Pythagorean Theorem Practice Problems

What this problem type covers

Each problem gives two sides of a right triangle and asks for the third. The first bands use Pythagorean triples so the answer is a whole number; later bands leave an irrational answer that has to be written as a simplified radical.

Example problems

One example from each difficulty band, with worked solutions.

These problems are generated automatically. The answers are derived as each problem is built and are checked by tests, but mistakes are still possible — please tell us if you find one.

1. 1 · ★1
   A right triangle has legs of 60 and 63. Find the hypotenuse.
   • ①c = 123
   • ②c = 60
   • ③c = 7569
   • ④c = 3√41
   • ⑤c = 87
   Show solution

   ⑤c = 87

   1. Use the Pythagorean theorem: a² + b² = c²
   2. Substitute the known sides: 60² + 63² = c²
   3. So the unknown side squared is 3600 + 3969 = 7569.
   4. Taking the positive square root: c = 87
2. 2 · ★2
   A right triangle has legs of 12 and 35. Find the hypotenuse.
   • ①c = 1369
   • ②c = 37
   • ③c = 12
   • ④c = 47
   • ⑤c = √1081
   Show solution

   ②c = 37

   1. Use the Pythagorean theorem: a² + b² = c²
   2. Substitute the known sides: 12² + 35² = c²
   3. So the unknown side squared is 144 + 1225 = 1369.
   4. Taking the positive square root: c = 37
3. 3 · ★3
   A right triangle has a hypotenuse of 50 and one leg of 48. Find the other leg.
   • ①b = 2
   • ②b = 98
   • ③b = 196
   • ④b = 14
   • ⑤b = 2√1201
   Show solution

   ④b = 14

   1. Use the Pythagorean theorem: a² + b² = c²
   2. Substitute the known sides: 48² + b² = 50²
   3. So the unknown side squared is 2500 − 2304 = 196.
   4. Taking the positive square root: b = 14
4. 4 · ★4
   A right triangle has legs of 4 and 2. Find the hypotenuse.
   • ①c = 20
   • ②c = 2√5
   • ③c = 6
   • ④c = 4
   • ⑤c = 2√3
   Show solution

   ②c = 2√5

   1. Use the Pythagorean theorem: a² + b² = c²
   2. Substitute the known sides: 4² + 2² = c²
   3. So the unknown side squared is 16 + 4 = 20.
   4. Taking the positive square root: c = 2√5
5. 5 · ★5
   A right triangle has legs of 5 and 3. Find the hypotenuse.
   • ①c = 34
   • ②c = 8
   • ③c = 4
   • ④c = 5
   • ⑤c = √34
   Show solution

   ⑤c = √34

   1. Use the Pythagorean theorem: a² + b² = c²
   2. Substitute the known sides: 5² + 3² = c²
   3. So the unknown side squared is 25 + 9 = 34.
   4. Taking the positive square root: c = √34

Where people usually go wrong

• Treating the hypotenuse as just another leg. a² + b² = c² only holds with c as the hypotenuse — the longest side.
• Adding when you should subtract. Finding a leg means c² − a², not c² + a².
• Forgetting the square root at the end. c² = 169 is not the answer; c = 13 is.
• Rounding √50 to 7.07. The exact answer is 5√2, and rounding early loses it for good.
• Leaving a radical unsimplified. √72 and 6√2 are the same number, but only one is finished.