Worked examples
From the legs. A triangle with legs 3 and 4 has a hypotenuse of √(9 + 16) = 5. Angle A = arctan(3 / 4) = 36.8699° (36° 52′ 12″), angle B = 53.1301°.
From the hypotenuse. A 10 mm slanted edge at 30° from the horizontal rises 10 × sin 30° = 5 mm and runs 10 × cos 30° = 8.6603 mm.
Where it comes up in the shop
- Chamfers: a 45° chamfer of 2 mm on each leg has a face of 2 × √2 = 2.83 mm. A chamfer given by length and angle (say 1.5 mm at 30°) has the other leg 1.5 × tan 30° = 0.87 mm.
- Dovetails and V-grooves: the width at the top from the depth and the flank angle is 2 × depth × tan(half angle) plus the bottom width.
- Hole positions from a datum at an angle: use the hypotenuse form to split a distance along an angled line into X and Y – for a full circle of holes the bolt circle calculator does it for every hole.
- Setting an angle: a sine bar or a compound slide taper uses the same relations.
Frequently asked questions
How do I find the hypotenuse of a right triangle?
c = √(a² + b²). Legs of 3 and 4 give √25 = 5.
How do I find an angle from two sides?
Angle A, opposite leg a, is arctan(a / b). For legs 3 and 4 that is 36.87°; the other angle is 90° − 36.87° = 53.13°.
How do I find the legs from the hypotenuse and an angle?
a = c × sin A (opposite) and b = c × cos A (adjacent). For c = 10 and A = 30°: a = 5 and b = 8.660.
What is the length of a 45° chamfer face?
The leg length times √2, about 1.414. A chamfer of 2 mm on each leg has a face of 2.83 mm.
More free machinist calculators
- Inch to mm – Convert inches to millimetres and back, fractions like 1 3/8 included. Exact factor 25.4, with a chart of every 1/64 inch.
- RPM and feed – Calculate spindle RPM from surface speed and diameter, surface speed from RPM, and feed rate from chip load – metric and imperial, with formulas.
- Taper – Taper ratio, half angle and taper per foot from the large and small diameter and the length, or the small diameter from the angle.
- Bolt circle – X and Y coordinates of evenly spaced holes on a bolt circle, with the hole spacing, start angle and centre offset. Copy the table into your program.
- Sine bar – Gauge block height for a sine bar angle, or the angle from a known stack, for 5 in, 10 in, 100 mm, 200 mm or any other bar length.
- Metal removal rate – Metal removal rate in cm³/min or in³/min from depth, width and feed – milling and turning, metric and imperial, with the formulas.
- Drill point depth – Length of a drill's point cone from diameter and point angle, and the total depth needed for a full-diameter hole to depth.
- Machining time – Cutting time from length, feed and spindle speed – turning with feed per revolution, milling with feed rate and number of passes.
- Decimal to fraction – Turn a decimal inch or a millimetre value into the nearest fraction – 8ths to 128ths – with the error in millimetres.
- Degrees to DMS – Convert decimal degrees to degrees, minutes and seconds and back, with the angle in radians.
- SFM to m/min – Convert surface feet per minute to metres per minute and back, and inches per minute to mm per minute, with the exact factors.
- TPI to pitch – Convert a thread pitch in millimetres to threads per inch (TPI) and back, for matching metric and inch threads.
- Cpk calculator – Paste measurements from Excel, enter USL and LSL and get Cp, Cpk, Pp and Ppk. Says plainly when 25 values are too few for an index.
- Tolerance stack-up – Add up the tolerances of a chain of parts: closing gap, worst case and statistical RSS for up to eight dimensions. Free, in your browser.
- Fit clearance – Enter hole and shaft limits and get minimum and maximum clearance and the kind of fit – clearance, transition or interference.
- Chip thinning – Feed per tooth to program when the radial width of cut is below half the cutter diameter, so the real chip thickness stays right.
- Ball-nose stepover – Scallop height left by a ball-nose cutter from the stepover, or the stepover for a wanted scallop height, on a flat surface.
- Surface finish – Theoretical Rt and estimated Ra of a turned surface from the feed per revolution and the tool nose radius.
- Torque and power – Torque from power and speed in newton-metres or pound-feet, and power from torque and speed. The formulas with worked examples.
- Spur gear – Pitch and outside diameter, centre distance, ratio and circular pitch of a pair of external spur gears from the module and the tooth numbers.
Making parts for a customer who wants paperwork?
Automotive and aerospace customers ask for proof that the parts match the drawing. See first article inspection (FAI), PPAP explained and the true position calculator.