Practice set
Pythagorean Theorem Examples
12 solved examples with full step-by-step working, finding the hypotenuse, finding a missing leg, verifying right triangles, and special triangle cases.
Examples in This Set
Group A: Find the Hypotenuse
- Example 1 — 3-4-5 (Basic)
- Example 2 — 5-12-13 (Basic)
- Example 3 — Decimal legs (Intermediate)
- Example 4 — Irrational hypotenuse (Intermediate)
Group B: Find the Missing Leg
Group C: Verify a Right Triangle
Group D: Special Triangles
Quick reference
Formula Quick Reference
These are the three formulas used throughout the examples below. Pick the card that matches what you need, then jump straight to that example group.
Each example below is labeled with the formula it uses. Click any formula card to jump to the corresponding example group.
Worked examples
12 Solved Problems with Answers
These examples move from straightforward triples to irrational answers and special-triangle cases. If you want real-world context problems instead of pure number practice, use Pythagorean Theorem Word Problems.
Group A: Find the Hypotenuse
Given both legs a and b, find c using c = √(a² + b²).
The 3-4-5 Triangle
Given: a = 3, b = 4. Find: c.
| Apply formula | c = √(a² + b²) = √(3² + 4²) |
|---|---|
| Square the legs | = √(9 + 16) |
| Add | = √25 |
| Result | c = 5 |
3-4-5 is the smallest Pythagorean triple. Multiples also work: 6-8-10, 9-12-15, and so on.
Try in Calculator →The 5-12-13 Triangle
Given: a = 5, b = 12. Find: c.
| Apply formula | c = √(5² + 12²) |
|---|---|
| Square the legs | = √(25 + 144) |
| Add | = √169 |
| Result | c = 13 |
5-12-13 is the second most common Pythagorean triple after 3-4-5.
Try in Calculator →Decimal Legs
Given: a = 2.5, b = 6. Find: c.
| Apply formula | c = √(2.5² + 6²) |
|---|---|
| Square the legs | = √(6.25 + 36) |
| Add | = √42.25 |
| Result | c = 6.5 |
2.5-6-6.5 is a scaled version of the 5-12-13 triple, divided by 2.
Try in Calculator →Irrational Hypotenuse
Given: a = 4, b = 7. Find: c.
| Apply formula | c = √(4² + 7²) |
|---|---|
| Square the legs | = √(16 + 49) |
| Add | = √65 |
| Result | c = √65 ≈ 8.062 |
Not every right triangle produces a whole-number hypotenuse. √65 is irrational.
Try in Calculator →Group B: Find the Missing Leg
Given the hypotenuse c and one leg, find the other leg using a = √(c² - b²).
The 8-15-17 Triangle
Given: a = 8, c = 17. Find: b.
| Apply formula | b = √(c² - a²) = √(17² - 8²) |
|---|---|
| Square | = √(289 - 64) |
| Subtract | = √225 |
| Result | b = 15 |
8-15-17 is a Pythagorean triple. Verify: 8² + 15² = 64 + 225 = 289 = 17² ✓
Try in Calculator →The 7-24-25 Triangle
Given: b = 24, c = 25. Find: a.
| Apply formula | a = √(c² - b²) = √(25² - 24²) |
|---|---|
| Square | = √(625 - 576) |
| Subtract | = √49 |
| Result | a = 7 |
7-24-25 is a Pythagorean triple. Notice how close 24 and 25 are, a thin, elongated triangle.
Try in Calculator →Decimal Hypotenuse
Given: a = 9, c = 15. Find: b.
| Apply formula | b = √(15² - 9²) |
|---|---|
| Square | = √(225 - 81) |
| Subtract | = √144 |
| Result | b = 12 |
9-12-15 is a multiple of the 3-4-5 triple, scaled by 3.
