Answer first
What this calculator tells you
Calculate the hypotenuse, both acute angles and the area of a right triangle from its two legs. Get the angle to cut or set from two measured sides. Formula: c = √(a² + b²); angle A = arctan(a ÷ b); angle B = 90° − A; area = ½ab. At the worked-example inputs, the hypotenuse (c) is 13. Holding every other input steady, moving leg a from 4 to 6 moves the result from 12.6 to 13.4.
Transparent method
The formula
Get the angle to cut or set from two measured sides.
Worked example
Example inputs
How to interpret the result
Two legs of a right triangle fix everything else. The hypotenuse follows from the Pythagorean theorem, the angles from the ratio of the legs, and the area from half their product. A 5 by 12 triangle has a 13 unit hypotenuse, angles of about 22.6 and 67.4 degrees, and an area of 30. That angle is what you set on a saw or a level when you have only tape measure readings. For comparison, a 3 by 4 triangle has a hypotenuse of 5 and an 8 by 15 triangle has 17. These whole-number sets, 3-4-5, 5-12-13 and 8-15-17, are why builders can check a square corner with a tape measure alone.
At the worked-example inputs the hypotenuse (c) is 13. It rises with leg b and leg a.
Formulas assume ideal geometric shapes. Real-world measurements carry their own error, and every input must use the same unit: mixing feet and inches silently produces a wrong answer.
Before you rely on it
What to check
Confirm the corner really is square. The formulas assume a 90 degree angle, and a slightly off corner gives an angle that only looks right.
The common error
Where people go wrong with right triangle calculator
Mixing up which leg goes with which angle. Angle A sits opposite leg a, so the longer leg faces the larger angle.
Sensitivity evidence
How leg a changes the hypotenuse (c)
Holding every other input at the worked-example value, moving leg a from 4 to 6 moves the hypotenuse (c) from 12.6 to 13.4: a spread of 0.767, or 6% of the worked-example result.
| Leg a | Hypotenuse (c) | Angle A (degrees) | Angle B (degrees) | Area |
|---|---|---|---|---|
| 4 | 12.6 | 18.4 | 71.6 | 24 |
| 4.5 | 12.8 | 20.6 | 69.4 | 27 |
| 5worked example | 13 | 22.6 | 67.4 | 30 |
| 5.5 | 13.2 | 24.6 | 65.4 | 33 |
| 6 | 13.4 | 26.6 | 63.4 | 36 |
Every input, tested
Which input moves the hypotenuse (c) most
Of the 2 inputs, leg b moves the hypotenuse (c) most (1.8 across the range tested) and leg a moves it least (0.384).
| Input | Tested from | To | Hypotenuse (c) at each end | Swing |
|---|---|---|---|---|
| Leg b | 11 | 13 | 12.1 to 13.9 | 1.8 (14%) |
| Leg a | 4.5 | 5.5 | 12.8 to 13.2 | 0.384 (3.0%) |
Two variables at once
Hypotenuse (c) by leg a and leg b
Across the grid the hypotenuse (c) runs from 10.8 to 15.2. Moving leg a from 4 to 6 shifts it by 0.767 at the middle column, and moving leg b from 10 to 14 shifts it by 3.7 at the middle row, so leg b is the bigger lever here.
| Leg a \ Leg b | 10 | 12 | 14 |
|---|---|---|---|
| 4 | 10.8 | 12.6 | 14.6 |
| 4.5 | 11 | 12.8 | 14.7 |
| 5 | 11.2 | 13 | 14.9 |
| 5.5 | 11.4 | 13.2 | 15 |
| 6 | 11.7 | 13.4 | 15.2 |
The highlighted cell is the worked example: 13.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Leg a | 5 | Enter the leg a used in this calculation. |
| Leg b | 12 | Enter the leg b used in this calculation. |
| Hypotenuse (c) | 13 | |
| Angle A (degrees) | 22.6 | |
| Angle B (degrees) | 67.4 | |
| Area | 30 | |
Inputs, definitions and assumptions
Leg a
Enter the leg a used in this calculation. The prefilled worked-example value is 5.
Leg b
Enter the leg b used in this calculation. The prefilled worked-example value is 12.
How to use this calculator
- 1Verify the inputs. Gather leg a and leg b from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the hypotenuse (c) at 13. Store your own version of it as Scenario A.
- 3Test one change. Start with leg b, the input with the biggest effect here: moving leg b from 11 to 13 takes the hypotenuse (c) from 12.1 to 13.9, a swing of 14% of the worked-example figure.
- 4Check the extremes. At half the example leg b (6) the hypotenuse (c) is 7.8; at double (24) it is 24.5.
People also ask
Frequently asked questions
How do you calculate right triangle?
c = √(a² + b²); angle A = arctan(a ÷ b); angle B = 90° − A; area = ½ab. At the worked-example inputs the hypotenuse (c) is 13.
What does the right triangle result mean?
Get the angle to cut or set from two measured sides. At the worked-example inputs the hypotenuse (c) is 13. It rises with leg b and leg a.
How much does leg a change the hypotenuse (c)?
Holding every other input at the worked-example value, moving leg a from 4 to 6 moves the hypotenuse (c) from 12.6 to 13.4, a spread of 0.767.
What are the limits of this right triangle calculator?
Formulas assume ideal geometric shapes. Real-world measurements carry their own error, and every input must use the same unit: mixing feet and inches silently produces a wrong answer. The tables on this page test leg a only from 4 to 6; a value outside that range is not tabulated here.
Which input moves the hypotenuse (c) most in the right triangle calculator?
Ranked by how far each moves the hypotenuse (c) across the range tested: leg b (1.8, 14%) and leg a (0.384, 3.0%).
If I double leg b in the right triangle calculator, does the hypotenuse (c) double?
Doubling it from 12 to 24 takes the hypotenuse (c) from 13 to 24.5, which is 1.89 times the worked-example figure. So it grows, but by less than double. Halving it to 6 gives 7.8.
How much does leg b matter in the right triangle calculator?
The worked example uses 12. Holding every other input at its worked-example value, moving leg b from 11 to 13 takes the hypotenuse (c) from 12.1 to 13.9, a swing of 14% of the worked-example figure.
Which inputs change the angle a (degrees) in the right triangle calculator?
At the worked-example inputs it is 22.6. Leg a takes it from 20.6 to 24.6 and leg b takes it from 24.4 to 21.
Which inputs change the angle b (degrees) in the right triangle calculator?
At the worked-example inputs it is 67.4. Leg a takes it from 69.4 to 65.4 and leg b takes it from 65.6 to 69.
Which inputs change the area in the right triangle calculator?
At the worked-example inputs it is 30. Leg a takes it from 27 to 33 and leg b takes it from 27.5 to 32.5.
How do I find the area of an irregular shape?
Split it into rectangles, triangles, trapezoids and circles, work out each one, and add the areas. Subtract any cut-outs. The more pieces you use, the closer the total gets to the true area.
Sources and evidence
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