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Math

Pythagorean Theorem Calculator

Pick whether you want the hypotenuse or the missing leg, enter two positive numbers, and get the third side, the triangle area (½ a b), perimeter (a + b + c) and the two non-right angles (computed via atan2). The classic 3-4-5, 5-12-13 and 8-15-17 are all integer Pythagorean triples that round-trip exactly.

Side a

3

Given

Side b

4

Given

Hypotenuse c

5

Computed

Area

6

Perimeter

12

Angle opposite a (∠A)

36.87°

Angle opposite b (∠B)

53.13°

Formula

c = √(a² + b²) ; b = √(c² − a²) ; area = ½ a b ; ∠A = atan(a / b)

Side units can be anything (cm, m, in, ft) — the result stays in the same unit since this is pure geometric scaling.

Formula

a² + b² = c² c = √(a² + b²) b = √(c² − a²) Area = ½ a b ; Perimeter = a + b + c ∠A = atan(a / b) ; ∠B = 90° − ∠A

Frequently asked

When do I pick "solve for hypotenuse" vs "solve for leg"?

If you know the two legs (the sides next to the right angle), solve for the hypotenuse: c = √(a² + b²). If you know one leg and the hypotenuse (the longest side, opposite the right angle), solve for the missing leg: b = √(c² − a²). The hypotenuse is always the longest side.

Why does it complain when I enter leg = 5 and hypotenuse = 3?

The Pythagorean theorem requires c² ≥ a², i.e. the hypotenuse must be at least as long as either leg. If you enter a hypotenuse smaller than the leg, c² − a² becomes negative and there is no real square root — such a triangle cannot exist. Double-check which side is the hypotenuse (the longest one, opposite the right angle).

How are the two angles computed — do I need a trig table?

No table needed. The angle opposite leg a is ∠A = atan(a / b) (the inverse tangent), and ∠B = 90° − ∠A. The widget uses Math.atan2(a, b) so b = 0 doesn't divide-by-zero. A 3-4-5 triangle has angles 36.87° and 53.13°.

Does Pythagoras only apply to right triangles?

Yes. For non-right triangles you need the more general Law of Cosines: c² = a² + b² − 2ab·cos(C), where C is the angle opposite side c. When C = 90°, cos(C) = 0 and the formula collapses back to the Pythagorean theorem.

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