BuilderGrade 8⏱ 20 min

The Pythagorean theorem

Drag a right triangle, watch the squares on its sides, and see a proof that needs no algebra.

Standards: 8.G.B.68.G.B.7

1Wonder

A ladder, a ramp, the diagonal of your phone screen. If you know two sides of a right triangle, can you find the third without measuring it?

People have known a beautiful trick for over 2,500 years. Let’s discover it with squares.

Squares on a triangle

This activity needs a bigger screen

Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.

✓ Right angle: the squares fit exactly!

Leg a 3 Leg b 4 Angle at C 90 °

Whole numbers Show unit squares

Try this

Make legs 3 and 4. Add up the green and blue areas. Compare them with the purple area.
Try 6 and 8, then 5 and 12. Does the pattern still hold?
Change the angle at C to 80° or 100°. What happens to the equation?
Turn off whole numbers and drag freely. Is the purple area always the sum of the other two when the angle is 90°?

Step 1 of 5

Take a right triangle with legs a and b and the long side (the hypotenuse) c. Make four identical copies.

The Pythagorean theorem

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

\htmlClass{c-green}{a^2} + \htmlClass{c-blue}{b^2} = \htmlClass{c-purple}{c^2}

It also works backwards (the converse): if $a^2 + b^2 = c^2$ , then the triangle has a right angle.

Find the hypotenuse. Legs 6 and 8: $c^2 = 36 + 64 = 100$ , so $c = \sqrt{100} = 10$ .

Find a leg. Hypotenuse 13, one leg 5: $b^2 = 13^2 - 5^2 = 169 - 25 = 144$ , so $b = 12$ .

Question 1 of 5

The legs are 3 and 4. How long is the hypotenuse c?

In a right triangle, the squares on the two legs together have the same area as the square on the hypotenuse.
$a^2 + b^2 = c^2$ . The hypotenuse $c$ is opposite the right angle and is the longest side.
Find a missing side: $c = \sqrt{a^2 + b^2}$ or $b = \sqrt{c^2 - a^2}$ .
If the angle is smaller than 90°, then $c^2 < a^2 + b^2$ ; if it is bigger, then $c^2 > a^2 + b^2$ .
Famous whole-number triples: 3-4-5, 5-12-13, 6-8-10, 8-15-17.