Try in Calculator →Irrational Missing Leg
Given: b = 5, c = 9. Find: a.
| Apply formula | a = √(9² - 5²) |
|---|---|
| Square | = √(81 - 25) |
| Subtract | = √56 = 2√14 |
| Result | a = 2√14 ≈ 7.483 |
The result is irrational. Check: 5² + (2√14)² = 25 + 56 = 81 = 9² ✓
Try in Calculator →Group C: Verify a Right Triangle
Given three sides, check whether a² + b² = c² holds. If it does, the triangle is a right triangle.
Is 6-8-10 a Right Triangle?
Given: sides 6, 8, 10. Is this a right triangle?
| Identify hypotenuse | Largest side = 10, so c = 10 |
|---|---|
| Check a² + b² | 6² + 8² = 36 + 64 = 100 |
| Check c² | 10² = 100 |
| Compare | 100 = 100 ✓ |
6-8-10 is a multiple of the 3-4-5 triple, scaled by 2.
Try in Calculator →Is 5-7-9 a Right Triangle?
Given: sides 5, 7, 9. Is this a right triangle?
| Identify hypotenuse | Largest side = 9, so c = 9 |
|---|---|
| Check a² + b² | 5² + 7² = 25 + 49 = 74 |
| Check c² | 9² = 81 |
| Compare | 74 ≠ 81 ✗ |
| Conclusion | Since a² + b² < c², the triangle is obtuse. |
If a² + b² > c², the triangle is acute. If a² + b² < c², it is obtuse.
Group D: Special Triangle Examples
These examples use the fixed ratios of 45-45-90 and 30-60-90 triangles.
45-45-90 Triangle: Find the Hypotenuse
Given: a = b = 6 (isosceles right triangle). Find: c.
| Apply formula | c = √(6² + 6²) |
|---|---|
| Square the legs | = √(36 + 36) |
| Add | = √72 = √(36 × 2) |
| Simplify | = 6√2 |
| Decimal | c = 6√2 ≈ 8.485 |
For any 45-45-90 triangle with leg a, the hypotenuse is always
a√2. Use the
45-45-90 Triangle Calculator.
30-60-90 Triangle: Find the Missing Leg
Given: short leg a = 5, hypotenuse c = 10. Find: long leg b.
| Apply formula | b = √(c² - a²) = √(10² - 5²) |
|---|---|
| Square | = √(100 - 25) |
| Subtract | = √75 = √(25 × 3) |
| Simplify | = 5√3 |
| Decimal | b = 5√3 ≈ 8.660 |
In a 30-60-90 triangle, the long leg is always (short leg) × √3. Here
5 × √3 = 5√3 ✓. Use the
30-60-90 Triangle Calculator.
Calculator
Try Your Own Numbers
Enter any values into the calculators below to get instant answers with the same step-by-step working shown above.
Find the Hypotenuse
Enter legs a and b → get c.
Calculate →Find a Missing Leg
Enter c and one leg → get the other.
Calculate →Verify a Right Triangle
Enter all three sides → check a² + b² = c².
Verify →FAQ
Frequently Asked Questions
The 3-4-5 triangle is the most commonly used example. It is the smallest set of positive
integers satisfying a² + b² = c², 9 + 16 = 25, and it appears in construction,
surveying, and standardized tests worldwide. Other common examples include 5-12-13, 8-15-17,
and 7-24-25.
The hypotenuse is always the longest side and always opposite the right angle. When verifying
a right triangle, always assign the largest of the three values to c. If you
assign the wrong side to c, the equation will not balance even for a valid right
triangle.
That is completely normal. Most right triangles do not produce integer side lengths. When the
result is irrational, like √65 or 2√14, you can leave it in exact
form or convert to a decimal approximation. Both forms are correct; exact form is preferred
in algebra, decimal form is more useful for measurement.
Pythagorean triples are sets of three positive integers, a, b, and
c, where a² + b² = c². Examples include 3-4-5, 5-12-13, 8-15-17,
and 7-24-25. Any multiple of a triple is also a triple, such as 6-8-10 and 9-12-15.
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