OneTaskBy Tim King · Updated
Right Angle Triangle Calculator
SOHCAHTOA Solver for Sides, Angles, Area and Heights, With Working
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How the Right Triangle Calculator works
This right angle triangle calculator solves a triangle with one 90° corner from just two values: both legs, a leg and the hypotenuse, or one side and one of the other angles. It returns all three sides, both acute angles, the area, perimeter and the height to the hypotenuse, with each step of the trigonometry written out.
In a right triangle the two sides that meet at the square corner are the legs, here a and b, and the long side opposite the right angle is the hypotenuse c. Angle A sits opposite leg a, and the two acute angles always add up to 90°, so one angle gives you the other for free.
How the right angle triangle calculator works
Two sides use Pythagoras, with the formula c² = a² + b², and then an inverse tangent for the angle. A side and an angle use SOH CAH TOA: sin A = a ÷ c, cos A = b ÷ c and tan A = a ÷ b. Rearrange whichever ratio holds the known side and solve for the rest.
First example. Legs of 3 and 4 give a hypotenuse of 5, angles of 36.87° and 53.13°, and an area of 6.
Second example. A 4 m ladder leaning at 75° to the ground reaches 4 × sin 75° ≈ 3.86 m up the wall, with its foot 4 × cos 75° ≈ 1.04 m out.
How many metres of ramp do you need to rise 0.5 m at 5°? The ramp is the hypotenuse: 0.5 ÷ sin 5° ≈ 5.74 m.
Sides of a right triangle with hypotenuse 1 for angle A from 4° to 88°
| Angle A | Angle B | Opposite a (sin A) | Adjacent b (cos A) | a ÷ b (tan A) | Area |
|---|---|---|---|---|---|
| 4° | 86° | 0.0698 | 0.9976 | 0.0699 | 0.0348 |
| 8° | 82° | 0.1392 | 0.9903 | 0.1405 | 0.0689 |
| 12° | 78° | 0.2079 | 0.9781 | 0.2126 | 0.1017 |
| 16° | 74° | 0.2756 | 0.9613 | 0.2867 | 0.1325 |
| 20° | 70° | 0.3420 | 0.9397 | 0.3640 | 0.1607 |
| 24° | 66° | 0.4067 | 0.9135 | 0.4452 | 0.1858 |
| 28° | 62° | 0.4695 | 0.8829 | 0.5317 | 0.2073 |
| 32° | 58° | 0.5299 | 0.8480 | 0.6249 | 0.2247 |
| 36° | 54° | 0.5878 | 0.8090 | 0.7265 | 0.2378 |
| 40° | 50° | 0.6428 | 0.7660 | 0.8391 | 0.2462 |
| 44° | 46° | 0.6947 | 0.7193 | 0.9657 | 0.2498 |
| 48° | 42° | 0.7431 | 0.6691 | 1.1106 | 0.2486 |
| 52° | 38° | 0.7880 | 0.6157 | 1.2799 | 0.2426 |
| 56° | 34° | 0.8290 | 0.5592 | 1.4826 | 0.2318 |
| 60° | 30° | 0.8660 | 0.5000 | 1.7321 | 0.2165 |
| 64° | 26° | 0.8988 | 0.4384 | 2.0503 | 0.1970 |
| 68° | 22° | 0.9272 | 0.3746 | 2.4751 | 0.1737 |
| 72° | 18° | 0.9511 | 0.3090 | 3.0777 | 0.1469 |
| 76° | 14° | 0.9703 | 0.2419 | 4.0108 | 0.1174 |
| 80° | 10° | 0.9848 | 0.1736 | 5.6713 | 0.0855 |
| 84° | 6° | 0.9945 | 0.1045 | 9.5144 | 0.0520 |
| 88° | 2° | 0.9994 | 0.0349 | 28.6363 | 0.0174 |
Tips
- Opposite and adjacent depend on the angle. Leg a is opposite angle A but adjacent to angle B.
- Set angles in degrees. All angle inputs here are degrees, not radians.
- Scale the table. Multiply the hypotenuse-1 values by your hypotenuse to get real sides.
FAQ
What is SOH CAH TOA?
It is a memory aid for the three trig ratios in a right triangle. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. Pick the ratio that links the side you know with the side you want, then rearrange it.
How do you find an angle of a right triangle from two sides?
Use the inverse of the ratio that matches the sides you have. With both legs, A = arctan(a ÷ b). With a leg and the hypotenuse, A = arcsin(a ÷ c). Legs of 3 and 4 give arctan(0.75) ≈ 36.87°, and the other angle is 90° minus that.
What is the height to the hypotenuse?
It is the perpendicular distance from the right-angle corner to the hypotenuse, equal to a × b ÷ c. For the 3-4-5 triangle that is 12 ÷ 5 = 2.4. It splits the triangle into two smaller right triangles that are both similar to the original one.
How is this different from the Pythagorean theorem calculator?
The Pythagorean theorem calculator only finds a missing side from the other two. This one also accepts an angle as an input and returns both angles, the area, the perimeter and the in and circumradius, so it suits trigonometry problems as well as plain side lengths.
See also: all Algebra and Geometry tools, Arithmetic, Number Theory, Statistics, Trigonometry and Calculus.
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