Layout and shop math
Right triangle calculator for shop layout
c = sqrt(a^2 + b^2); a hole 1.2500 in up and 2.7500 in over sits 3.0208 in from the datum at 24.4440° (24° 26' 38").
Technical review pending. Formulas and values are cited. This notice comes down after a machinist review.
Solve a triangle
Hypotenuse c
in
c = sqrt(a^2 + b^2) = sqrt(1.2500^2 + 2.7500^2) = 3.0208 in
3.0208 in (76.727 mm). Angle A 24.4440° (24° 26' 38"), angle B 65.5560° (65° 33' 22").
| Value | Result, inch | Result, metric | Formula with your numbers |
|---|---|---|---|
| Side a, opposite A | 1.2500 in | 31.750 mm | Given |
| Side b, adjacent to A | 2.7500 in | 69.850 mm | Given |
| Hypotenuse c | 3.0208 in | 76.727 mm | c = sqrt(a^2 + b^2) = sqrt(1.2500^2 + 2.7500^2) = 3.0208 in |
| Angle A | 24.4440° (24° 26' 38") | 24.4440° (24° 26' 38") | A = atan(a / b) = atan(1.2500 / 2.7500) = 24.4440° |
| Angle B | 65.5560° (65° 33' 22") | 65.5560° (65° 33' 22") | B = 90° - A = 90° - 24.4440° = 65.5560° |
| Area | 1.7188 in² | 1,108.87 mm² | area = a x b / 2 = 1.2500 x 2.7500 / 2 = 1.7188 in² |
| Perimeter | 7.0208 in | 178.327 mm | perimeter = a + b + c = 1.2500 + 2.7500 + 3.0208 = 7.0208 in |
| Altitude to the hypotenuse, h | 1.1380 in | 28.904 mm | h = a x b / c = 1.2500 x 2.7500 / 3.0208 = 1.1380 in |
Preset: chamfer legs
Other leg
in
other leg = L x tan(A) = 0.04 x tan(30°) = 0.0231 in
Chamfer face: face = L / cos(A) = 0.04 / cos(30°) = 0.0462 in. Angle measured from the other surface: other leg = L / tan(A) = 0.04 / tan(30°) = 0.0693 in.
Preset: taper angle and rise
Half angle (compound setting)
°
half angle = atan((D - d) / (2 x L)) = atan((1.25 - 0.75) / (2 x 4)) = 3.5763° (3° 34' 35")
Compound setting 3° 34' 35". Included angle 7.1527° (7° 9' 10"). Rise = (D - d) / 2 = (1.25 - 0.75) / 2 = 0.2500 in. Flank = sqrt(L^2 + rise^2) = sqrt(4^2 + 0.25^2) = 4.0078 in. TPF = 12 x (D - d) / L = 12 x (1.25 - 0.75) / 4 = 1.500 in/ft.
Preset: hole on an angle
X and Y from the reference
in
X = R x cos(A) = 2 x cos(35°) = 1.6383 in; Y = R x sin(A) = 2 x sin(35°) = 1.1472 in
From the reference point, angle 35° counterclockwise from +X. X 1.6383 in (41.613 mm), Y 1.1472 in (29.138 mm).
Preset: sine bar stack
Stack height
in
H = L x sin(A) = 5 x sin(12°) = 1.0396 in
1.0396 in (26.405 mm) stack. Angle 12° 0' 0".
Source: Machinery's Handbook (trigonometry; solution of right triangles); plane trigonometry as shown. Taper and sine bar presets follow the formulas on the taper and sine bar calculators.
Results are starting points for layout. Check the first feature against the drawing before cutting the rest.
How it works
Right triangle formulas
c = sqrt(a^2 + b^2); sin A = a / c; cos A = b / c; tan A = a / b; B = 90° - A
| Symbol | Meaning | Unit |
|---|---|---|
| a | Side opposite angle A | in or mm |
| b | Side adjacent to angle A | in or mm |
| c | Hypotenuse, across from the 90° corner | in or mm |
| A, B | The two acute angles | ° |
| h | Altitude from the 90° corner to the hypotenuse: h = a x b / c | in or mm |
| area | a x b / 2 | in² or mm² |
| You know | Sides | Angle A |
|---|---|---|
| a and b | c = sqrt(a^2 + b^2) | A = atan(a / b) |
| a and c | b = sqrt(c^2 - a^2) | A = asin(a / c) |
| b and c | a = sqrt(c^2 - b^2) | A = acos(b / c) |
| A and c | a = c x sin(A); b = c x cos(A) | Given |
| A and b | a = b x tan(A); c = b / cos(A) | Given |
| A and a | b = a / tan(A); c = a / sin(A) | Given |
Then B = 90° - A, area = a x b / 2, perimeter = a + b + c and h = a x b / c. D-M-S converts with A = D + M / 60 + S / 3600. Inch results show 4 places, mm 3 places, angles 4 places and D-M-S to the second.
