Pythagorean Theorem: Formula, Proof, and Applications

2026.04.24

by QANDA

The Pythagorean theorem describes the relationship between the three sides of a right triangle. It's one of the most famous results in all of mathematics and serves as the foundation for geometry, trigonometry, and coordinate math.

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The Pythagorean theorem is the backbone of nearly every geometry problem involving distances, heights, or diagonals. One formula unlocks them all.

Understanding the Pythagorean Theorem

The Formula

In a right triangle with legs $a$ and $b$ and hypotenuse $c$ :

$a^2 + b^2 = c^2$

The hypotenuse is always the longest side — the one opposite the right angle.

Key Terminology

Term	Description	Label
Legs	The two sides forming the right angle	$a$ , $b$
Hypotenuse	The side opposite the right angle	$c$ (always the longest)

Proof of the Pythagorean Theorem

Step 1: Area-Based Proof (Euclid's Method)

Arrange four identical right triangles inside a square with side length $(a + b)$ :

Area of large square = 4 triangles + inner square

$(a + b)^2 = 4 \times \frac{1}{2}ab + c^2$

Step 2: Expand and Simplify

$a^2 + 2ab + b^2 = 2ab + c^2$

Subtract $2ab$ from both sides:

$a^2 + b^2 = c^2$

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The Pythagorean theorem applies only to right triangles. For acute or obtuse triangles, use the law of cosines instead. Always verify the right angle first!

Applications

Distance Between Two Points

The distance between points $A(x_1, y_1)$ and $B(x_2, y_2)$ on a coordinate plane:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This formula is derived directly from the Pythagorean theorem.

Special Right Triangles

Name	Side Ratios	Angles
Pythagorean triple 3-4-5	$3 : 4 : 5$	Right triangle
Isosceles right triangle	$1 : 1 : \sqrt{2}$	$45°$ - $45°$ - $90°$
Half equilateral triangle	$1 : \sqrt{3} : 2$	$30°$ - $60°$ - $90°$

Practice Problems

Example 1: Finding the Hypotenuse

Problem: A right triangle has legs of 6 cm and 8 cm. Find the hypotenuse.

Show Solution

Apply the Pythagorean theorem:

$c^2 = 6^2 + 8^2 = 36 + 64 = 100$

$c = \sqrt{100} = 10$

Answer: $10$ cm

Example 2: Identifying a Right Triangle

Problem: Determine whether a triangle with sides 5, 12, and 13 is a right triangle.

Show Solution

Check if $a^2 + b^2 = c^2$ using the longest side (13) as $c$ :

$5^2 + 12^2 = 25 + 144 = 169 = 13^2$

Since $a^2 + b^2 = c^2$ , it is a right triangle.

Answer: Yes, it is a right triangle.

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Memorize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples (6-8-10, 9-12-15) to quickly identify right triangles.

Top 3 Common Mistakes

Putting the hypotenuse on the wrong side — $c$ is always the longest side (opposite the right angle). Identify it first before substituting.
Forgetting to take the square root — After finding $c^2 = 100$ , the answer is $c = 10$ , not $100$ . And since length is positive, $c = -10$ is not valid.
Applying to non-right triangles — The theorem only works for right triangles. Always confirm the right angle exists in the problem.

Related Concepts

Trigonometry — trig ratios
Vectors — coordinate geometry
Quadratic Equations — algebraic skills

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