# Pythagorean Theorem Calculator

Calculate any missing side of a right triangle using the Pythagorean theorem. Find the hypotenuse or a leg given the other two sides.

## What this calculates

Find the missing side of any right triangle using the Pythagorean theorem (a² + b² = c²). Enter any two sides and this calculator solves for the third, along with the triangle's area and perimeter.

## Inputs

- **Solve For** — options: Hypotenuse (c), Side a, Side b — Select which side of the right triangle to calculate.
- **Side a** — One leg of the right triangle. Leave blank if solving for this side.
- **Side b** — The other leg of the right triangle. Leave blank if solving for this side.
- **Side c (Hypotenuse)** — The longest side, opposite the right angle. Leave blank if solving for this side.

## Outputs

- **Missing Side** — The calculated length of the missing side.
- **All Sides** — formatted as text — The lengths of all three sides.
- **Area of Triangle** — The area of the right triangle (1/2 x a x b).
- **Perimeter** — The total length of all three sides.
- **Verification (a² + b² = c²)** — formatted as text — Confirms the Pythagorean theorem holds for the calculated values.

## Details

The Pythagorean theorem is one of the most fundamental and widely-used theorems in all of mathematics. For any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.

The Formula: a² + b² = c²

Where a and b are the legs (the sides forming the right angle) and c is the hypotenuse (always the longest side).

Solving for Each Side

- Hypotenuse: c = sqrt(a² + b²). Example: If a = 3 and b = 4, then c = sqrt(9 + 16) = sqrt(25) = 5.

- Leg a: a = sqrt(c² - b²). Example: If c = 13 and b = 5, then a = sqrt(169 - 25) = sqrt(144) = 12.

- Leg b: b = sqrt(c² - a²). Same process as above, solving for the other leg.

Pythagorean Triples

Some right triangles have all integer side lengths. These are called Pythagorean triples. Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (20, 21, 29). Any multiple of a Pythagorean triple is also a triple: (6, 8, 10) is a multiple of (3, 4, 5).

Applications

The Pythagorean theorem is used in construction (ensuring right angles), navigation (distance calculations), computer graphics (pixel distances), physics (vector magnitudes), and countless other fields.

## Frequently Asked Questions

**Q: What is the Pythagorean theorem?**

A: The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the two legs (a and b): a² + b² = c². It only applies to right triangles (triangles with a 90-degree angle).

**Q: How do I find the hypotenuse of a right triangle?**

A: To find the hypotenuse, square each leg, add the squares together, and take the square root. For example, if the legs are 6 and 8: c = sqrt(6² + 8²) = sqrt(36 + 64) = sqrt(100) = 10.

**Q: How do I find a missing leg of a right triangle?**

A: If you know the hypotenuse (c) and one leg (say b), find the other leg with: a = sqrt(c² - b²). For example, if c = 10 and b = 6: a = sqrt(100 - 36) = sqrt(64) = 8. Note that the hypotenuse must always be longer than either leg.

**Q: What are Pythagorean triples?**

A: Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². The most well-known is (3, 4, 5). Others include (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any scalar multiple of a triple is also a triple, such as (6, 8, 10) = 2 x (3, 4, 5).

**Q: Can the Pythagorean theorem be used for non-right triangles?**

A: No, the Pythagorean theorem only applies to right triangles. For non-right triangles, use the Law of Cosines: c² = a² + b² - 2ab*cos(C), which generalizes the Pythagorean theorem. When angle C = 90 degrees, cos(90) = 0, and it reduces to the Pythagorean theorem.

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Source: https://vastcalc.com/calculators/math/pythagorean
Category: Math
Last updated: 2026-04-21