CalcRover

Pythagorean Theorem Calculator

Find the missing side of a right triangle: both legs give the hypotenuse, or a leg plus the hypotenuse gives the other leg. The squaring, subtracting and square-root steps are written out the way homework wants them.

The Pythagorean theorem

In any right triangle, the legs a and b and the hypotenuse c (the side opposite the right angle, always the longest) satisfy a² + b² = c². Solving for a leg just rearranges: a² = c² − b² — which is why the hypotenuse must be longer than either leg.

The theorem only holds for right triangles — and that's also its builder's use in reverse: if the sides satisfy the equation, the corner is square. The 3-4-5 triangle (and its multiples 6-8-10, 9-12-15) is the classic site check.

c = √(a² + b²)
a = √(c² − b²)

Worked examples

Example 1: the 3-4-5 triangle

Legs 3 and 4 — find the hypotenuse.

  1. Square and add.
    3² + 4² = 9 + 16 = 25
  2. Take the root.
    c = √25 = 5

c = 5 — the oldest right angle check in construction.

Example 2: how high does the ladder reach?

A 20 ft ladder, base 6 ft from the wall. Height on the wall?

  1. The ladder is the hypotenuse; solve for a leg.
    √(20² − 6²) = √(400 − 36)
  2. Evaluate.
    √364 ≈ 19.1 ft

About 19.1 feet up the wall.

Example 3: TV screen size

A "65-inch" TV is its diagonal. With a 16:9 shape, how wide is it?

  1. Sides are 16k and 9k; diagonal fixes k.
    √(16² + 9²)k = 65 → 18.36k = 65 → k = 3.54
  2. Width is 16k.
    16 × 3.54 ≈ 56.7 in

About 56.7 inches wide (and 31.9 tall) — check the cabinet before buying.

FAQ

Which side is the hypotenuse?+

The side opposite the right angle — always the longest. If your "hypotenuse" comes out shorter than a leg, the labels are swapped (the calculator refuses that input and says why).

Does it work for non-right triangles?+

No — general triangles need the law of cosines: c² = a² + b² − 2ab·cos(C). The Pythagorean theorem is its special case at C = 90°.

What are Pythagorean triples?+

Whole-number side sets: (3,4,5), (5,12,13), (8,15,17), (7,24,25) and all their multiples. Builders use them for square corners without measuring angles.

How do I check a deck or wall is square with this?+

Mark 3 ft on one edge, 4 ft on the other; the diagonal between the marks should be exactly 5 ft (or use 6-8-10 for more accuracy). Off by much — the corner isn't 90°.

Related calculators