Pythagorean Theorem Calculator
The Pythagorean Theorem is a fundamental concept in mathematics, widely used in geometry. It explains the relationship between the sides of a right triangle. This article will break down the theorem in a clear, easy-to-understand way.
What is the Pythagorean Theorem?
The Pythagorean Theorem states:
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
This can be written as:
\[
a^2 + b^2 = c^2
\]
Where:
- \(a\) and \(b\) are the lengths of the two shorter sides (legs).
- \(c\) is the length of the hypotenuse (the side opposite the right angle).
How Does It Work?
Step 1: Identify the Triangle
The theorem applies only to right triangles. A right triangle has one angle equal to 90 degrees.
Step 2: Label the Sides
- The hypotenuse is always the longest side.
- The other two sides are the legs.
Step 3: Apply the Formula
If you know the lengths of two sides, you can calculate the third.
Examples of Using the Theorem
Example 1: Find the Hypotenuse
A triangle has legs \(a = 3\) and \(b = 4\). What is the hypotenuse (\(c\))?
\[
a^2 + b^2 = c^2
\]
\[
3^2 + 4^2 = c^2
\]
\[
9 + 16 = c^2
\]
\[
c^2 = 25 \implies c = \sqrt{25} = 5
\]
Answer: The hypotenuse is 5.
Example 2: Find a Missing Leg
A triangle has a hypotenuse \(c = 10\) and one leg \(a = 6\). What is the other leg (\(b\))?
\[
a^2 + b^2 = c^2
\]
\[
6^2 + b^2 = 10^2
\]
\[
36 + b^2 = 100
\]
\[
b^2 = 100 – 36 = 64
\]
\[
b = \sqrt{64} = 8
\]
Answer: The missing leg is 8.
Why is the Pythagorean Theorem Important?
The theorem is a cornerstone of geometry. It helps us solve problems involving distances, angles, and shapes. Its simplicity makes it an essential tool for students and professionals alike.