Pythagorean Theorem Word Problems Worksheets - Free Printable

Let's solve each of these Pythagorean Theorem word problems step by step. The Pythagorean Theorem states:

$$
a^2 + b^2 = c^2
$$

where $ a $ and $ b $ are the legs of a right triangle, and $ c $ is the hypotenuse (the longest side, opposite the right angle).

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1) Dan secures a ladder firmly to the wall...

- The foot of the ladder is 6 feet from the wall.
- The top of the ladder rests 8 feet up the wall.
- We need to find the length of the ladder (the hypotenuse).

This forms a right triangle:
- $ a = 6 $ ft
- $ b = 8 $ ft
- $ c = ? $

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

✔ Answer: 10.0 feet

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2) Diagonal of a square carom board with side 28 inches

A square has equal sides, and its diagonal forms a right triangle with two sides.

So:
- $ a = 28 $
- $ b = 28 $
- $ c = ? $

$$
c^2 = 28^2 + 28^2 = 784 + 784 = 1568
$$
$$
c = \sqrt{1568} \approx 39.6
$$

✔ Answer: 39.6 inches

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3) Carlos flies his drone 5 feet above ground, then 12 feet to the right

The drone starts at ground level, goes up 5 feet vertically, then moves 12 feet horizontally.

We want the straight-line distance from start point to current position — this is the hypotenuse of a right triangle:
- $ a = 5 $ ft (vertical)
- $ b = 12 $ ft (horizontal)
- $ c = ? $

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

✔ Answer: 13.0 feet

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4) Ship travels 9 miles north, then 20 miles west — find shortest route

The ship’s path forms a right triangle:
- North leg: 9 miles
- West leg: 20 miles
- Shortest route (hypotenuse) from start to island?

$$
c^2 = 9^2 + 20^2 = 81 + 400 = 481
$$
$$
c = \sqrt{481} \approx 21.9
$$

✔ Answer: 21.9 miles

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5) Dahlia cuts fabric in shape of right triangle

- Longest side (hypotenuse): 8 inches
- One leg: 4 inches
- Find the other leg?

Let $ a = 4 $, $ c = 8 $, $ b = ? $

$$
a^2 + b^2 = c^2
\Rightarrow 4^2 + b^2 = 8^2
\Rightarrow 16 + b^2 = 64
\Rightarrow b^2 = 48
\Rightarrow b = \sqrt{48} \approx 6.9
$$

✔ Answer: 6.9 inches

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✔ Final Answers (rounded to nearest tenth):

1) 10.0 feet
2) 39.6 inches
3) 13.0 feet
4) 21.9 miles
5) 6.9 inches

Let me know if you'd like a visual explanation or diagram for any of them!