Study Guide: Understanding the Pythagorean Theorem

The Pythagorean Theorem is a rule used in math to find the length of a side in a right triangle. It connects the lengths of all three sides in a simple equation. In this guide, you will learn what it means, how it works, and how to practice it with confidence.

Type of Triangle
It only works for right triangles.
Main Formula
a² + b² = c²
Longest Side
The longest side is called the hypotenuse.
Common Use
It helps find missing side lengths.

What the Pythagorean Theorem Is

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The Pythagorean Theorem is a mathematical rule that applies only to right triangles. A right triangle has one angle that measures 90 degrees.

The theorem states that the square of one leg plus the square of the other leg equals the square of the hypotenuse.

  • a and b are the shorter sides, called legs.
  • c is the longest side, called the hypotenuse.
  • The equation is written as a² + b² = c².

How It Works and What to Expect

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When solving a problem, you are usually given two side lengths and asked to find the third.

You square the known side lengths, add them if they are the legs, and then take the square root to find the missing side.

If the hypotenuse is missing, you add the squares of the legs. If a leg is missing, you subtract one squared value from the other before taking the square root.

  • Always check that the triangle is a right triangle.
  • Make sure the hypotenuse is the longest side.
  • Use careful arithmetic when squaring and finding square roots.

How to Study Effectively

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Understanding the formula is important, but practice helps you use it correctly.

Work through different examples where the missing side changes. This helps you learn when to add and when to subtract.

  • Write the formula before starting each problem.
  • Label the sides clearly as a, b, and c.
  • Check your answer to see if it makes sense for the size of the triangle.

Flashcards can help you remember the formula and key terms, but solving full problems builds stronger understanding.

Frequently Asked Questions

Does the Pythagorean Theorem work for all triangles?

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No. It only works for right triangles, which have one 90-degree angle.

How do I know which side is the hypotenuse?

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The hypotenuse is always opposite the 90-degree angle and is the longest side of the triangle.

What if my answer is a decimal?

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It is normal to get decimals when taking square roots. Round your answer only if your teacher or test instructions say to round.

Why do I need to square the numbers?

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Squaring the numbers follows the rule of the theorem. The relationship between the sides is based on their squared lengths.