What is the Pythagorean theorem?

For any right triangle, a² + b² = c², where a and b are the legs (the two sides forming the right angle) and c is the hypotenuse (the longest side, opposite the right angle).

How do I find the hypotenuse of a right triangle?

Square both legs, add them, then take the square root: c = √(a² + b²). For example, with legs 6 and 8: c = √(36 + 64) = √100 = 10.

How do I find a missing leg?

Rearrange the theorem: a = √(c² − b²). The hypotenuse c must be greater than the known leg b — otherwise no real triangle exists.

What is a Pythagorean triple?

Three positive integers (a, b, c) that satisfy a² + b² = c². The smallest is 3-4-5. Others include 5-12-13, 8-15-17, 7-24-25. Any integer multiple of a triple is also a triple.

Can the Pythagorean theorem be used for distance?

Yes. The 2D distance formula d = √((x₂−x₁)² + (y₂−y₁)²) is a direct application: the horizontal and vertical differences form the two legs of a right triangle whose hypotenuse is the straight-line distance.

How do I verify a triangle is a right triangle?

Check whether the square of the longest side equals the sum of squares of the other two. If a² + b² = c² exactly, it's a right triangle; if a² + b² > c², it's acute; if a² + b² < c², it's obtuse.

Does the theorem work in 3D?

Yes — the space diagonal of a rectangular box with edges x, y, z is d = √(x² + y² + z²), obtained by applying the theorem twice.

Pythagorean Theorem Calculator

The Pythagorean theorem is the cornerstone of right-triangle geometry: for any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs (a² + b² = c²). This calculator solves for any missing side — hypotenuse or either leg — and also doubles as a 2D distance calculator using the same identity. It shows full step-by-step algebra, computes triangle area (½ab), perimeter, the acute angles via arctangent, and flags exact Pythagorean triples like 3-4-5, 5-12-13, 8-15-17, 7-24-25, and 9-40-41. Useful for homework, carpentry and framing, squaring a foundation, navigation, physics vector magnitudes, screen-diagonal sizing, and any engineering task that reduces to a right triangle.

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calendar_today Updated: April 2026

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calculate a² + b² = c²

What do you want to find?

Leg a

Leg b

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Hypotenuse c

Area

Perimeter

Angle A

36.87°

Angle B

53.13°

Pythagorean triple: Yes

Step-by-step

c = sqrt(3^2 + 4^2) = sqrt(25) = 5

tips_and_updates Tips

• The hypotenuse is always the longest side — it's opposite the 90° angle.
• Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, 20-21-29.
• Any multiple of a triple is also a triple (e.g., 6-8-10, 9-12-15).
• To verify a right angle on-site, measure 3 ft, 4 ft, and 5 ft — if the diagonal is exactly 5, the corner is square.
• The theorem extends to 3D: d = √(x² + y² + z²) for diagonals of a box.
• Distance between two points (x₁,y₁) and (x₂,y₂): d = √((x₂−x₁)² + (y₂−y₁)²).

The Formula

Rearrange to solve for the unknown: c = √(a² + b²), a = √(c² − b²), or b = √(c² − a²). When solving for a leg, the hypotenuse must be larger than the known leg.

a² + b² = c²

lightbulb Variables Explained

a, b The two legs (shorter sides) of the right triangle
c The hypotenuse — the side opposite the right angle

tips_and_updates Pro Tips

The hypotenuse is always the longest side — it's opposite the 90° angle.

Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, 20-21-29.

Any multiple of a triple is also a triple (e.g., 6-8-10, 9-12-15).

To verify a right angle on-site, measure 3 ft, 4 ft, and 5 ft — if the diagonal is exactly 5, the corner is square.

The theorem extends to 3D: d = √(x² + y² + z²) for diagonals of a box.

Distance between two points (x₁,y₁) and (x₂,y₂): d = √((x₂−x₁)² + (y₂−y₁)²).