Pythagorean Theorem — Geometry Practice Worksheet

Grades 8–10 Medium difficulty 10 problems

The Pythagorean theorem states that in a right triangle with legs $a$ and $b$ and hypotenuse $c$ , $a^2 + b^2 = c^2$ . It applies only to right triangles — triangles containing a $90°$ angle.

The hypotenuse is always the longest side, opposite the right angle. When finding a leg, rearrange to $a^2 = c^2 - b^2$ . Always take the positive square root for side lengths.

The converse helps determine if a triangle is right: if $a^2 + b^2 = c^2$ , the triangle is a right triangle. This is useful in construction and navigation.

Skills practiced

Identifying hypotenuse and legs
Finding missing side lengths
Applying the theorem to word problems
Using the converse to test for right triangles

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Practice Worksheet: Pythagorean Theorem

Instructions: Solve each problem carefully. Show all work clearly. Write your final answer in the space provided or on a separate sheet as directed.

1.A right triangle has legs of lengths 6 cm and 8 cm. Find the length of the hypotenuse.
2.A 13-foot ladder is leaning against a wall. The base of the ladder is 5 feet from the wall. How high up the wall does the ladder reach?
3.Find the missing side length of a right triangle with a hypotenuse of 25 inches and one leg of 7 inches.
4.A rectangular field is 40 meters long and 30 meters wide. What is the distance from one corner to the opposite corner?
5.The legs of a right triangle are 9 cm and 12 cm. What is the perimeter of the triangle?
6.A television screen measures 32 inches diagonally and 24 inches in height. What is the width of the screen?
7.A right triangle has a hypotenuse of 17 feet and one leg of 15 feet. Find the length of the other leg.
8.A ramp rises 3 feet over a horizontal distance of 4 feet. What is the length of the ramp?
9.Two friends start at the same point. One walks 5 miles north, the other walks 12 miles east. How far apart are they?
10.A right triangle has legs of lengths $x$ and $x+1$ , and a hypotenuse of length 5. Find the value of $x$ .

Answer Key

1.

$c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm

Final answer: 10 cm
2.

$h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ feet

Final answer: 12 feet
3.

$b = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24$ inches

Final answer: 24 inches
4.

$d = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50$ meters

Final answer: 50 meters
5.

Hypotenuse $= \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$ cm. Perimeter $= 9 + 12 + 15 = 36$ cm

Final answer: 36 cm
6.

$w = \sqrt{32^2 - 24^2} = \sqrt{1024 - 576} = \sqrt{448} \approx 21.2$ inches

Final answer: 21.2 inches
7.

$b = \sqrt{17^2 - 15^2} = \sqrt{289 - 225} = \sqrt{64} = 8$ feet

Final answer: 8 feet
8.

$r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ feet

Final answer: 5 feet
9.

$d = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ miles

Final answer: 13 miles
10.

$x^2 + (x+1)^2 = 5^2$ ; $x^2 + x^2 + 2x + 1 = 25$ ; $2x^2 + 2x - 24 = 0$ ; $x^2 + x - 12 = 0$ ; $(x+4)(x-3)=0$ ; $x=3$ (positive solution)

Final answer: 3

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Common mistakes to avoid

Adding legs when the hypotenuse is needed (should subtract)
Forgetting the square root at the end
Applying the theorem to non-right triangles
Using the wrong side as the hypotenuse

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Last updated: 2026

Pythagorean Theorem — Geometry Practice Worksheet

Skills practiced

Answer Key

Common mistakes to avoid

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