Pythagorean theorem worksheet

Grade 8 math, Common Core 8.G.B.7. 10 problems with an answer key and worked steps. Free to print, and you can make new problems any time.

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Pythagorean theorem

Grade 8 math, Common Core 8.G.B.7

Use the Pythagorean theorem. Round to the nearest tenth if needed.

1.
A 5-foot ladder leans against a wall. Its foot is 3 feet from the wall. How high up the wall does it reach?
2.
The legs of a right triangle are 7 ft and 5 ft. How long is the hypotenuse?
3.
Is a triangle with sides 6, 8 and 10 a right triangle?
4.
The legs of a right triangle are 12 cm and 16 cm. How long is the hypotenuse?
5.
The legs of a right triangle are 5 cm and 12 cm. How long is the hypotenuse?
6.
The legs of a right triangle are 7 cm and 8 cm. How long is the hypotenuse?
7.
A 10-foot ladder leans against a wall. Its foot is 6 feet from the wall. How high up the wall does it reach?
8.
A right triangle has a hypotenuse of 50 ft and a leg of 14 ft. How long is the other leg?
9.
The legs of a right triangle are 4 cm and 8 cm. How long is the hypotenuse?
10.
The legs of a right triangle are 3 m and 4 m. How long is the hypotenuse?

Answer key: Pythagorean theorem

Grade 8 math. Each answer has a few worked steps, to help check the method as well as the answer.

1. 4 feet
5² − 3² = 25 − 9 = 16√16 = 4
2. √74 ≈ 8.6 ft
7² + 5² = 74√74 ≈ 8.6
3. Yes. 6² + 8² = 100 = 10².
4. 20 cm
12² + 16² = 144 + 256 = 400√400 = 20
5. 13 cm
5² + 12² = 25 + 144 = 169√169 = 13
6. √113 ≈ 10.6 cm
7² + 8² = 113√113 ≈ 10.6
7. 8 feet
10² − 6² = 100 − 36 = 64√64 = 8
8. 48 ft
50² − 14² = 2,500 − 196 = 2,304√2,304 = 48
9. √80 ≈ 8.9 cm
4² + 8² = 80√80 ≈ 8.9
10. 5 m
3² + 4² = 9 + 16 = 25√25 = 5

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About pythagorean theorem

In a right triangle, the longest side (the hypotenuse) is opposite the right angle, and a² + b² = c², where a and b are the two legs and c is the hypotenuse. Use it to find a missing side, or to check whether three lengths make a right triangle.

Worked example 1

A right triangle has a hypotenuse of 30 m and a leg of 18 m. How long is the other leg?

Step 1.30² − 18² = 900 − 324 = 576
Step 2.√576 = 24

Answer: 24 m

Worked example 2

The legs of a right triangle are 9 m and 40 m. How long is the hypotenuse?

Step 1.9² + 40² = 81 + 1,600 = 1,681
Step 2.√1,681 = 41

Answer: 41 m

Worked example 3

A 20-foot ladder leans against a wall. Its foot is 12 feet from the wall. How high up the wall does it reach?

Step 1.20² − 12² = 400 − 144 = 256
Step 2.√256 = 16

Answer: 16 feet

Watch out for

c is always the longest side, across from the right angle.
Square first, then add (or subtract), then take the square root.
To find a leg, subtract: a² = c² − b².

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