What is the Pythagorean theorem formula?

The Pythagorean theorem states a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse. Given any two sides, you can calculate the third.

How to calculate the hypotenuse?

Given two legs a and b, the hypotenuse c = √(a² + b²). For example, if a=3 and b=4, then c=√(9+16)=√25=5.

How to verify a right triangle?

If three sides satisfy a² + b² = c² (where c is the longest side), then the triangle is a right triangle. For example, sides 3, 4, 5 satisfy 9+16=25, so it's a right triangle.

📐 Pythagorean Theorem Calculator

Enter any two sides to automatically calculate the third

Enter any two side lengths; leave the third blank to auto-calculate

Leg a

Leg b

Hypotenuse c

Longest side

Calculate Hypotenuse c

c = 5

Leg a

Leg b

Hypotenuse c

Calculation Process

a² = 3² = 9

b² = 4² = 16

a² + b² = 25

c² = 5² = 25

Pythagorean Theorem Calculator Guide

This calculator uses the Pythagorean theorem to compute the sides of a right triangle. Enter any two sides (legs a, b or hypotenuse c) to automatically calculate the third. When all three sides are entered, it determines whether they form a right triangle.

The Pythagorean theorem is one of the most fundamental theorems in geometry, widely applied in construction, surveying, and navigation.

Pythagorean Theorem Explained

Pythagorean Theorem: In a right triangle, the sum of squares of the two legs equals the square of the hypotenuse.

Formula: a^2 + b^2 = c^2 Where a, b are legs and c is the hypotenuse

Derived formulas: - c = sqrt(a^2 + b^2) (given two legs, find hypotenuse) - a = sqrt(c^2 - b^2) (given hypotenuse and one leg, find the other) - b = sqrt(c^2 - a^2)

Common Pythagorean triples: - 3, 4, 5 - 5, 12, 13 - 8, 15, 17 - 7, 24, 25

Triangle type test: If a^2 + b^2 = c^2 → right triangle If a^2 + b^2 > c^2 → acute triangle If a^2 + b^2 < c^2 → obtuse triangle

FAQ

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Pythagorean Theorem Calculator Guide

The Pythagorean Theorem Calculator solves for the side relationships in right triangles. Enter any two side lengths and the tool automatically calculates the third. You can also enter all three sides to verify whether they form a right triangle.

The Pythagorean theorem is one of the oldest and most important theorems in mathematics, with the earliest records found in China's Zhoubi Suanjing ("3-4-5 right triangle"). It has been applied in architecture, navigation, and engineering surveying for thousands of years. This tool supports both modes: finding the third side from any two, and three-side verification.

Pythagorean Theorem Core Formulas

【Basic Formula】a² + b² = c²
Where a, b are legs, c is the hypotenuse

【Given two legs, find hypotenuse】c = √(a² + b²)
【Given hypotenuse and one leg, find the other】a = √(c² - b²)

【Common Pythagorean Triples】(integer solutions)
3-4-5, 5-12-13, 6-8-10, 7-24-25
8-15-17, 9-12-15, 9-40-41, 20-21-29

【Verification Method】If a²+b²=c² (c is the longest side), then it's a right triangle

Practical Examples

📋

【Example 1】Ladder against wall Ladder base is 3m from wall, ladder is 5m long Find: How high can the ladder reach? Known: hypotenuse 5m and leg 3m Height = √(5² - 3²) = √16 = 4m Result: Ladder reaches 4 meters up the wall 【Example 2】Diagonal distance Rectangular room 8m × 6m, diagonal length? d = √(8² + 6²) = √100 = 10m 【Example 3】Verify right triangle Sides: 5, 12, 13 Verify: 5²+12²=25+144=169=13² ✓ Conclusion: Forms a right triangle 【Example 4】Not a right triangle Sides: 2, 3, 4 Verify: 2²+3²=13 ≠ 16=4² ✗ Conclusion: Not a right triangle (difference is 3)

Important Notes

💡

- The hypotenuse c is always the longest side of a right triangle - Given two legs, the hypotenuse is always greater than each leg - Given the hypotenuse and one leg, the hypotenuse must be greater than that leg to calculate - The Pythagorean theorem only applies to right triangles, not arbitrary triangles - Results are rounded to 4 decimal places, meeting general engineering needs

Application Scenarios

- Construction: Calculate roof pitch, stair stringer length, room diagonal distance
- Woodworking: Cut 45° angled materials, calculate frame diagonal bracing dimensions
- Navigation: Planar distance between two points d=√[(x₂-x₁)²+(y₂-y₁)²]
- Graphic design: Circle tangent length, ellipse focal distance calculations
- Everyday: Optimal TV viewing distance, ladder safety angle estimation

History and Generalizations of the Pythagorean Theorem

There are over 400 known proofs of the Pythagorean theorem — from the ancient Chinese chord diagram proof (Zhao Shuang), Euclid's Elements proof in ancient Greece, to US President Garfield's trapezoid proof. It is considered one of the most fundamental theorems in mathematics.

Generalizations: (1) Law of Cosines c²=a²+b²-2ab·cosC — applicable to any triangle; (2) 3D Pythagorean theorem d²=x²+y²+z² — distance from a point to the origin in space; (3) n-dimensional Euclidean distance formula. These generalizations ensure the Pythagorean theorem's ideas permeate all of geometry and analysis.

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