48 Pythagorean Theorem Worksheet with Answers [Word + PDF] - Free Printable

Let’s solve each problem step by step using the Pythagorean Theorem:
> In a right triangle, $ a^2 + b^2 = c^2 $, where c is the hypotenuse (the longest side, opposite the right angle).

We’ll also calculate area and perimeter for problems 1–4 as instructed.

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Problem 1:

Sides: 17 (base), ? (height), 19 (hypotenuse)

Use Pythagoras:
$ 17^2 + b^2 = 19^2 $
→ $ 289 + b^2 = 361 $
→ $ b^2 = 361 - 289 = 72 $
→ $ b = \sqrt{72} ≈ 8.5 $

✔ Missing length = 8.5

Area = $ \frac{1}{2} × base × height = \frac{1}{2} × 17 × 8.5 = 72.25 → \boxed{72.3} $ (rounded to nearest tenth)

Perimeter = 17 + 8.5 + 19 = 44.5

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Problem 2:

Sides: 5 (leg), 13 (leg), ? (hypotenuse)

$ 5^2 + 13^2 = c^2 $
→ $ 25 + 169 = 194 $
→ $ c = \sqrt{194} ≈ 13.9 $

✔ Missing length = 13.9

Area = $ \frac{1}{2} × 5 × 13 = 32.5 $

Perimeter = 5 + 13 + 13.9 = 31.9

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Problem 3:

Sides: 20 (hypotenuse), 10 (leg), ? (other leg)

$ a^2 + 10^2 = 20^2 $
→ $ a^2 + 100 = 400 $
→ $ a^2 = 300 $
→ $ a = \sqrt{300} ≈ 17.3 $

✔ Missing length = 17.3

Area = $ \frac{1}{2} × 10 × 17.3 = 86.5 $

Perimeter = 10 + 17.3 + 20 = 47.3

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Problem 4:

Sides: 5 (leg), 14 (leg), ? (hypotenuse)

$ 5^2 + 14^2 = c^2 $
→ $ 25 + 196 = 221 $
→ $ c = \sqrt{221} ≈ 14.9 $

✔ Missing length = 14.9

Area = $ \frac{1}{2} × 5 × 14 = 35.0 $

Perimeter = 5 + 14 + 14.9 = 33.9

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Problems 5–9: Use $ a^2 + b^2 = c^2 $, with c = hypotenuse

#### Problem 5:
a = 12, b = 5 → find c

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

✔ c = 13

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#### Problem 6:
a = 8, c = 10 → find b

$ 8^2 + b^2 = 10^2 $
→ $ 64 + b^2 = 100 $
→ $ b^2 = 36 $
→ $ b = 6 $

✔ b = 6

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#### Problem 7:
a = 15, c = 17 → find b

$ 15^2 + b^2 = 17^2 $
→ $ 225 + b^2 = 289 $
→ $ b^2 = 64 $
→ $ b = 8 $

✔ b = 8

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#### Problem 8:
b = 40, c = 50 → find a

$ a^2 + 40^2 = 50^2 $
→ $ a^2 + 1600 = 2500 $
→ $ a^2 = 900 $
→ $ a = 30 $

✔ a = 30

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#### Problem 9:
b = 2, c = 4 → find a

$ a^2 + 2^2 = 4^2 $
→ $ a^2 + 4 = 16 $
→ $ a^2 = 12 $
→ $ a = \sqrt{12} ≈ 3.5 $

✔ a = 3.5

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Final Answer:

Problem 1: missing = 8.5, area = 72.3, perimeter = 44.5
Problem 2: missing = 13.9, area = 32.5, perimeter = 31.9
Problem 3: missing = 17.3, area = 86.5, perimeter = 47.3
Problem 4: missing = 14.9, area = 35.0, perimeter = 33.9
Problem 5: c = 13
Problem 6: b = 6
Problem 7: b = 8
Problem 8: a = 30
Problem 9: a = 3.5