Pythagorean Theorem Explained Step-by-Step

Learn how the Pythagorean theorem works, how to solve right triangles, and how this important geometry formula is used.

The Pythagorean theorem is one of the most important ideas in geometry. It helps students solve right triangles and appears throughout algebra, trigonometry, calculus, physics, engineering, and computer science.

What is the Pythagorean theorem?

The Pythagorean theorem describes the relationship between the sides of a right triangle.

\[ a^2 + b^2 = c^2 \]

Here:

\(a\) and \(b\) are the legs of the triangle
\(c\) is the hypotenuse

What is the hypotenuse?

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

Example: Finding the hypotenuse

Suppose a right triangle has legs of length \(3\) and \(4\). Find the hypotenuse.

Step 1: Write the formula.

\[ a^2 + b^2 = c^2 \]

Step 2: Substitute the values.

\[ 3^2 + 4^2 = c^2 \]

Step 3: Simplify.

\[ 9 + 16 = c^2 \] \[ 25 = c^2 \]

Step 4: Take the square root.

\[ c = 5 \]

Example: Finding a missing leg

Suppose the hypotenuse is \(13\) and one leg is \(5\). Find the other leg.

\[ 5^2 + b^2 = 13^2 \]

\[ 25 + b^2 = 169 \]

\[ b^2 = 144 \]

\[ b = 12 \]

Common Pythagorean triples

Some right triangles appear frequently and are useful to memorize.

\(3,4,5\)
\(5,12,13\)
\(8,15,17\)

Applications of the Pythagorean theorem

The Pythagorean theorem appears throughout mathematics and science.

Finding distances
Coordinate geometry
Construction and architecture
Physics and engineering
Computer graphics

Connection to the distance formula

The distance formula in coordinate geometry comes directly from the Pythagorean theorem.

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

Common mistakes students make

Using the theorem on non-right triangles
Mixing up the hypotenuse and legs
Forgetting to square values
Making arithmetic mistakes
Forgetting the square root at the end

Need help with geometry?

Geometry often combines algebra, diagrams, formulas, and visual reasoning all at once. Working through examples carefully step-by-step can make concepts much easier to understand.

I provide online geometry tutoring for students learning triangles, circles, proofs, coordinate geometry, and related topics.