Pythagorean Theorem Word Problems- Practice Examples

What Pythagorean Theorem Word Problems Actually Look Like

These problems show up everywhere—construction, navigation, sports, home improvement. The setup is always the same: you're given two sides of a right triangle and asked to find the third. The twist is they hide the triangle inside a story.

Here's the pattern you'll see:

• A ladder leaning against a wall
• A boat traveling north and east
• A baseball diamond diagonal distance
• A screen's diagonal measurement
• Someone walking in two perpendicular directions

Once you know what to look for, these become straightforward. The hard part is extracting the triangle from the narrative.

The Formula You Need

The Pythagorean Theorem states:

a² + b² = c²

Where a and b are the legs (the shorter sides) and c is the hypotenuse (the longest side, opposite the right angle).

That's it. Memorize it. You'll use it constantly.

How to Solve Any Word Problem in 4 Steps

Step 1: Find the Right Angle

Look for words like "turn," "corner," "perpendicular," or "at a 90° angle." The corner of the right angle is where the two legs meet.

Step 2: Identify the Two Known Sides

One will be labeled as the hypotenuse (longest side, often "diagonal," "distance," or "straight-line"). The other two are the legs.

Step 3: Plug Into the Formula

Square the known numbers, add or subtract depending on what's missing, then take the square root.

Step 4: Check Your Answer

Does the hypotenuse actually look like the longest side? Does the answer make sense in context? If your ladder is 50 feet tall but the wall is only 10 feet, something's wrong.

Practice Problems with Full Solutions

Problem 1: The Ladder

A 15-foot ladder leans against a building. The base of the ladder is 9 feet from the building. How far up the building does the ladder reach?

The ladder is the hypotenuse. The distance from the building is one leg. You need the height.

Solution:

9² + b² = 15²

81 + b² = 225

b² = 144

b = 12 feet

Problem 2: The Shortcut

Marcus walks 8 blocks east and then 15 blocks north to get to school. How far would he walk if he took a straight diagonal path instead?

East and north are perpendicular. The diagonal is the hypotenuse.

Solution:

8² + 15² = c²

64 + 225 = c²

289 = c²

c = 17 blocks

Problem 3: The Baseball Diamond

A baseball diamond is a square with 90 feet between bases. How far is it from home plate to second base?

Home plate to first base and first base to second base form a right angle. You're finding the diagonal of a square.

Solution:

90² + 90² = c²

8100 + 8100 = c²

16200 = c²

c = √16200 ≈ 127.28 feet

Problem 4: The Broken Pole

A telephone pole snapped and the top hit the ground 12 meters from the base. The broken part is 13 meters long. How tall was the original pole?

The ground distance (12m) and the remaining upright portion form a right angle. The broken top (13m) is the hypotenuse.

Solution:

12² + b² = 13²

144 + b² = 169

b² = 25

b = 5 meters (the remaining height)

Original height = 5 + 13 = 18 meters

Common Pythagorean Theorem Mistakes

• Using the wrong side as the hypotenuse. The hypotenuse is always the longest side. If your answer is smaller than one of your given sides, you messed up.
• Forgetting to square root. You found c², not c. Take the square root or your answer is wrong.
• Not checking if it's actually a right triangle. The theorem only works on right triangles. If there's no right angle, you can't use it.
• Mixing up units. If one side is in feet and another in meters, convert first.

Quick Reference: When to Use the Theorem

Final Note

These problems follow a predictable pattern. Once you've solved 10 of them, you'll recognize the structure instantly. The story changes, the math doesn't. Practice with real numbers until the process becomes automatic.