The Pythagorean Theorem for Kids

The Pythagorean theorem is a rule about triangles that have a square corner, called right triangles. If you build a square on each side, the two smaller squares together hold exactly as many tiles as the biggest square. In math language that is a² + b² = c². For a triangle with sides 3, 4 and 5: 9 + 16 = 25.

First, what is a right triangle?

A right angle is a square corner, like the corner of a book, a door frame or a sheet of paper. It measures exactly 90 degrees (90°). A right triangle is a triangle with one of these square corners. In drawings, a tiny square in the corner tells you it is a right angle.

The sides of a right triangle have names:

• The two sides that make the square corner are the legs. We call them a and b.
• The side across from the square corner is the hypotenuse (say "hy-POT-en-yoos"). We call it c. It is always the longest side.

The rule only works for right triangles. If a triangle has no square corner, the rule doesn't apply.

What "squared" means

To square a number, multiply it by itself. We write a small 2 up high: 3² means 3 × 3 = 9. It is called "squared" because a square that is 3 tiles wide and 3 tiles tall has 9 tiles in it.

Be careful: squaring is not times 2. 3² is 9, not 6. Here are the first few squares:

The 3-4-5 tile-counting activity

You can see the rule for yourself with grid paper. You need grid paper, a pencil, colored pencils and scissors.

1. Draw the triangle. Draw a line 3 squares tall going up, and a line 4 squares long going across from the bottom of it. Join their ends with a slanted line. That is a right triangle with legs 3 and 4.
2. Build a square on each leg. On the 3 side, draw a 3 × 3 square and color it blue. On the 4 side, draw a 4 × 4 square and color it orange.
3. Count the tiles. The blue square has 9 tiles and the orange square has 16. Together that is 25 tiles.
4. Measure the slanted side. Use the edge of another sheet of grid paper as a ruler. The slanted side is exactly 5 squares long.
5. Build the big square. A 5 × 5 square has 25 tiles. Cut out your 9 blue and 16 orange tiles. They fill the big square exactly, with none missing and none left over.

Here is the same idea drawn on the triangle itself. Each square sits on one side.

This works for every right triangle, not just this one. Mathematicians have proved it in hundreds of different ways. You can see some of them, with moving pictures, on the proofs page.

The shortcut across the park

Maya walks from home to school. The streets go around a rectangular park: 300 meters up one side, then 400 meters along the top. That's 700 meters in total.

One day she cuts straight across the park instead. How far is that? The two streets meet at a square corner, so her shortcut is the hypotenuse of a right triangle.

The shortcut is 500 meters, so Maya saves 200 meters. Going straight across is always shorter than going around the corner. That's true for every right triangle, and it's a good way to check your answers: the hypotenuse is longer than each leg but shorter than the two legs added together.

How to use the rule in three steps

When you know the two legs and want the hypotenuse, do this:

1. Square each leg. For legs 6 and 8: 6² = 36 and 8² = 64.
2. Add the squares. 36 + 64 = 100. That is how many tiles are in the big square.
3. Find the square root. Ask: "Which number times itself makes 100?" The answer is 10, so the hypotenuse is 10. This step is called taking the square root, written √100 = 10. A calculator's √ button does it for you.

Not every answer is a whole number. A triangle with legs 1 and 1 has a hypotenuse of √2, which is about 1.41. That's fine; it is still a real length you can draw.

What if you know the hypotenuse and one leg? Then you subtract instead of adding. A 5-meter ladder leans on a wall with its foot 3 meters from the wall. The ladder is the hypotenuse, so the height is √5² − 3² = √25 − 9 = √16 = 4 meters. Older students can learn more on finding a missing side.

Real-life examples: screens and pizza boxes

How big is that screen?

TV, tablet and phone screens are measured from one corner to the opposite corner, along the diagonal. A tablet screen that is 8 inches wide and 6 inches tall has a diagonal of √64 + 36 = √100 = 10 inches, so it is called a 10-inch screen. Try the screen size calculator with your own TV.

Will the breadstick fit?

A square pizza box is 30 cm on each side. You have a breadstick that is 40 cm long. It won't fit along a side, because 40 is more than 30. But will it fit if you lay it from corner to corner?

The diagonal is about 42.4 cm, so the 40 cm breadstick fits, with a little room to spare.

Five easy practice questions

Try each one before you open the answer. Use a calculator for square roots if you like.

Want more? Try the printable worksheets or the quiz, and check answers with the Pythagorean theorem calculator.

Tips for parents and teachers

• Start with tiles, not the formula. Counting squares on grid paper, or laying out square crackers or sticky notes, makes a² mean "a square of tiles" before it means anything else. Many children first mix up 3² with 3 × 2, and tiles fix that.
• Make a 3-4-5 corner. Mark a string at 12 equal spaces and tie it into a loop. Stretch it into a triangle with sides of 3, 4 and 5 spaces, and the corner between the 3 and the 4 is a right angle. Builders still use 3-4-5 measurements to check square corners. More hands-on ideas are on the classroom activities page.
• Measure real things. Measure the width and height of a TV, phone or picture frame, predict the diagonal, then check it with a tape measure. Small differences come from rounding and from the frame around the screen.
• Let them use a calculator for square roots. The big idea is "square, add, square root". Finding √169 by hand is a separate skill and can come later.
• Know where it fits. In the United States the theorem is part of the 8th-grade Common Core standards (see the 8th grade standards page), but younger children can enjoy the tile version as soon as they can multiply.
• Watch for two classic mix-ups: adding the sides instead of their squares (3 + 4 is not 5) and forgetting the square root at the end (25 is the tiles, 5 is the side).

For a slightly more grown-up explanation of the same ideas, read what is the Pythagorean theorem?