Quick answer: With legs 3 and 4 the hypotenuse is 5, angle A = atan(3 ÷ 4) = 36.8699°, angle B = 53.1301°, the area is 6 and the altitude to the hypotenuse is 3 × 4 ÷ 5 = 2.4.
How it's calculated
Label the right angle C, the legs a and b, and the hypotenuse c; angle A is opposite a and angle B is opposite b, so A + B = 90°.
Two sides: the third comes from a² + b² = c², and the angles from sin A = a ÷ c or tan A = a ÷ b. A side and an angle: use SOH-CAH-TOA, for example c = a ÷ sin A and b = a ÷ tan A, or a = c·sin A and b = c·cos A.
The area is a × b ÷ 2 (the legs are base and height), and the altitude from the right angle to the hypotenuse is h = a × b ÷ c.
Worked examples
- Legs 3 and 4: c = 5, A = 36.8699°, B = 53.1301°, area 6
- Hypotenuse 10, angle A = 30°: a = 10 × sin 30° = 5, b = 10 × cos 30° = 8.6603
- Leg a = 5, angle A = 45°: b = 5, c = 7.0711
Frequently asked questions
What is SOH-CAH-TOA?
A memory aid: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent.
Can I solve it from two angles?
No. Two angles fix the shape but not the size; you need at least one side.
What about triangles without a right angle?
Use the triangle calculator, which applies the laws of sines and cosines.
What are the angles in a 3-4-5 triangle?
90°, about 36.87° and about 53.13°. The two smaller ones always add up to 90°.
How do I get a ramp's angle from its rise and length?
Take the inverse tangent of rise ÷ run. A ramp that climbs 1 m over 12 m is about 4.8°.
Can I work out a building's height from where I stand?
Yes. Standing 30 m away and looking up 40° to the top, the building rises about 30 × tan 40° = 25.2 m above your eyes. Add your eye height, roughly 1.6 m.