Shop presets
other leg = L x tan(A); face = L / cos(A); half angle = atan((D - d) / (2 x L)); X = R x cos(A); Y = R x sin(A); H = L x sin(A)
| Symbol | Meaning | Unit |
|---|---|---|
| L | Chamfer length, taper length or sine bar length (center to center of the rolls), per preset | in or mm |
| D, d | Large and small diameter of the taper | in or mm |
| R | Distance from the reference point to the hole | in or mm |
| X, Y | Coordinates from the reference point | in or mm |
| H | Gage block stack height | in or mm |
| A | Chamfer angle, hole angle from +X, or sine bar angle, per preset | ° |
| rise, flank | Taper rise per side, (D - d) / 2, and flank length along the taper | in or mm |
| TPF | Taper per foot, inch taper preset only | in/ft |
Worked example: hole offset from two legs (inch)
A hole sits 1.2500 in up (side a) and 2.7500 in over (side b) from the datum corner.
- c = sqrt(1.2500^2 + 2.7500^2) = sqrt(1.5625 + 7.5625) = sqrt(9.1250) = 3.0208 in (76.727 mm).
- A = atan(1.2500 / 2.7500) = atan(0.454545) = 24.4440° (24° 26' 38").
- B = 90° - 24.4440° = 65.5560° (65° 33' 22").
- Area = 1.2500 x 2.7500 / 2 = 1.7188 in² (1,108.87 mm²).
- Perimeter = 1.2500 + 2.7500 + 3.0208 = 7.0208 in (178.327 mm).
- h = 1.2500 x 2.7500 / 3.0208 = 1.1380 in (28.904 mm).
Result: 3.0208 in (76.727 mm) from the datum at 24.4440°. Metric legs: a 31.750 mm, b 69.850 mm.
Worked example: ramp at 15° 30' over 80 mm (metric, D-M-S)
A ramp milled at 15° 30' over 80 mm of X travel. A = 15 + 30 / 60 = 15.5000°; b = 80 mm.
- tan(15.5°) = 0.277325; a = 80 x 0.277325 = 22.186 mm of rise (0.8735 in).
- cos(15.5°) = 0.963630; c = 80 / 0.963630 = 83.019 mm along the ramp (3.2685 in).
- B = 90° - 15.5° = 74.5000° (74° 30' 0").
- Area = 22.186 x 80 / 2 = 887.44 mm² (1.3755 in²); perimeter = 185.205 mm; h = 22.186 x 80 / 83.019 = 21.379 mm.
- Check: sqrt(22.185964^2 + 80^2) = 83.019377 mm, the same hypotenuse.
Result: 22.186 mm of rise and 83.019 mm along the ramp.
Preset checks
- Chamfer 0.030 in x 45°: other leg 0.0300 in, face 0.0424 in. 1 mm x 45°: face 1.414 mm.
- Chamfer 0.040 in x 30°, angle from the surface the 0.040 in runs along: other leg 0.040 x 0.577350 = 0.0231 in, face 0.040 / 0.866025 = 0.0462 in. Angle from the other surface: 0.040 / 0.577350 = 0.0693 in.
- Taper 1.250 in to 0.750 in over 4.000 in: rise 0.2500 in; half angle atan(0.0625) = 3.5763° (3° 34' 35"); included 7.1527° (7° 9' 10"); flank 4.0078 in; TPF 1.500 in/ft.
- Hole 2.000 in at 35°: X = 2.000 x 0.819152 = 1.6383 in; Y = 2.000 x 0.573576 = 1.1472 in. Metric, 50 mm at 22.5°: X 46.194 mm, Y 19.134 mm.
- Sine bar, 5 in bar at 12°: H = 5 x 0.207912 = 1.0396 in. A 1.2500 in stack under a 5 in bar: A = asin(0.2500) = 14.4775° (14° 28' 39").
Shop notes
- Side a is always opposite angle A. Most wrong answers come from swapping a and b.
- Check the calculator mode before trusting a pocket calculator: sin(30) in radians is -0.9880, not 0.5000.
- 15° 30' is 15.5°, not 15.30°. Enter minutes in the minutes field, or convert on the degree converter.
- Key coordinates from the hole preset as absolute values from one zero; rounding cannot stack that way.
- For a chamfer on a CNC, the chamfer face length is not what a chamfer mill's depth is set to; the leg along the axis of the tool is. Read the drawing callout before choosing which leg to cut to.
FAQ
How do you find the hypotenuse from two legs?
c = sqrt(a^2 + b^2). Legs of 1.2500 in and 2.7500 in give 3.0208 in; legs of 3.000 in and 4.000 in give 5.0000 in.
How deep is a 0.040 in x 30° chamfer?
It depends on which surface the angle is measured from. Measured from the surface the 0.040 in runs along, the other leg is 0.0231 in; measured from the other surface, it is 0.0693 in. A 45° chamfer has equal legs: 0.030 in x 45° is 0.030 in each way.
What compound angle cuts a taper from 1.250 in to 0.750 in over 4.000 in?
The half angle: atan(0.2500 / 4.000) = 3.5763° (3° 34' 35"), with a 0.2500 in radial rise.
What are X and Y for a hole 2.000 in from center at 35°?
X 1.6383 in, Y 1.1472 in: 2.000 x cos(35°) and 2.000 x sin(35°).
What gage block stack sets 12° on a 5 in sine bar?
1.0396 in: 5 x sin(12°) = 5 x 0.207912.
Is 15° 30' the same as 15.30°?
No. 30 minutes is half a degree, so 15° 30' is 15.5000°.
Does this solver match the short one on the trig hub?
Yes. A 3.000 by 4.000 in triangle gives c 5.0000 in and A 36.8699°; 30° on a 5.000 in hypotenuse gives b 4.3301 in, the same as the geometry and trigonometry hub.
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