Pythagorean theorem: learn and practice

Who this lesson is for

This Pythagorean theorem lesson is for grade 8. Students find the hypotenuse of a right triangle from the two shorter sides. The catalog skill also appears in grade 9, and this site's grade hubs stop at grade 8, so only the grade 8 hub links here. The practice link opens that skill.

The right angle decides the hypotenuse

The theorem applies to a right triangle, a triangle with one 90° angle. The hypotenuse is the side opposite that right angle, and it is the longest side. If the shorter sides are a and b, and the hypotenuse is c, then a² + b² = c². You square the two shorter sides, add those squares, and the sum is the square of the hypotenuse. The side lengths themselves do not add to the hypotenuse.

A known triple checks the arithmetic

Some whole-number triples satisfy the theorem exactly. 3² + 4² = 9 + 16 = 25 = 5². 5² + 12² = 25 + 144 = 169 = 13². Doubling a triple keeps it a triple: 6, 8, and 10 is double 3, 4, and 5. These are the cases the catalog draws from. If your squares do not add to another square in the list, recheck which side is the hypotenuse.

Worked example 1: sides 3 and 4

The two shorter sides are 3 and 4. Square them: 3² = 9 and 4² = 16. Add: 9 + 16 = 25. The hypotenuse is the number whose square is 25, which is 5, because 5² = 25. So the hypotenuse is 5. Adding the sides, 3 + 4 = 7, is not the theorem. The triangle inequality would even allow 7, and the right angle forces 5 instead. Check: 9 + 16 = 25.

Worked example 2: sides 6 and 8

The two shorter sides are 6 and 8. This is double the 3-4-5 triangle, so the hypotenuse should be double 5, which is 10. The squares confirm it: 6² = 36 and 8² = 64. 36 + 64 = 100, and 10² = 100. The hypotenuse is 10. Scaling every side by the same number keeps the right angle and the theorem. You do not scale only one side.

Worked example 3: sides 5 and 12

The two shorter sides are 5 and 12. Square them: 5² = 25 and 12² = 144. Add: 25 + 144 = 169. The number whose square is 169 is 13, because 13 × 13 = 169. The hypotenuse is 13. A nearby miss is 17, from adding 5 + 12. That sum is longer than the straight-line hypotenuse this right angle produces. The catalog figure marks the unknown side opposite the right angle.

Checks before you leave the problem

Square the two shorter sides, add, and ask which whole number squares to that sum. Do not add the side lengths. The hypotenuse is the side opposite the right angle. Then practice the skill on the board.

Common questions about the Pythagorean theorem

Does 3 + 4 equal the hypotenuse?

No. 3 + 4 = 7, and the hypotenuse of a 3-4 right triangle is 5. The theorem adds the squares of the sides, not the sides.

Which side is the hypotenuse?

The side opposite the right angle. It is the longest side. The two sides that form the right angle are the ones you square and add.

Why is there no grade 9 hub for this lesson?

The practice catalog continues the skill into grade 9. The published hubs on this site run through grade 8, so the lesson keeps a single grade 8 address instead of a second copy.