Pythagorean Theorem Practice Worksheet

Practice finding missing side lengths in right triangles using the Pythagorean Theorem.

Try each problem on your own first. Then click Show solution to check the setup, substitution, and final answer.

Formula Reminder

For a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\):

\[ a^2+b^2=c^2 \]
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Level 1: Find the Hypotenuse

Problem 1

A right triangle has legs of length \(3\) and \(4\). Find the hypotenuse.

\[ a=3,\quad b=4,\quad c=? \]
Show solution

Use the Pythagorean Theorem.

\[ a^2+b^2=c^2 \] \[ 3^2+4^2=c^2 \] \[ 9+16=c^2 \] \[ 25=c^2 \] \[ c=5 \]

The hypotenuse is \(5\).

Problem 2

A right triangle has legs of length \(5\) and \(12\). Find the hypotenuse.

\[ a=5,\quad b=12,\quad c=? \]
Show solution
\[ 5^2+12^2=c^2 \] \[ 25+144=c^2 \] \[ 169=c^2 \] \[ c=13 \]

The hypotenuse is \(13\).

Problem 3

A right triangle has legs of length \(8\) and \(15\). Find the hypotenuse.

\[ a=8,\quad b=15,\quad c=? \]
Show solution
\[ 8^2+15^2=c^2 \] \[ 64+225=c^2 \] \[ 289=c^2 \] \[ c=17 \]

The hypotenuse is \(17\).

Level 2: Find a Missing Leg

Problem 4

A right triangle has one leg of length \(6\) and a hypotenuse of length \(10\). Find the missing leg.

\[ a=6,\quad b=?,\quad c=10 \]
Show solution

Since \(c\) is the hypotenuse, use:

\[ a^2+b^2=c^2 \] \[ 6^2+b^2=10^2 \] \[ 36+b^2=100 \] \[ b^2=64 \] \[ b=8 \]

The missing leg is \(8\).

Problem 5

A right triangle has one leg of length \(9\) and a hypotenuse of length \(15\). Find the missing leg.

\[ a=9,\quad b=?,\quad c=15 \]
Show solution
\[ 9^2+b^2=15^2 \] \[ 81+b^2=225 \] \[ b^2=144 \] \[ b=12 \]

The missing leg is \(12\).

Problem 6

A right triangle has one leg of length \(7\) and a hypotenuse of length \(25\). Find the missing leg.

\[ a=7,\quad b=?,\quad c=25 \]
Show solution
\[ 7^2+b^2=25^2 \] \[ 49+b^2=625 \] \[ b^2=576 \] \[ b=24 \]

The missing leg is \(24\).

Level 3: Decimal Answers

Problem 7

A right triangle has legs of length \(4\) and \(7\). Find the hypotenuse. Round to the nearest tenth.

\[ a=4,\quad b=7,\quad c=? \]
Show solution
\[ 4^2+7^2=c^2 \] \[ 16+49=c^2 \] \[ 65=c^2 \] \[ c=\sqrt{65} \] \[ c\approx 8.1 \]

The hypotenuse is approximately \(8.1\).

Problem 8

A right triangle has one leg of length \(5\) and a hypotenuse of length \(11\). Find the missing leg. Round to the nearest tenth.

\[ a=5,\quad b=?,\quad c=11 \]
Show solution
\[ 5^2+b^2=11^2 \] \[ 25+b^2=121 \] \[ b^2=96 \] \[ b=\sqrt{96} \] \[ b\approx 9.8 \]

The missing leg is approximately \(9.8\).

Need more help with the Pythagorean Theorem?

The Pythagorean Theorem becomes easier when students remember that it only works for right triangles and that the hypotenuse is always the longest side. For a full explanation, visit my Pythagorean Theorem guide.

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