Right triangle calculator
Legs of 8 and 15 make a hypotenuse of 17 in the right triangle calculator, with acute angles of 28.07° and 61.93° and an area of 60.
Hypotenuse
- Hypotenuse a squared plus b squared equals c squared: 64 + 225 = 289.
- 17
- A Pythagorean triple All three sides are whole numbers, which is rare and is why a handful of triples (3-4-5, 5-12-13, 8-15-17) turn up in every textbook and on every building site.
- 8, 15, 17
- Sides
- a = 8, b = 15, c = 17
- Angles Summing to 180 degrees, as every plane triangle must.
- A = 28.07deg, B = 61.93deg, C = 90deg
- Area
- 60
- Perimeter
- 40
- Type
- scalene, right
- Heights The perpendicular from each side to the opposite vertex. Area is half the base times the height for any of the three.
- to a: 15, to b: 8, to c: 7.0588
- Circumradius The radius of the circle through all three vertices.
- 8.5
- Inradius The radius of the largest circle that fits inside.
- 3
| a | b | c | Multiples |
|---|---|---|---|
| 3 | 4 | 5 | 6-8-10, 9-12-15, 30-40-50 |
| 5 | 12 | 13 | 10-24-26, 15-36-39 |
| 8 | 15 | 17 | 16-30-34 |
| 7 | 24 | 25 | 14-48-50 |
| 20 | 21 | 29 | 40-42-58 |
| 9 | 40 | 41 | 18-80-82 |
- The Pythagorean theorem holds only for right triangles. For any other triangle the law of cosines generalizes it: c squared equals a squared plus b squared minus 2ab cos(C), which reduces to Pythagoras when C is 90 degrees and cos(C) is zero.
- Any two sides must add to more than the third, or the shape does not close. That triangle inequality is checked here before anything else is computed.
How the Triangle Calculator works
To calculate a right triangle you need any two of its sides. The third comes from the Pythagorean theorem, c = √(a² + b²) for the hypotenuse or a = √(c² - b²) for a leg, and everything else follows from the three sides.
With legs a = 8 and b = 15, c = √(64 + 225) = √289 = 17. The angle opposite a is A = arctan(a ÷ b) = arctan(8 ÷ 15) = 28.07°, and the other acute angle is B = 90° - 28.07° = 61.93°. The area is half the product of the legs, ½ × 8 × 15 = 60, and the perimeter is 8 + 15 + 17 = 40.
The height from the right angle to the hypotenuse is a × b ÷ c = 120 ÷ 17 = 7.06. It splits the triangle into two smaller right triangles, both similar to the first.
The trig ratios are the other way in. sin A = opposite ÷ hypotenuse = 8/17, cos A = adjacent ÷ hypotenuse = 15/17 and tan A = opposite ÷ adjacent = 8/15. Any one of them with any one side solves the triangle.
Two circles come out simply too. The circumradius of a right triangle is half the hypotenuse, 8.5, because the hypotenuse is a diameter of the circle through all three corners. The inradius is (a + b - c) ÷ 2 = 3.
Triangle Calculator FAQ
What are the special right triangles?
The 45-45-90 triangle, with sides in the ratio 1 : 1 : √2, and the 30-60-90 triangle, with sides in the ratio 1 : √3 : 2. A 30-60-90 triangle with a hypotenuse of 10 has legs of 5 and 8.660.
How do I find the sides from one side and an angle?
Use sine and cosine. With a hypotenuse of 20 and an angle of 40°, the side opposite the angle is 20 × sin 40° = 12.856 and the side next to it is 20 × cos 40° = 15.321.
How do I tell whether three sides make a right triangle?
Check whether the squares of the two shorter sides add up to the square of the longest. 8² + 15² = 289 = 17², so 8, 15, 17 is a right triangle; 8, 15, 16 is not.
More from the Triangle Calculator
- Law of cosines calculator
- Law of sines calculator
- Equilateral triangle calculator
- Isosceles triangle solver
- Pythagorean theorem calculator
- Triangle area calculator