Pythagorean Theorem Calculator

Units

Leg a

Leg b

Hypotenuse c

Enter any two sides to compute the third.

Side a—

Side b—

Hypotenuse c—

Angle A (opposite a)—

Angle B (opposite b)—

Area—

Perimeter—

Tip: a is drawn horizontally, b vertically, and c is the hypotenuse. The right angle is shown at the corner where a and b meet.

Pythagorean Theorem Calculator – Hypotenuse, Missing Side, Angles & Area

Solve any right triangle in seconds. Enter any two sides to calculate the third using a² + b² = c². You’ll also get angles, area, perimeter, and a clear step-by-step explanation—ideal for students, teachers, and professionals.

How the Calculator Works

This right-triangle tool uses the Pythagorean theorem to solve for any missing side and then derives angles, area, and perimeter.

1) Enter Inputs

Provide any two sides: leg a, leg b, or hypotenuse c.
Use any length unit (mm, cm, m, in, ft), but keep units consistent.

2) Core Formula

The calculator applies a² + b² = c² (Pythagorean theorem):

Hypotenuse: c = √(a² + b²)
Missing leg: a = √(c² − b²) or b = √(c² − a²)

3) Angles, Area & Perimeter

Angles: A = sin⁻¹(a / c), B = sin⁻¹(b / c)
Area: ½ × a × b
Perimeter: a + b + c

4) Validation & Errors

If all three sides are entered, the tool checks that a² + b² = c².
It warns if the hypotenuse isn’t the longest side or if inputs are incomplete.

5) Rounding & Display

Results are rounded to a sensible number of decimal places for readability.
You can copy results to your clipboard for homework or reports.

6) Tips

Make sure the triangle is right-angled (90°) for these formulas to apply.
Keep units consistent—don’t mix mm with m or in with ft.

What Is the Pythagorean Theorem?

The Pythagorean theorem states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: a² + b² = c². Here, a and b are the legs, and c is the side opposite the right angle (the hypotenuse).

Formulas You’ll Use

Hypotenuse: c = √(a² + b²)
Missing leg (given c and the other leg): a = √(c² – b²) or b = √(c² – a²)
Area: Area = ½ × a × b
Angles: A = sin⁻¹(a / c), B = sin⁻¹(b / c)

Step-by-Step Example

Example: a = 3, b = 4

Square the legs: 3² = 9, 4² = 16
Add: 9 + 16 = 25
Square root: c = √25 = 5

Result: c = 5, Area = ½ × 3 × 4 = 6, Perimeter = 3 + 4 + 5 = 12

Common Mistakes to Avoid

Using a triangle that is not a right triangle
Making a leg longer than the hypotenuse
Mixing units (e.g. mm vs m vs ft)

Frequently Asked Questions

What is the equation for the Pythagorean theorem?

The equation is a² + b² = c², where a and b are the legs and c is the hypotenuse.

How do I find the hypotenuse?

Square both legs, add them, then take the square root: c = √(a² + b²).

How do I find a missing leg?

Subtract the square of the known leg from the square of the hypotenuse, then take the square root: a = √(c² − b²) or b = √(c² − a²).

Can I check if three sides form a right triangle?

Yes. With the longest side as c, if a² + b² = c² holds (within rounding), the triangle is right-angled.

What units can I use?

Any consistent length unit (mm, cm, m, in, ft). Don’t mix units in one calculation.

What else does the calculator show?

Angles, area, and perimeter:
Angles: A = sin⁻¹(a / c), B = sin⁻¹(b / c)
Area: ½ · a · b
Perimeter: a + b + c

Why does my result look slightly different from manual math?

Displayed values are rounded for readability. The calculator uses full precision internally, but rounding inputs or results can introduce small differences.