Pythagorean Theorem Explained: a² + b² = c² with Examples

The Pythagorean theorem is one of the most fundamental and widely used principles in mathematics. Named after the Greek mathematician Pythagoras (though known to Babylonian mathematicians centuries earlier), this theorem describes the relationship between the three sides of a right triangle.

The Formula

For any right triangle (a triangle with one 90-degree angle):

a² + b² = c²

Where:

• c = the length of the hypotenuse (the side opposite the right angle, always the longest side)
• a and b = the lengths of the other two sides (legs)

Why It Works

The theorem states that the area of the square built on the hypotenuse equals the sum of the areas of the squares built on the other two sides. This geometric interpretation helps visualize why the formula works.

Example 1: Finding the Hypotenuse

A right triangle has legs of length 3 and 4. Find the hypotenuse.

a² + b² = c²

3² + 4² = c²

9 + 16 = c²

25 = c²

c = 5

The hypotenuse is 5 units. This is the classic 3-4-5 triangle, one of the Pythagorean triples.

Example 2: Finding a Missing Leg

A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.

a² + b² = c²

5² + b² = 13²

25 + b² = 169

b² = 144

b = 12

Common Pythagorean Triples

These are sets of three integers that satisfy a² + b² = c²:

• 3, 4, 5
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25
• 9, 40, 41
• 6, 8, 10 (multiply 3-4-5 by 2)
• 9, 12, 15 (multiply 3-4-5 by 3)

Proofs of the Pythagorean Theorem

There are over 400 known proofs. The most intuitive is the rearrangement proof:

• Arrange four identical right triangles around a square
• The total area can be calculated two ways
• Equating them yields a² + b² = c²

Real-World Applications

Construction and Carpentry:

Builders use the 3-4-5 rule to ensure corners are perfectly square. Measure 3 feet along one wall, 4 feet along the other — if the diagonal is exactly 5 feet, the corner is square.

Navigation and GPS:

The theorem is used in GPS triangulation. A GPS receiver calculates its position by finding the intersection of spheres whose radii are determined by distance from multiple satellites.

Architecture and Engineering:

Structural engineers use the theorem to calculate diagonal bracing lengths, roof pitch, stair stringer lengths, and truss dimensions.

Computing and Graphics:

In computer graphics, the theorem calculates distances between pixels, collision detection, and rendering perspective.

Distance Formula:

The distance between any two points (x₁, y₁) and (x₂, y₂) is:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

This is the Pythagorean theorem applied to the coordinate plane.

FAQ: Pythagorean Theorem

No, it only works for right triangles (triangles with a 90-degree angle). For non-right triangles, use the Law of Cosines.

Use the formula a² = c² − b² or b² = c² − a², then take the square root.

A set of three positive integers (a, b, c) that satisfy a² + b² = c². The most common is (3, 4, 5).

The hypotenuse is always the side opposite the right angle and is always the longest side of the triangle.

Yes. For a rectangular box with dimensions a, b, c, the space diagonal d = √(a² + b² + c²).

The theorem was known to Babylonian mathematicians over 1,000 years before Pythagoras. However, Pythagoras is credited with the first recorded proof.