Pythagorean Theorem Calculator
Find any missing side of a right triangle instantly. Enter two known sides and get the third with a step-by-step solution.
Step-by-step solution:
What Is the Pythagorean Theorem?
The Pythagorean theorem is one of the most fundamental principles in mathematics. Named after the ancient Greek mathematician Pythagoras (c. 570–495 BC), it describes the relationship between the three sides of a right triangle. The theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this formula, a and b represent the two shorter sides (called legs), and c is the longest side (called the hypotenuse). This relationship holds true for every right triangle, regardless of its size.
How to Use This Calculator
Simply choose what you want to find — the hypotenuse or one of the legs — then enter the two known values. The calculator instantly computes the missing side and shows a step-by-step solution.
- To find the hypotenuse (c): Enter both legs (a and b). Formula: c = √(a² + b²).
- To find a leg (a or b): Enter the hypotenuse and the known leg. Formula: b = √(c² − a²).
Pythagorean Theorem Examples
Example 1: Find the Hypotenuse
Given: a = 3, b = 4
c = √(9 + 16) = √25 = 5
Example 2: Find a Leg
Given: c = 13, a = 5
b = √(169 − 25) = √144 = 12
Example 3: Decimal Values
Given: a = 7.5, b = 10
c = √(56.25 + 100) = √156.25 = 12.5
Example 4: Large Triangle
Given: a = 20, b = 21
c = √(400 + 441) = √841 = 29
Common Pythagorean Triples
A Pythagorean triple consists of three positive integers that satisfy a² + b² = c². The most well-known triples are:
- 3, 4, 5 — the most basic and widely used triple
- 5, 12, 13 — common in geometry problems
- 8, 15, 17 — useful for construction measurements
- 7, 24, 25 — appears in advanced problems
- 20, 21, 29 — a less obvious but valid triple
Any multiple of a Pythagorean triple is also a valid triple. For example, 6-8-10 is the 3-4-5 triple multiplied by 2.
Real-World Applications
The Pythagorean theorem is used in construction (ensuring walls are square), navigation (calculating shortest distance), physics (resolving force vectors), architecture (roof pitch calculations), and computer graphics (distance between pixels). Surveyors, engineers, and pilots rely on it daily.
History of the Pythagorean Theorem
Although named after Pythagoras, evidence suggests that the Babylonians knew the theorem over 1,000 years before Pythagoras. The famous clay tablet Plimpton 322 (c. 1800 BC) contains Pythagorean triples. Ancient Indian, Chinese, and Egyptian mathematicians also independently discovered this relationship. Today, over 400 distinct proofs exist, including one by U.S. President James Garfield.
Frequently Asked Questions
The Pythagorean theorem states that in a right triangle, a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle) and a and b are the two legs.
Square both legs (a² and b²), add them together, and take the square root: c = √(a² + b²). For example, if a = 6 and b = 8, then c = √(36 + 64) = √100 = 10.
Yes! Switch to “Find a Leg” mode, enter the hypotenuse and the known leg, and the calculator finds the missing side using b = √(c² − a²).
No, only for right triangles (with a 90° angle). For other triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C).
A set of three whole numbers that satisfy a² + b² = c². Examples: 3-4-5, 5-12-13, 8-15-17. Any multiple of a triple is also a valid triple.