ganak.appPythagorean Theorem Calculator
Free Pythagorean theorem calculator. Enter any 2 sides of a right triangle to find the missing side, angles, area, and perimeter. Shows step-by-step solution, detects Pythagorean triples, and includes a dynamic diagram. Works with a² + b² = c².
Pythagorean Theorem
a² + b² = c²
Enter 2 sides to solve the right triangle
Enter any two side lengths (a, b, or c) to find the missing side, angles, area, and perimeter. Enter all three to verify a right triangle.
What Is the Pythagorean Theorem?
The most famous equation in geometry, linking the sides of any right triangle
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Written as a² + b² = c², where a and b are the legs and c is the hypotenuse.
Named after the ancient Greek mathematician Pythagoras (c. 570–495 BC), although the relationship was known to Babylonian and Indian mathematicians centuries earlier. It is one of the most widely used theorems in mathematics, with applications in construction, navigation, physics, and computer graphics.
Key rule: The Pythagorean theorem only applies to right triangles (triangles with one 90° angle). For other triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C).
Pythagorean Theorem Formulas
All forms of a² + b² = c² for solving any missing side
Find the Hypotenuse (c)
c = √(a² + b²)
When you know both legs, square each one, add them, and take the square root to find the hypotenuse.
Find Leg a
a = √(c² − b²)
When you know the hypotenuse and one leg, subtract the leg squared from the hypotenuse squared, then take the square root.
Find Leg b
b = √(c² − a²)
Same approach as finding leg a. Rearrange the theorem and solve for the unknown leg.
Area of a Right Triangle
Area = ½ × a × b
The area of a right triangle is half the product of its two legs. No need for height calculations.
Common Pythagorean Triples
Integer sets (a, b, c) where a² + b² = c² exactly
A Pythagorean triple is a set of three positive integers that satisfy the theorem perfectly — no decimals, no rounding. The smallest and most famous is 3, 4, 5. Any multiple of a triple (like 6, 8, 10 or 9, 12, 15) is also valid.
| a | b | c | a² + b² | c² |
|---|---|---|---|---|
| 3 | 4 | 5 | 25 | 25 |
| 5 | 12 | 13 | 169 | 169 |
| 8 | 15 | 17 | 289 | 289 |
| 7 | 24 | 25 | 625 | 625 |
| 20 | 21 | 29 | 841 | 841 |
| 9 | 40 | 41 | 1681 | 1681 |
| 12 | 35 | 37 | 1369 | 1369 |
| 11 | 60 | 61 | 3721 | 3721 |
| 28 | 45 | 53 | 2809 | 2809 |
| 33 | 56 | 65 | 4225 | 4225 |
30-60-90 Triangle
Side ratio: 1 : √3 : 2
Example: 5, 8.66, 10
45-45-90 Triangle
Side ratio: 1 : 1 : √2
Example: 5, 5, 7.07
Real-World Applications
Practical uses of the Pythagorean theorem in everyday life
Construction & Carpentry
Builders use the 3-4-5 rule to check right angles. Roof pitch, stair stringers, and diagonal bracing all rely on the Pythagorean theorem.
Navigation & Distance
The distance formula d = √((x₂−x₁)² + (y₂−y₁)²) is a direct application. Pilots, sailors, and GPS systems use it to calculate straight-line distances.
Screen & Display Sizing
TV and monitor diagonals are calculated from width and height using the Pythagorean theorem. A 16:9 screen with 52" width and 29.25" height has a 59.7" diagonal.
Ladder & Height Problems
Determine how high a ladder reaches against a wall, or how far from the base to place it. If a 10 ft ladder is 6 ft from the wall, it reaches 8 ft high.
Frequently Asked Questions
Common questions about the Pythagorean theorem, formulas, and applications