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Pythagorean Theorem Calculator

Solve the missing side of a right triangle using a^2 + b^2 = c^2, with area and perimeter.

  • pythagorean theorem
  • hypotenuse
  • right triangle
  • triangle side
  • a squared plus b squared

Choose which right-triangle side is missing.

One leg of the right triangle. Required unless solving for side a.

The other leg of the right triangle. Required unless solving for side b.

The side opposite the right angle. Required when solving for a leg.

Status: initial

Results

Awaiting calculation

Explore your result

Right-triangle proportions

Explore the pythagorean theorem calculator result visually while keeping this page's detailed guidance and examples.

Right-triangle proportions
Side a3
Side b4
Hypotenuse5
Area6

Use this result well

  1. 1Verify the calculator inputs.
  2. 2Compare the key result relationships.
  3. 3Review the page guidance before acting.

The Pythagorean theorem applies only to right triangles.

Calculator guide

Introduction

The Pythagorean Theorem Calculator solves the missing side of a right triangle, shows the substitution steps, and adds useful right-triangle details such as area, perimeter, acute angles, altitude to the hypotenuse, exact radical form, and Pythagorean triple checks.

What is the Pythagorean theorem?

The theorem says that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. It applies only when the angle between the two legs is exactly 90 degrees.

Right triangle method

Choose the side you need to find, enter the other two side lengths, and the calculator rearranges a^2 + b^2 = c^2. The result keeps the side labels clear: a and b are legs, while c is always the hypotenuse.

Variable explanations

Understand what each input and result means before calculating.

Side a

One leg of the right triangle. It meets side b at the right angle.

Side b

The other leg of the right triangle.

Hypotenuse c

The longest side, opposite the right angle. It must be longer than either leg.

Missing side

The side selected in the solve-for control.

Exact missing side

A simplified radical is shown when the squared value is an integer and the result is not already a whole number.

Area

For a right triangle, area = a x b / 2.

Perimeter

The sum of all three side lengths.

Altitude h

The perpendicular height from the right-angle corner to the hypotenuse.

Reviewed by the Calculator.org.in Editorial Team

Formula behavior, validation cases, explanatory examples, and cited sources are checked before publication. This review supports educational accuracy and is not a substitute for qualified professional advice.

Last reviewed: 2026-08-03

Review process

Formula guide

See the calculation logic, variable definitions, and practical meaning.

Find hypotenuse

c = sqrt(a^2 + b^2)

  • a and b are the known legs.
  • c is the hypotenuse opposite the right angle.

Use this for diagonals, ladders, ramps, screens, rooms, or any right triangle where both perpendicular sides are known.

Find leg a

a = sqrt(c^2 - b^2)

  • b is the known leg.
  • c is the hypotenuse and must be greater than b.

Use this when the diagonal or hypotenuse is known and one leg is missing.

Find leg b

b = sqrt(c^2 - a^2)

  • a is the known leg.
  • c is the hypotenuse and must be greater than a.

Use this when side a and the hypotenuse are known.

Right triangle area

Area = a x b / 2

  • a and b are the perpendicular legs.

The legs act as base and height, so the area does not require the hypotenuse directly.

Perimeter

Perimeter = a + b + c

  • a, b, and c are the three side lengths.

This helps with edging, trim, boundaries, and geometry homework checks.

Altitude to hypotenuse

h = a x b / c

  • h is the altitude from the right angle to the hypotenuse.

This is useful when comparing right-triangle height, area, and similar-triangle relationships.

Acute angles

A = atan(a / b), B = 90 - A

  • A and B are the two acute angles.

The calculator estimates the non-right angles after the side lengths are known.

Worked examples

Follow realistic inputs through the calculation step by step.

