Pythagorean Theorem Calculator
Find Hypotenuse (c)
Given sides a and b, find hypotenuse c. Formula: c = √(a² + b²)
Find Missing Side
Given one side and hypotenuse, find the other side. Enter one of a or b and c.
Verify Right Triangle
Enter three sides to verify if it forms a right triangle
Calculate All Properties
Enter two sides and get all triangle properties
Pythagorean Theorem Results
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The Pythagorean theorem states that in a right triangle: a² + b² = c², where c is the hypotenuse (longest side) and a, b are the other two sides. This theorem only applies to right triangles (90° angle).
Complete Pythagorean Theorem Guide
The Pythagorean theorem is one of the most important relationships in geometry. This guide explains the theorem, its applications, and how to solve right triangle problems.
What is the Pythagorean Theorem?
The Pythagorean theorem states that in any right triangle (triangle with one 90° angle), the square of the hypotenuse (longest side) equals the sum of squares of the other two sides.
Where:
a = length of first leg
b = length of second leg
c = length of hypotenuse (opposite the right angle)
Example: If a=3 and b=4, then c² = 9 + 16 = 25, so c = 5
Common Pythagorean Triples
| a | b | c | Name/Example |
|---|---|---|---|
| 3 | 4 | 5 | 3-4-5 triangle (most common) |
| 5 | 12 | 13 | 5-12-13 triangle |
| 8 | 15 | 17 | 8-15-17 triangle |
| 7 | 24 | 25 | 7-24-25 triangle |
| 6 | 8 | 10 | Double of 3-4-5 |
| 9 | 12 | 15 | Triple of 3-4-5 |
How to Use Pythagorean Theorem
- Find Hypotenuse: If you know both legs (a and b), use c = √(a² + b²)
- Find a Leg: If you know hypotenuse (c) and one leg, use a = √(c² - b²)
- Verify Right Triangle: If a² + b² = c², then it's a right triangle
- Find Angles: Use trigonometry (sine, cosine, tangent) along with Pythagorean theorem
Real-World Applications
- Construction & Building: Checking if corners are square (90°). Mark 3 feet along one wall, 4 feet along adjacent wall, measure diagonal - should be 5 feet
- Navigation & Distance: Finding straight-line distance between two points. If you go 3 miles east and 4 miles north, you're 5 miles from start
- Ladder Safety: 10-foot ladder should be 6 feet from wall for safe angle. Distance along wall = √(100 - 36) = 8 feet
- Diagonal Measurements: TV screen diagonal size from width and height. 40" diagonal with 16:9 aspect ratio uses Pythagorean theorem
- Sports & Games: Diamond distances in baseball, tennis court, soccer field dimensions
- Roof Pitch Calculation: Finding rafter length from rise and run measurements
Pythagorean Theorem Variations
| Need to Find | Formula | Example |
|---|---|---|
| Hypotenuse (c) | c = √(a² + b²) | c = √(3² + 4²) = √25 = 5 |
| Side a | a = √(c² - b²) | a = √(5² - 4²) = √9 = 3 |
| Side b | b = √(c² - a²) | b = √(5² - 3²) = √16 = 4 |
| Verify Right Triangle | a² + b² = c²? | 3² + 4² = 9 + 16 = 25 = 5² ✓ |
Triangle Properties & Angles
- Right Angle: Always 90°
- Other Angles: Sum of all angles = 180°, so other two angles = 90°
- 3-4-5 Triangle Angles: 90°, 53.13°, 36.87° (approximately)
- Isosceles Right Triangle: 45°-45°-90°. If legs = 1, hypotenuse = √2
- 30-60-90 Triangle: Sides in ratio 1:√3:2
Pythagorean Theorem in 3D
The theorem extends to 3D for finding diagonal of rectangular box: d = √(a² + b² + c²), where a, b, c are the three dimensions and d is the space diagonal.
Common Mistakes to Avoid
- Confusing which side is c: c is always the hypotenuse (longest side, opposite right angle)
- Forgetting to take square root: c² = 25 means c = 5, not c = 25
- Using on non-right triangles: Theorem only works for triangles with 90° angle
- Mixing units: All sides must use same units before applying theorem
- Rounding too early: Keep precision until final answer
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Frequently Asked Questions
1. What is hypotenuse?
Hypotenuse is the longest side of a right triangle, opposite the 90° angle. In formula a² + b² = c², c is the hypotenuse.
2. When can I use Pythagorean theorem?
Only for right triangles (triangles with one 90° angle). Won't work for other triangle types.
3. What's a Pythagorean triple?
Three whole numbers that satisfy Pythagorean theorem. Example: 3, 4, 5 (since 3² + 4² = 5²). Other examples: 5-12-13, 8-15-17.
4. How do I find the missing side?
If you know hypotenuse and one side: a = √(c² - b²). If you know both legs: c = √(a² + b²).
5. Is 3-4-5 the only common triple?
No. Common ones include 5-12-13, 8-15-17, 7-24-25. Multiples also work: 6-8-10, 9-12-15, etc.
6. How is this used in construction?
To verify corners are square (90°): measure 3 feet along one wall, 4 feet along other, diagonal should be 5 feet.
7. What if I only know the hypotenuse?
You need at least one other side. With just hypotenuse, infinite right triangles are possible (different leg combinations).
8. Does order matter? Is a² + b² different from b² + a²?
No, order doesn't matter mathematically (addition is commutative). 3² + 4² = 4² + 3² = 25. Both are correct.
9. What's the relationship between Pythagorean theorem and trigonometry?
Both describe right triangles. Pythagorean theorem relates side lengths. Trigonometry (sine, cosine, tangent) relates sides and angles.
10. How do I check if three sides form right triangle?
Check if a² + b² = c² (where c is longest side). Example: 3² + 4² = 9 + 16 = 25 = 5², so 3-4-5 is right triangle.
11. Can sides be decimals or just whole numbers?
Sides can be any positive number: whole numbers, decimals, fractions. Example: 1.5² + 2² = 6.25, so c ≈ 2.5.
12. What if a² + b² doesn't equal c² exactly?
Then it's not a right triangle. It might be slightly off due to rounding, so allow small tolerance (within 0.01).