Pythagorean Theorem Calculator

Math

Solve for the hypotenuse (c) or any leg (a, b) using a² + b² = c². Includes exact radical simplification, decimal approximations, step-by-step working, and Pythagorean triple verification.

Presets:

Solved Hypotenuse (c) Pythagorean Triple ✓

c = 5 ≈ 5.000000

Pythagorean Squares & Measurements

a² 9.00 (a = 3.00)

b² 16.00 (b = 4.00)

c² 25.00 (c = 5.00)

Area: 6 Perimeter: 12

Algebraic Solution Steps

1 1. State the Pythagorean Theorem

c² = a² + b²

The square of the hypotenuse is equal to the sum of the squares of the legs.

2 2. Substitute Known Leg Values

c² = (3)² + (4)² = 9 + 16 = 25

Square each leg length and sum the results.

3 3. Take the Square Root

c = √25 ≈ 5.000000

Take the positive principal square root to obtain hypotenuse length.

How This Calculator Works

The Pythagorean Theorem states that in any right-angled triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two legs.

📐 The Pythagorean Equation

a² + b² = c² ⟹ c = √(a² + b²), a = √(c² - b²), b = √(c² - a²)

a, b: Legs of the right triangle (the sides forming the 90° angle)

c: Hypotenuse (the longest side opposite the 90° angle)

Pythagorean Triple: A set of three positive integers (a, b, c) that satisfy a² + b² = c² (e.g. 3-4-5, 5-12-13, 8-15-17)

Primitive Triple: A triple where gcd(a, b, c) = 1 (coprime integers)

Key Mathematical Rules & Properties

Identify Hypotenuse vs Legs

The hypotenuse is always the longest side c facing the 90° angle.

Simplify Square Roots

Factor out the largest perfect square from the radicand (e.g. √50 = √(25 × 2) = 5√2).

Understanding your calculation

Method, example and limitations

How the calculation works

For a right triangle, the hypotenuse c satisfies c² = a² + b², where a and b are the perpendicular legs.

Example to check

Legs 6 and 8 give hypotenuse √(36 + 64) = 10.

Before using the result

This relationship applies to right triangles in Euclidean geometry. The hypotenuse must be longer than either leg.

Calculation methodology · Report a result that needs review