1

Worked example

Classic 3-4-5 triangle

  1. 1Enter a = 3 and b = 4.
  2. 2Use c = sqrt(3^2 + 4^2) = sqrt(25).
  3. 3The missing hypotenuse is 5, area is 6, and perimeter is 12.
2

Worked example

Missing leg from a 5-12-13 triangle

  1. 1Choose leg b, enter a = 5 and c = 13.
  2. 2Use b = sqrt(13^2 - 5^2) = sqrt(144).
  3. 3The missing leg is 12.
3

Worked example

Rectangle diagonal

  1. 1A rectangle is 6 by 8 in the same unit.
  2. 2The diagonal is the hypotenuse: c = sqrt(6^2 + 8^2).
  3. 3The diagonal is 10.
4

Worked example

Exact radical result

  1. 1Enter a = 2 and b = 3.
  2. 2The hypotenuse is sqrt(13).
  3. 3The decimal result is about 3.605551, while the exact radical remains sqrt(13).

Common mistakes

Avoid these common input and interpretation errors.

Using it for non-right triangles

The theorem applies to right triangles only. For non-right triangles, use trigonometry or the law of cosines.

Treating a leg as the hypotenuse

The hypotenuse is always the longest side and always opposite the right angle.

Entering a hypotenuse shorter than a leg

A valid right triangle cannot have c less than or equal to a known leg.

Forgetting the square root

After adding or subtracting squared values, take the square root to return to a side length.

Mixing units

Use the same unit for both known sides. Do not mix feet and inches unless you convert first.

Rounding too early

For construction or geometry proofs, keep more decimal places until the final answer.

Assuming every whole-number answer is primitive

A triangle such as 6-8-10 is a Pythagorean triple, but it is a scaled version of 3-4-5.

Frequently asked questions

Quick answers to the questions users ask most often.

What does this Pythagorean Theorem Calculator do?
It solves a missing side of a right triangle using a^2 + b^2 = c^2 and also reports area, perimeter, angles, altitude, steps, and triple checks.
What is the Pythagorean theorem formula?
The formula is a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse.
Which side is the hypotenuse?
The hypotenuse is the longest side of a right triangle and is opposite the 90 degree angle.
Can this find the hypotenuse?
Yes. Choose hypotenuse c and enter both legs.
Can this find a missing leg?
Yes. Choose leg a or leg b and enter the hypotenuse plus the other known leg.
Why does the hypotenuse have to be longer than a leg?
In a right triangle, the hypotenuse is opposite the largest angle, so it is the longest side.
Does this calculator work for every triangle?
No. It is only valid for right triangles. For other triangles, use a general triangle calculator or the law of cosines.
Can I use decimals?
Yes. Decimal side lengths are supported.
What units should I use?
Use any consistent length unit. The side, perimeter, and altitude results use that unit, while area uses the square of that unit.
How is the area calculated?
For a right triangle, area = a x b / 2 because the two legs are perpendicular.
How is perimeter calculated?
Perimeter is a + b + c.
What is altitude to the hypotenuse?
It is the perpendicular height from the right-angle vertex to the hypotenuse. This calculator uses h = a x b / c.
What are Pythagorean triples?
They are integer side lengths that satisfy a^2 + b^2 = c^2, such as 3-4-5, 5-12-13, and 8-15-17.
What is a primitive Pythagorean triple?
It is a Pythagorean triple where the three side lengths have no common factor greater than 1.
Can this show exact radicals?
Yes, when the squared value is an integer, the calculator shows a simplified radical such as sqrt(13) or 3sqrt(5).
Why is my exact result different from the decimal result?
The radical is exact, while the decimal is rounded for readability.
Can this check a construction diagonal?
Yes. If two runs meet at a right angle, the diagonal between their endpoints is the hypotenuse.
Can this help with the 3-4-5 rule?
Yes. The calculator identifies integer triples and scaled triples that are often used for right-angle checks.
What if my side lengths are very large?
Each side must be greater than 0 and no more than 1,000,000,000 to keep results practical and reliable.
Do a and b have to be entered in a specific order?
No. The two legs can be swapped. Only c must be the hypotenuse.
Does this replace a full right triangle calculator?
No. It focuses on Pythagorean side solving, but it adds common supporting measurements after the sides are known.

References

Sources used to support the calculator guidance.

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Version history

A transparent record of calculator content updates.

Updated 2026-08-03
  • 1.1.0 · 2026-08-03

    Added exact radical output, angle estimates, altitude, triple classification, richer step tables, and expanded SEO content.

  • 1.0.0 · 2026-07-06

    Initial Pythagorean theorem calculator release.