Pythagorean Theorem

Learn the famous relationship between the sides of a right triangle and use it to solve real-world problems.

Intermediate25 minLesson
Printable worksheet

Definition

The Pythagorean Theorem describes the relationship between the three sides of a right triangle:
Where:
  • and are the lengths of the two legs (the sides that form the right angle)
  • is the length of the hypotenuse (the side opposite the right angle)
The hypotenuse is always the longest side of a right triangle.
Key insight: The sum of the squares of the two shorter sides equals the square of the longest side.

Try it now

In a right triangle, which side is always the longest?

Worked Examples

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

1

Write the Pythagorean theorem

→ Formula ready

2

Substitute the known values

→ Values plugged in

3

Calculate the squares

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4

Take the square root

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Common Mistakes

Using the formula on non-right triangles

Why it's wrong: The Pythagorean theorem ONLY works for right triangles (triangles with a 90° angle).

Correct: Always check that the triangle has a right angle before applying .

Putting the hypotenuse in the wrong position

Why it's wrong: Students sometimes add the hypotenuse with a leg: . This gives wrong answers.

Correct: The hypotenuse () is ALWAYS alone on one side: . The hypotenuse is opposite the right angle.

Forgetting to take the square root

Why it's wrong: After finding , students write instead of .

Correct: Always take the square root as the final step: if , then .

Adding instead of squaring first

Why it's wrong: Writing then squaring, instead of .

Correct: Square EACH leg individually first: , not .

Interactive Visual

Triangle Explorer

a = 3b = 4c = 5.00

a² + b² = c²

3² + 4² = 5.00²

9 + 16 = 25.00

Change the leg lengths to see the Pythagorean theorem in action.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy

In a right triangle, which side is always the longest?

Why It Matters

The Pythagorean theorem is one of the most useful tools in mathematics:
  • Construction: Builders use it to ensure corners are perfectly square
  • Navigation: Calculate the shortest distance between two points
  • Sports: Determine diagonal distances on fields and courts
  • Architecture: Design stable structures with precise measurements
  • Video games: Calculate distances and movements in 2D and 3D spaces
This 2,500-year-old theorem is used daily by engineers, architects, and scientists worldwide!

Real World Applications

Construction and Carpentry

Builders use the 3-4-5 rule to check if corners are perfectly square (90°).

Example:

Measure 3 feet along one wall, 4 feet along the other. If the diagonal is exactly 5 feet, the corner is a perfect right angle.

1Try It Yourself

A carpenter is building a rectangular frame. One side is 8 feet, another is 15 feet.

How long should the diagonal brace be?

Step 1: Write the mathematical expression

Use the Pythagorean theorem:

Navigation and GPS

Finding the shortest (straight-line) distance between two locations.

Example:

If you walk 400 meters east and then 300 meters north, the direct distance back is meters.

2Try It Yourself

A ship sails 12 km south, then 9 km west.

How far is the ship from its starting point?

Step 1: Write the mathematical expression

The path forms a right triangle.

Sports Field Measurements

Calculate diagonal distances across rectangular fields and courts.

Example:

A soccer field is 100 meters long and 70 meters wide. The diagonal distance is meters.

3Try It Yourself

A rectangular basketball court is 28 meters long and 15 meters wide.

A player runs diagonally from one corner to the opposite corner. How far does she run? (Round to the nearest whole number)

Step 1: Write the mathematical expression

Calculate the diagonal using the Pythagorean theorem.

Key Takeaways

  • 1The Pythagorean theorem states that in a right triangle,
  • 2The legs ( and ) are the two sides that form the right angle
  • 3The hypotenuse () is the longest side, opposite the right angle
  • 4Use this theorem to find any missing side when you know the other two
  • 5Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25

Frequently Asked Questions

Pythagoras was a Greek mathematician who lived around 500 BCE. While the theorem bears his name, similar knowledge existed in Babylon and India even earlier. Pythagoras (or his school) is credited with the first formal proof.
Pythagoras was a Greek mathematician who lived around 500 BCE. While the theorem bears his name, similar knowledge existed in Babylon and India even earlier. Pythagoras (or his school) is credited with the first formal proof.
A Pythagorean triple is a set of three whole numbers that satisfy the theorem. Examples: 3-4-5, 5-12-13, 8-15-17. Any multiple of a triple also works (6-8-10, 9-12-15, etc.).
Yes! If the sides are and , the hypotenuse is . Not all right triangles have whole-number sides.
Yes! If three sides satisfy , the triangle MUST be a right triangle. This is called the converse of the Pythagorean theorem.

Glossary

Right triangle
A triangle with one 90° angle
Hypotenuse
The longest side of a right triangle, opposite the right angle
Leg
Either of the two shorter sides of a right triangle that form the right angle
Pythagorean triple
A set of three positive integers , , such that
Square root
A number that, when multiplied by itself, gives the original number. because

Formula Card

Pythagorean Theorem

Where $a$ and $b$ are the legs and $c$ is the hypotenuse

Finding the Hypotenuse

Use when you know both legs and need to find the hypotenuse

Finding a Leg

Use when you know the hypotenuse and one leg

More in This Topic

Types of Triangles

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Triangle Angle Sum

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