About the Right Triangle Calculator
This right triangle calculator solves a triangle with a 90° corner from just two known values. Give it both legs, a leg and the hypotenuse, or one side plus an acute angle, and it returns every side and angle, the area, the perimeter, the altitude to the hypotenuse, and the radii of the inscribed and circumscribed circles.
Right triangles are everywhere in practical measuring: a ladder leaning on a wall, the rise and run of a roof or ramp, the diagonal of a rectangular room, or the height of a tree from its shadow. Students use the same maths for SOH-CAH-TOA trigonometry problems and Pythagorean theorem questions.
Labelling: legs a and b meet at the right angle C, the hypotenuse c is opposite it, angle A is opposite leg a and angle B is opposite leg b, so A + B = 90°. Angles are in degrees and lengths in any single unit.
How to use the right triangle calculator
- 1Choose which two values you know.
- 2Enter the lengths in one unit and any angle in degrees.
- 3Read the missing sides and angles from the solution line.
- 4Use the area, perimeter and altitude for follow-up questions.
Formula and method
Any right triangle is fixed by two independent values (the right angle is the third). With two sides, the Pythagorean theorem gives the third side and the inverse tangent gives angle A. With a side and an angle, SOH-CAH-TOA gives the other sides: the side opposite A is c·sin A, the adjacent leg is c·cos A, and a = b·tan A.
The other angle is always B = 90° − A. The area is half the product of the legs; the altitude from the right angle to the hypotenuse is ab ÷ c; the inscribed circle has radius (a + b − c) ÷ 2; and the circumscribed circle has its centre at the midpoint of the hypotenuse, so its radius is c ÷ 2.
- a, b
- Legs (the sides that form the right angle)
- c
- Hypotenuse (longest side, opposite the right angle)
- A, B
- Acute angles opposite a and b (A + B = 90°)
- h
- Altitude from the right angle to the hypotenuse
Worked examples
Legs 3 and 4
c = √(3² + 4²) = 5. A = arctan(3/4) ≈ 36.87° and B ≈ 53.13°. The area is 3 × 4 ÷ 2 = 6, and the altitude to the hypotenuse is 12 ÷ 5 = 2.4.
A 20 ft ladder at 75° to the ground
The ladder is the hypotenuse. Its top reaches a = 20 × sin 75° ≈ 19.32 ft up the wall, and its foot sits b = 20 × cos 75° ≈ 5.18 ft from the base.
Roof rafter: 12 ft run at a 30° pitch
The run is the leg adjacent to the pitch angle. The rise is 12 × tan 30° ≈ 6.93 ft and the rafter length is 12 ÷ cos 30° ≈ 13.86 ft (before overhang).
Leg 5 and hypotenuse 13
b = √(169 − 25) = 12 — the 5-12-13 Pythagorean triple. A = arcsin(5/13) ≈ 22.62°, and the area is 5 × 12 ÷ 2 = 30.
Frequently asked questions
How many values do I need to solve a right triangle?+
Two, as long as at least one of them is a side. Two angles alone only fix the shape, not the size, because any scaled copy has the same angles.
What does SOH-CAH-TOA mean?+
Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent. These ratios link an acute angle of a right triangle to its sides.
How do I find the hypotenuse?+
Square both legs, add them and take the square root: c = √(a² + b²). For legs 6 and 8 the hypotenuse is √100 = 10.
What is the altitude to the hypotenuse?+
It is the perpendicular line from the right angle to the hypotenuse. Its length is ab ÷ c, and it splits the triangle into two smaller triangles similar to the original.
What are common Pythagorean triples?+
3-4-5, 5-12-13, 8-15-17 and 7-24-25 are whole-number right triangles, and any multiple (6-8-10, 9-12-15) works too. Builders use 3-4-5 to check corners are square.