Pythagorean Theorem Calculator
Calculate Triangle Sides
Find missing sides of right triangles, verify right angles, and solve geometry problems using the Pythagorean theorem with step-by-step solutions.
Pythagorean Theorem Result
Hypotenuse: 5.00
Step-by-Step Calculation:
Triangle Verification:
Triangle Visualization:
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
What is the Pythagorean Theorem?
Pythagorean Theorem is a fundamental principle in geometry that describes the relationship between the three sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (called legs or cathetus).
Pythagorean Theorem Formulas
Standard Formula
Most common form
Hypotenuse calculation
Find Leg
Missing leg
Rearranged formula
Verification
Right triangle check
Equation validation
Distance Formula
Coordinate geometry
Pythagorean application
Pythagorean Theorem Rules
1. Basic Pythagorean Theorem
For any right triangle with legs a and b, and hypotenuse c:
a² + b² = c²
2. Finding the Hypotenuse
When both legs are known, calculate the hypotenuse:
c = √(a² + b²)
3. Finding a Missing Leg
When hypotenuse and one leg are known, find the other leg:
a = √(c² - b²)
Real-World Applications
Construction & Architecture
- Roof framing: Calculating rafter lengths and roof pitches
- Stair construction: Determining stair stringer lengths and riser heights
- Foundation layout: Ensuring perfect right angles for building corners
- Cabinet making: Calculating diagonal braces and shelf supports
Navigation & Surveying
- GPS and mapping: Calculating straight-line distances between coordinates
- Land surveying: Measuring property boundaries and diagonal distances
- Aviation: Calculating flight paths and navigation routes
- Marine navigation: Determining shortest routes across water bodies
Technology & Engineering
- Computer graphics: Calculating distances and rendering 3D objects
- Robotics: Path planning and obstacle avoidance algorithms
- Electrical engineering: Calculating impedance in AC circuits
- Mechanical engineering: Designing mechanical components and linkages
Everyday Life
- Home improvement: Measuring diagonal distances for square corners
- Sports: Calculating throwing distances and field dimensions
- Art and design: Creating proportional layouts and compositions
- Gardening: Planning garden layouts and measuring plot diagonals
Common Pythagorean Triples
| Triple Name | Leg A (a) | Leg B (b) | Hypotenuse (c) | Verification |
|---|---|---|---|---|
| 3-4-5 Triangle | 3 | 4 | 5 | 3² + 4² = 9 + 16 = 25 = 5² |
| 5-12-13 Triangle | 5 | 12 | 13 | 5² + 12² = 25 + 144 = 169 = 13² |
| 8-15-17 Triangle | 8 | 15 | 17 | 8² + 15² = 64 + 225 = 289 = 17² |
| 7-24-25 Triangle | 7 | 24 | 25 | 7² + 24² = 49 + 576 = 625 = 25² |
Pythagorean Theorem Properties
| Property | Description | Mathematical Expression | Practical Significance |
|---|---|---|---|
| Right Angle Requirement | Only applies to right triangles | ∠C = 90° | Fundamental limitation of the theorem |
| Hypotenuse Longest Side | Hypotenuse is always the longest side | c > a, c > b | Useful for identifying the hypotenuse |
| Converse Theorem | If a² + b² = c², then triangle is right | a² + b² = c² ⇒ ∠C = 90° | Used to verify right angles |
| Scaled Triples | Multiples of triples also work | ka, kb, kc also satisfy theorem | Allows for similar triangle calculations |
Step-by-Step Calculation Process
Example 1: Find hypotenuse of 3-4-5 triangle
- Identify legs: a = 3, b = 4
- Square the legs: 3² = 9, 4² = 16
- Sum the squares: 9 + 16 = 25
- Take square root: √25 = 5
- Hypotenuse c = 5
- Verify: 3² + 4² = 9 + 16 = 25 = 5² ✓
Example 2: Find missing leg when hypotenuse=13, leg=5
- Identify known sides: c = 13, a = 5
- Square known sides: 13² = 169, 5² = 25
- Subtract squares: 169 - 25 = 144
- Take square root: √144 = 12
- Missing leg b = 12
- Verify: 5² + 12² = 25 + 144 = 169 = 13² ✓
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Frequently Asked Questions (FAQs)
Q: Does the Pythagorean theorem work for all triangles?
A: No, the Pythagorean theorem only applies to right triangles (triangles with one 90-degree angle). For other triangles, different formulas like the Law of Cosines are used.
Q: What if my triangle sides don't satisfy a² + b² = c²?
A: If the sides don't satisfy the equation, the triangle is not a right triangle. It could be acute (all angles less than 90°) or obtuse (one angle greater than 90°).
Q: Can the Pythagorean theorem be used in 3D?
A: Yes! In 3D space, the distance between points (x₁,y₁,z₁) and (x₂,y₂,z₂) is √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²], which extends the Pythagorean theorem to three dimensions.
Q: What are Pythagorean triples?
A: Pythagorean triples are sets of three positive integers that satisfy a² + b² = c², such as (3,4,5), (5,12,13), and (8,15,17). These represent right triangles with integer side lengths.
Master Pythagorean theorem calculations with Toolivaa's free Pythagorean Theorem Calculator, and explore more mathematical tools in our Math Calculators collection.