Right Triangle Calculator — hypotenuse, area, perimeter and angles

How it works

A right triangle has one 90° angle. The two sides forming it are the legs (a and b); the side opposite, the longest, is the hypotenuse (c). Everything else follows from the two legs:

• Hypotenuse: c = √(a² + b²) — the Pythagorean theorem.
• Area: ½ · a · b — the legs are a base and its perpendicular height.
• Perimeter: a + b + c.
• Acute angles: the angle opposite leg a is arctan(a ÷ b); the other is its complement, since the three angles sum to 180° and one is already 90°.

Worked example

Take legs a = 3 and b = 4:

• Hypotenuse: c = √(3² + 4²) = √25 = 5
• Area: ½ × 3 × 4 = 6
• Perimeter: 3 + 4 + 5 = 12
• Angle opposite a: arctan(3 ÷ 4) ≈ 36.87°; the other ≈ 53.13° (they sum to 90°)

That's the 3-4-5 triangle. Equal legs give an isosceles right triangle — legs 1 and 1 make a hypotenuse of √2 ≈ 1.414 and two 45° angles.

Frequently asked questions

How do I find the hypotenuse of a right triangle?

Square both legs, add them, and take the square root: c = √(a² + b²). For legs 6 and 8, c = √(36 + 64) = √100 = 10. The hypotenuse is always the longest side and sits opposite the right angle.

How do I find the area?

Multiply the two legs and halve: area = ½ × a × b. Because the legs meet at a right angle, one serves as the base and the other as the height, so no separate height is needed. Legs 5 and 12 give ½ × 5 × 12 = 30.

How are the acute angles found?

With the inverse tangent. The angle opposite leg a is arctan(a ÷ b), and the angle opposite leg b is arctan(b ÷ a) — or just 90° minus the first, since the two acute angles are complementary. For legs 3 and 4 they're about 36.87° and 53.13°.

What if I know the hypotenuse and one leg instead?

Find the other leg first: b = √(c² − a²), rearranging the Pythagorean theorem. With that, the area and angles follow as above. This calculator takes the two legs; if you have the hypotenuse, compute the missing leg and use it.

What makes 3-4-5, 5-12-13 and 8-15-17 special?

They're **Pythagorean triples** — right triangles whose three sides are all whole numbers. Any multiple of a triple (like 6-8-10) is also one. Most leg pairs give an irrational hypotenuse instead, such as legs 1 and 2 giving √5.

Do the angles depend on the size of the triangle?

No — only on the *ratio* of the legs. Legs 3 and 4 give the same 36.87°/53.13° angles as legs 6 and 8 or 30 and 40, because arctan(3/4) = arctan(6/8). Scaling a right triangle changes its side lengths and area but not its angles.