Pythagorean Theorem Calculator

Calculate the sides of a right triangle using the Pythagorean theorem (a² + b² = c²). Find the hypotenuse or any leg given two sides.

Result

Hypotenuse c

5.0000

Pythagorean Triple!

3.00² + 4.00² = 5.00²

9.00 + 16.00 = 25.00

Area6.00 sq units
Perimeter12.00 units

What do you want to find?

Enter Known Values

Right Triangle Visualization

a = 3.00b = 4.00c = 5.0053.1°36.9°90°

Calculation Steps

Step 1: Square both legs

a² = 3² = 9.0000

b² = 4² = 16.0000

Step 2: Add the squares

a² + b² = 9.0000 + 16.0000 = 25.0000

Step 3: Take the square root

c = √25.0000 = 5.0000

Triangle Properties

6.00

Area

12.00

Perimeter

36.9°

Angle A

53.1°

Angle B

Note: Angle A is opposite side a, Angle B is opposite side b, and the right angle (90°) is opposite the hypotenuse c.

Common Pythagorean Triples

Click any triple to use those values in the calculator.

Result

Hypotenuse c

5.0000

Pythagorean Triple!

3.00² + 4.00² = 5.00²

9.00 + 16.00 = 25.00

Area6.00 sq units
Perimeter12.00 units

?How to Use the Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, a2 + b2 = c2, where c is the hypotenuse (longest side, opposite the right angle) and a and b are the two legs. To find the hypotenuse: c = sqrt(a2 + b2). To find a leg: a = sqrt(c2 - b2). Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17.

What is the Pythagorean Theorem?

The Pythagorean theorem is a fundamental principle in geometry stating 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. This relationship, expressed as a2 + b2 = c2, is used in construction, navigation, surveying, and countless mathematical applications.

Key Facts About the Pythagorean Theorem

  • Pythagorean theorem: a squared + b squared = c squared (works only for right triangles)
  • c (hypotenuse) is the longest side, opposite the 90-degree angle
  • To find hypotenuse: c = sqrt(a squared + b squared)
  • To find a leg: a = sqrt(c squared - b squared)
  • Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
  • Any multiple of a Pythagorean triple is also a Pythagorean triple (6-8-10, 9-12-15)
  • The distance formula d = sqrt((x2-x1)2 + (y2-y1)2) is based on the Pythagorean theorem
  • Named after ancient Greek mathematician Pythagoras (c. 570-495 BC)

Quick Answer

The Pythagorean theorem states that in a right triangle, a2 + b2 = c2, where c is the hypotenuse (longest side, opposite the right angle) and a and b are the two legs. To find the hypotenuse: c = sqrt(a2 + b2). To find a leg: a = sqrt(c2 - b2). Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17.

Frequently Asked Questions

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides. Written as: a² + b² = c², where c is the hypotenuse.
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Multiples of triples are also triples: (6, 8, 10).
To find the hypotenuse (c), use the formula c = √(a² + b²). Square both legs, add them together, and take the square root. For example, with legs 3 and 4: c = √(9 + 16) = √25 = 5.
To find a leg (a) when you know the other leg (b) and hypotenuse (c): a = √(c² - b²). Subtract the square of the known leg from the square of the hypotenuse, then take the square root.
Common uses include: calculating distances on maps, determining diagonal screen sizes, construction and carpentry (ensuring right angles), navigation, physics problems, and computer graphics. Any problem involving right triangles can use this theorem.

Last updated: 2025-01-15