Pythagorean Theorem Calculator - Find Hypotenuse, Legs & Angles

Free Online Pythagorean Theorem Calculator

Our Pythagorean theorem calculator is a powerful, user-friendly tool designed to solve right triangle problems quickly and accurately. Whether you're calculating the hypotenuse, finding missing legs, determining angles, or computing area and perimeter, this calculator provides instant results with detailed step-by-step solutions.

What is the Pythagorean Theorem?

The Pythagorean theorem is a fundamental principle in geometry that relates the three sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

The mathematical formula is: a² + b² = c²

Where:

a and b are the lengths of the two legs (sides adjacent to the right angle)
c is the length of the hypotenuse (side opposite the right angle)

What Can You Calculate?

Our Pythagorean calculator supports five main types of calculations:

1. Find Hypotenuse (c)

Calculate the length of the hypotenuse when you know the lengths of both legs. Perfect for finding the diagonal distance, rope length, or longest side of a right triangle.

2. Find Leg (a or b)

Calculate the length of a missing leg when you know one leg and the hypotenuse. Useful for finding the height of buildings, width of rectangles, or any missing side measurement.

3. Find Angles

Calculate the acute angles of a right triangle when you know all three side lengths. Essential for determining angles in construction, engineering, and navigation.

4. Find Area

Calculate the area of a right triangle using the two legs. Useful for determining the area of triangular sections, roof areas, or land measurements.

5. Find Perimeter

Calculate the total perimeter of a right triangle when you know all three side lengths. Helpful for fencing, trim work, or any project requiring the total length around a triangle.

How to Use the Calculator

Select Calculation Type: Choose from the dropdown menu which type of calculation you need.
Enter Values: Input the known side lengths in the appropriate fields.
Get Results: The calculator will automatically compute and display the result with step-by-step calculations.
Copy Results: Use the copy button to copy results to your clipboard.

Mathematical Formulas

Pythagorean Theorem

c² = a² + b²

To find hypotenuse: c = √(a² + b²)

To find leg: a = √(c² - b²) or b = √(c² - a²)

Area of Right Triangle

Area = (1/2) × a × b

Perimeter of Right Triangle

Perimeter = a + b + c

Angles in Right Triangle

Angle A = arctan(a/b)

Angle B = arctan(b/a)

Angle C = 90° (right angle)

Practical Applications

Construction and Engineering

Calculating diagonal measurements for square foundations
Determining roof pitch and rafter lengths
Finding the length of support beams and braces
Calculating distances for surveying and mapping

Everyday Life

Finding the diagonal of a TV screen or computer monitor
Calculating the shortest distance between two points
Determining the length of ladder needed to reach a certain height
Finding the diagonal of a rectangular room or garden

Mathematics and Science

Solving geometry problems in school and college
Calculating distances in coordinate geometry
Finding components in physics problems
Analyzing data in statistics and research

Common Right Triangle Ratios

Some right triangles have special ratios that are worth memorizing:

3-4-5 Triangle

If the legs are 3 and 4 units, the hypotenuse is 5 units. This works for any multiple: 6-8-10, 9-12-15, etc.

5-12-13 Triangle

If the legs are 5 and 12 units, the hypotenuse is 13 units. Multiples include 10-24-26, 15-36-39, etc.

8-15-17 Triangle

If the legs are 8 and 15 units, the hypotenuse is 17 units. Multiples include 16-30-34, 24-45-51, etc.

Tips for Accurate Calculations

Always verify your triangle is right-angled: The largest side should be the hypotenuse, and a² + b² should equal c².
Use appropriate units: Make sure all measurements are in the same units (inches, feet, meters, etc.).
Check your work: Verify your answer by plugging it back into the Pythagorean theorem.
Round appropriately: For practical applications, round to a reasonable number of decimal places.

Frequently Asked Questions

What is the Pythagorean theorem used for?

The Pythagorean theorem is used to find missing side lengths in right triangles. It's commonly used in construction, engineering, navigation, and everyday problem-solving when you need to calculate distances, diagonals, or verify that a triangle is right-angled.

Can I use this calculator for any triangle?

No, this calculator is specifically designed for right triangles (triangles with one 90-degree angle). For other types of triangles, you would need different formulas like the Law of Cosines or Law of Sines.

What if I get a negative number under the square root?

If you get a negative number under the square root when trying to find a leg, it means the values you entered don't form a valid right triangle. The hypotenuse must always be longer than either leg. Check your input values and try again.

How accurate are the calculations?

Our calculator provides results accurate to 6 decimal places. For most practical applications, this level of precision is more than sufficient. You can round the final answer to an appropriate number of decimal places based on your needs.

Question not found

Yes! When you select "Find Angles" and enter all three side lengths, the calculator will determine the two acute angles of the right triangle using inverse trigonometric functions. The right angle (90°) is always present in a right triangle.

What units should I use?

You can use any units you prefer (inches, feet, meters, centimeters, etc.), but make sure all measurements are in the same units. The calculator will work with any consistent unit system.

Is the Pythagorean theorem only for right triangles?

Yes, the Pythagorean theorem (a² + b² = c²) only applies to right triangles. For other types of triangles, you need different formulas. However, you can sometimes use the Pythagorean theorem as part of solving more complex problems involving non-right triangles.

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