Pythagorean Theorem Calculator
Solve Right Triangles in Seconds with the Pythagorean Theorem
Result
| Perimeter | |
| Semi-Perimeter | |
| Area | |
| Angle A | |
| Angle B | |
| Angle C | |
| Inradius (r) | |
| Circumradius (R) |
Pythagorean Theorem - Formula, Explanation & Examples
What is the Pythagorean Theorem?
The Pythagorean Theorem is a fundamental rule in geometry that applies to right-angled triangles. It states:
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
This principle is named after the Greek mathematician Pythagoras.
Pythagorean Theorem Formula
If a and b are the triangle's legs and c is the hypotenuse, then:
c² = a² + b²To solve for each side:
- Hypotenuse (c):
c = √(a² + b²) - Leg (a or b):
a = √(c² - b²)orb = √(c² - a²)
Conditions for Using the Theorem
- The triangle must be right-angled.
- Applies only in Euclidean geometry.
Why is the Pythagorean Theorem Important?
This theorem is widely used in mathematics, engineering, and real-life problem-solving. It's essential for:
- Calculating distances and lengths
- Solving problems in architecture and construction
- Working with coordinates and vectors
Distance Formula
The Pythagorean Theorem leads to the distance formula in coordinate geometry:
Distance = √((x2 - x1)2 + (y2 - y1)2)
This formula is essentially applying the Pythagorean Theorem to a triangle formed by two points on a coordinate plane.
Examples
Example 1: Find the Hypotenuse
Given: a = 3 cm, b = 4 cm
Solution:
c = √(3² + 4²) = √(9 + 16) = √25 = 5 cm
Example 2: Find One Leg
Given: c = 13 cm, a = 5 cm
Solution:
b = √(13² - 5²) = √(169 - 25) = √144 = 12 cm
Example 3: Distance Between Two Points
Points: (2, 3) and (7, 11)
Distance = √((7-2)² + (11-3)²) = √(25 + 64) = √89 ≈ 9.43 units
Real-Life Applications
- Construction & Carpentry: Ensuring structures have 90° angles
- Navigation & GPS: Calculating shortest paths
- Computer Graphics: Determining distances between pixels
- Physics & Astronomy: Solving motion equations
How to Use the Pythagorean Theorem Calculator
This calculator uses the Pythagorean Theorem formula: a² + b² = c², where:
- a and b are the lengths of the two shorter sides of a right triangle (the legs).
- c is the length of the longest side (the hypotenuse).
To use the calculator:
- Enter the known values for any two sides (a, b, or c).
- Leave the side you want to calculate blank.
- Click the "Calculate" button to find the missing side.
- The calculator will also display additional triangle properties such as area, perimeter, angles, inradius, and circumradius.
- Click "Reset" to clear all fields and start over.
Example: If you know sides a = 3 and b = 4, the calculator will find c = 5, completing the classic 3-4-5 triangle.
Frequently Asked Questions
The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically: c² = a² + b².
You can use the Pythagorean Theorem only in right-angled triangles. It helps to find the length of a side when the other two sides are known.
The hypotenuse is the longest side of a right triangle. It is always opposite the 90-degree angle and is represented as c in the formula c² = a² + b².
No. The Pythagorean Theorem is valid only for right-angled triangles. For non-right triangles, use the Law of Cosines.
The theorem is named after the Greek philosopher Pythagoras, but it was known to Babylonian and Indian mathematicians centuries before him.
If the sides of a triangle satisfy a² + b² = c², then the triangle is a right triangle. This is a useful test when you know all three sides.
It's used in construction to ensure corners are square, in navigation to calculate direct distances, and in computer graphics to measure pixel distances.