Pythagorean Theorem Calculator
Calculate any missing side of a right triangle using the Pythagorean theorem: a² + b² = c². Enter two known sides and solve for the third instantly.
Enter the Sides of a Right Triangle
Step-by-Step Solution:
1. Write the Pythagorean theorem:
a² + b² = c²
2. Substitute the known values:
3² + 4² = c²
3. Calculate the squares:
9 + 16 = c²
4. Add:
25 = c²
5. Take the square root:
c = √25 = 5
What is the Pythagorean Theorem?
The Pythagorean theorem is one of the most fundamental formulas in mathematics. It describes the relationship between the three sides of a right triangle.
Formula: a² + b² = c²
where: - a = one leg of the right triangle - b = the other leg of the right triangle - c = the hypotenuse (the longest side, opposite the right angle)
The theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides.
How to Use the Pythagorean Theorem
Solving for the Hypotenuse (c)
- Identify the two legs (a and b) of the right triangle
- Square each leg: calculate a² and b²
- Add the squares: a² + b²
- Take the square root of the sum to find c
Example: If a = 3 and b = 4, then c = √(3² + 4²) = √(9 + 16) = √25 = 5
Solving for a Leg (a or b)
- Identify the hypotenuse (c) and the known leg
- Rearrange the formula: a² = c² - b² (or b² = c² - a²)
- Square the known values and subtract
- Take the square root to find the missing leg
Example: If b = 5 and c = 13, then a = √(13² - 5²) = √(169 - 25) = √144 = 12
Understanding the Pythagorean Theorem
The theorem is named after the ancient Greek mathematician Pythagoras, although it was known to Babylonian and Indian mathematicians centuries earlier. The proof is beautifully geometric: if you draw squares on each side of a right triangle, the area of the square on the hypotenuse equals the combined areas of the squares on the two legs.
This relationship only holds for right triangles — triangles that contain exactly one 90-degree angle. The hypotenuse is always the side opposite the right angle and is always the longest side of the triangle.
Pythagorean Triples
Some right triangles have sides that are all whole numbers. These sets of integers are called Pythagorean triples:
- (3, 4, 5) — the most common triple
- (5, 12, 13)
- (8, 15, 17)
- (7, 24, 25)
Any multiple of a Pythagorean triple is also a triple. For example, doubling (3, 4, 5) gives (6, 8, 10).
Real-World Applications
The Pythagorean theorem is used across many fields:
- Construction: Checking if corners are square (the 3-4-5 rule)
- Navigation: Calculating straight-line distances
- Architecture: Determining roof pitch and rafter lengths
- Surveying: Finding distances across terrain
- Computer graphics: Calculating distances between pixels or objects
- Physics: Resolving vector components
- Sports: Finding diagonal distances on a field or court
Common Questions
What is the Pythagorean theorem?
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. The formula is a² + b² = c², where c is the hypotenuse.
How do you find the hypotenuse of a right triangle?
To find the hypotenuse, square both legs (a² and b²), add them together, and take the square root of the sum: c = √(a² + b²). For example, if a = 3 and b = 4, then c = √(9 + 16) = √25 = 5.
Can the Pythagorean theorem be used on any triangle?
No, the Pythagorean theorem only works for right triangles (triangles with one 90-degree angle). For other triangles, you would use the law of cosines: c² = a² + b² - 2ab·cos(C).
What are Pythagorean triples?
Pythagorean triples are sets of three whole numbers that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a Pythagorean triple is also a Pythagorean triple.