Lesson: The Pythagorean Theorem (Grade 8)

In a right triangle: small square + small square = big square.

The right triangle

A right triangle has one square corner (90°), like the corner of a book.

• The two sides that make the square corner are the legs. We call them a and b.
• The longest side, across from the square corner, is the hypotenuse. We call it c.

See the squares

Draw a square on each side of a triangle with sides 3, 4 and 5.

• Square on side 3: 3 × 3 = 9
• Square on side 4: 4 × 4 = 16
• Square on side 5: 5 × 5 = 25

Now look: 9 + 16 = 25. The two small squares fill the big square exactly!

The rule

This works for every right triangle:

a² + b² = c²

(a² means a × a.)

Find the long side

A right triangle has legs 6 and 8. How long is the hypotenuse?

1. Square the legs: 6² = 36 and 8² = 64
2. Add them: 36 + 64 = 100
3. Which number times itself is 100? 10

c = 10

Find a missing leg

The hypotenuse is 13 and one leg is 5. Find the other leg.

1. Square both: 13² = 169 and 5² = 25
2. Subtract (the big square minus the small one): 169 − 25 = 144
3. Which number times itself is 144? 12

b = 12

Real life: the ladder

A ladder leans on a wall. Its foot is 3 m from the wall and it reaches 4 m up. How long is the ladder? It's a 3, 4, 5 triangle, so the ladder is 5 m.

Practice

Try the Pythagorean Theorem (Intro) worksheets. Then try Missing Legs & Word Problems.