Use this Pythagorean Theorem Calculator to find the missing side of a right triangle. Enter the lengths of any two known sides to calculate the third side, view the formula, and follow the calculation step by step.
Triangle Details
Pythagorean Theorem Result
How to Use the Pythagorean Theorem Calculator
- Choose which side you want to calculate: the hypotenuse, side a, or side b.
- Enter the two known side lengths.
- Choose how many decimal places you want in the answer.
- The result updates automatically and shows the missing side along with the calculation steps.
Use positive numbers only. If you are calculating a missing leg, the hypotenuse must be longer than the known leg.
What Is the Pythagorean Theorem?
The Pythagorean theorem is a mathematical rule used to find the sides of a right triangle. It relates the two shorter sides, called legs, to the longest side, called the hypotenuse.
a² + b² = c²
The theorem works only for right triangles, which have one angle measuring 90 degrees.
Pythagorean Theorem Formulas
Depending on which side is missing, use the appropriate formula:
- To find the hypotenuse: Take the square root of the sum of the squares of sides a and b.
- To find side a: Take the square root of the hypotenuse squared minus side b squared.
- To find side b: Take the square root of the hypotenuse squared minus side a squared.
Pythagorean Theorem Examples
Example 1: Find the hypotenuse
Given: side a = 3 and side b = 4
Calculation: √(32 + 42) = √25 = 5
Answer: 5 units
Example 2: Find the missing leg
Given: hypotenuse = 13 and side b = 5
Calculation: √(132 − 52) = √144 = 12
Answer: 12 units
Example 3: Find another missing leg
Given: hypotenuse = 10 and side a = 6
Calculation: √(102 − 62) = √64 = 8
Answer: 8 units
Where Is the Pythagorean Theorem Used?
The Pythagorean theorem is useful in many everyday and academic situations, including:
- School mathematics: Solving right-triangle geometry problems.
- Construction: Checking right angles and calculating diagonal measurements.
- Architecture: Finding diagonal distances in building plans.
- Navigation: Calculating straight-line distances from perpendicular distances.
- DIY projects: Measuring the diagonal of a rectangular area or frame.
Frequently Asked Questions
1. What does the Pythagorean theorem calculate?
It helps you find a missing side of a right triangle when the other two side lengths are known.
2. What is the hypotenuse?
The hypotenuse is the longest side of a right triangle. It is always opposite the 90-degree angle.
3. Can I use this calculator for any triangle?
No. The Pythagorean theorem applies only to right triangles. It does not directly apply to triangles without a 90-degree angle.
4. Can the calculator find a missing leg?
Yes. Select either “Missing leg a” or “Missing leg b,” then enter the known leg and hypotenuse.
5. Can a hypotenuse be shorter than a leg?
No. The hypotenuse must be longer than either leg. If your values do not meet this condition, the calculator will display an error.
6. Does the calculator support decimal numbers?
Yes. You can enter decimal side lengths and choose the number of decimal places shown in the result.
7. What is a 3-4-5 triangle?
A 3-4-5 triangle is a right triangle with side lengths of 3, 4, and 5 units. It is a common example used to demonstrate the Pythagorean theorem.