it · hyp = the long slanted side.
opp = hyp × sin(A)  |  adj = hyp × cos(A)  |  opp = adj × tan(A)  |  hyp² = opp² + adj²

Triangle Solver — fill in what you know, leave the rest blank

Tip: the hypotenuse is always the longest side and sits opposite the 90° corner.

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Sine Bar / Sine Plate Angle Setups

A sine bar turns an angle into a stack of gauge blocks you can actually build. The bar is a known length (usually 5"). Tip it up on a block stack and the angle it sits at is locked in by simple trig — no protractor guessing.

Where you use it: holding a precise angle for grinding, inspection, or milling without a rotary table or protractor.

θHL = bar lengthgauge blocks
Bar of length L tilts on a gauge-block stack of height H to hold angle θ
Block stack = Bar length × sin(angle)
Backwards: angle = asin( stack ÷ bar length )

Sine Bar Calculator

Enter the angle to get the stack, or enter the stack to get the angle. Leave the one you want blank.

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Bolt Circle Hole Coordinates

A ring of evenly spaced holes is a bolt circle. The print gives you the circle diameter and how many holes — this spits out the X/Y location of every hole from the center so you can punch them straight into a DRO or program.

Where you use it: flanges, face plates, index plates, anything with a hole pattern on a diameter.

YXθbolt circle diameter
Each hole sits at angle θ on the circle — its X/Y are measured from the center
For each hole: angle = start + (360 ÷ holes) × hole#
X = (Dia ÷ 2) × cos(angle)  |  Y = (Dia ÷ 2) × sin(angle)   (center = 0,0)

Bolt Circle Calculator

0° is 3-o'clock and angles go counter-clockwise (standard math convention). Center is X0 Y0.

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Radius & Arc Chord / Sagitta

You almost never get to measure a radius directly. Instead you measure across the arc (the chord) and how deep it bulges (the height / sagitta) — and from those two the radius falls right out. Works the other way too.

Where you use it: finding the radius of a curved edge, checking a blend, setting a ball-nose step, reverse-engineering a worn part.

RchordHcenter of the radius
Chord goes across the arc · height H bulges from the chord · R runs from the center
Radius from chord + height: R = (chord² ÷ (8 × height)) + (height ÷ 2)
Height from R + chord: h = R − √(R² − (chord÷2)²)
Chord from R + height: chord = 2 × √(height × (2R − height))

Arc Calculator — enter any two

Fill in two, leave the third blank. "Height" is measured at the middle of the chord.

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Tap Drill Hole Before Threads

Before you tap a hole, you drill it — and the drill has to leave exactly the right amount of material for the threads to bite. Too big = weak threads, too small = snapped tap. This finds the drill size for a clean, strong ~75% thread (the sweet spot the shop uses).

Where you use it: every tapped hole. The easy shop rule for metric is "major diameter minus the pitch" — this gives you the precise version plus that rule.

Major Ø (thread crest)tap drill Øring between the circles = thread material
Drill the inner (tap-drill) hole — the threads are cut out to the major diameter
Inch (75% thread): tap drill = Major dia − (0.974 ÷ TPI)
Metric (75% thread): tap drill = Major dia − (0.812 × pitch)
Quick metric rule: tap drill ≈ Major − pitch

Tap Drill Calculator

Match the decimal to the nearest drill in your index (number / letter / fractional). For brittle material or blind holes, going a hair bigger (toward 65–70%) saves taps.

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3-Wire Thread Measuring Pitch Dia

You can't mic a thread's real size across the crests — you measure over three wires dropped into the grooves (two one side, one the other) and do a little math. This gives you the "measurement over wires" for a 60° thread and the best wire size to use.

Where you use it: checking the pitch diameter of a thread you're cutting on a lathe so it gages right.

M = measurement over wireswire3 wires in the grooves — mic over the outer two
Drop wires in the thread grooves, then measure across the outer wires (M)
Best wire = 0.57735 ÷ TPI
Measurement over wires: M = PitchDia + (3 × wire) − (1.5155 ÷ TPI)  (60° UN thread)

3-Wire Calculator

Leave wire blank and it uses the ideal "best wire" size. Formula is for standard 60° Unified/metric threads.

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Speeds & Feeds RPM & IPM

Two numbers run the cut: how fast the tool spins (RPM) and how fast it moves through the metal (feed, IPM). Spin comes from the material's surface speed and the tool diameter; feed comes from RPM, how many flutes, and how big a bite each flute takes.

Where you use it: every drill, mill, or lathe cut. Start conservative and open it up once the chips look right.

RPM (spin)feed (IPM)Ø D
Spin speed comes from the tool diameter · feed is how fast it moves through the metal
RPM = (SFM × 3.82) ÷ tool diameter
Feed (IPM) = RPM × chip load per tooth × number of flutes

Speeds & Feeds Calculator

Ballpark SFM — HSS: mild steel 80–110, aluminum 250–350, stainless 40–70, cast iron 50–80. Carbide: roughly 3–4× those. Chip load scales with tool size (tiny ~0.0005", 1/2" ~0.002–0.004").

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Tapers TPF ↔ Angle

A taper is just a controlled slope. Shops talk about it two ways: taper per foot (TPF) and included angle. This flips between them, and if you've got the two diameters and the length it works the taper out for you.

Where you use it: setting a compound or tailstock offset, checking a Morse/Jarno taper, grinding a draft angle.

DdL (length)angle
Big diameter D, small diameter d, over length L — the slope sets the included angle
From the part: TPF = ((big dia − small dia) ÷ length) × 12
Included angle = 2 × atan( (big dia − small dia) ÷ (2 × length) )
From TPF: included angle = 2 × atan( TPF ÷ 24 )

Taper Calculator

Give the three part dimensions for everything, or just TPF to get the angle. The compound is set to HALF the included angle.

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