Hypotenuse Calculator

Find the longest side of a right triangle from its two perpendicular sides. Enter the leg lengths to calculate the hypotenuse, area and perimeter with unit conversion and clear working.

Calculate Your Hypotenuse

Enter the two sides that meet at the 90° angle. Either leg can be labeled a or b.

One of the two perpendicular sides, not the hypotenuse.

The other side meeting leg a at the right angle.

a b c 90°
c is the hypotenuse, opposite the 90° angle. Diagram is illustrative and does not resize with your inputs.

The legs can use different units. Changing an input unit converts its entered length. Results update automatically as you type.

Only one side known? If your triangle has angles of 30°, 60° and 90°, use the 30 60 90 triangle calculator to find the other sides from its fixed ratio.

QUICK START

How to Use the Hypotenuse Calculator

Enter the two sides that form the right angle. The calculator applies the Pythagorean theorem to find the side opposite that angle.

  1. Identify the two perpendicular legs

    Find the sides meeting at the 90° angle and label them a and b. Either leg can go in either field. The sloping side opposite the right angle is the hypotenuse you are calculating.

  2. Enter both lengths and their units

    Enter positive measurements, including decimals. Choose the unit beside each leg. You can use different units for a and b; the calculator converts them before applying the formula.

  3. Choose result units and decimal places

    Select how you want the results displayed. Hypotenuse and perimeter use length units, while area uses square units. Choose two, four or six decimal places for rounding.

  4. Review the working and copy the result

    Follow the calculation from a² + b² = c² to the positive square root. Check the area and perimeter, then select Copy Result to copy your answers or Clear to enter another triangle.

Confirm the right angle: This tool assumes the entered sides meet at 90°. Two lengths alone do not determine the third side of a triangle when their included angle is unknown.

WORKED EXAMPLES

Hypotenuse Calculations Step by Step

Square the two perpendicular leg lengths, add their squares and take the positive square root. When the measurements use different units, convert them first.

EXAMPLE 01 · SAME UNITS

Find the hypotenuse from 5 cm and 12 cm

A right triangle has perpendicular legs measuring 5 centimeters and 12 centimeters. Both lengths already use the same unit.

GIVEN MEASUREMENTS
Leg a
5 cm
Leg b
12 cm
  1. Square each leg length

    a² = 5² = 25 cm²

    b² = 12² = 144 cm²

  2. Add the squares

    c² = 25 + 144 = 169 cm²

  3. Take the positive square root

    c = √169 = 13 cm

RESULT Hypotenuse = 13 cm

Area = 5 × 12 ÷ 2 = 30 cm². Perimeter = 5 + 12 + 13 = 30 cm.

EXAMPLE 02 · MIXED UNITS

Convert the legs before finding the diagonal

A rectangle measures 80 centimeters by 1.5 meters. Its diagonal forms the hypotenuse of a right triangle with those two dimensions as the legs.

GIVEN MEASUREMENTS
Leg a
80 cm
Leg b
1.5 m
  1. Convert centimeters to meters

    80 cm = 0.8 m

  2. Square and add in consistent units

    c² = 0.8² + 1.5²

    c² = 0.64 + 2.25 = 2.89 m²

  3. Take the positive square root

    c = √2.89 = 1.7 m

RESULT Diagonal = 1.7 m = 170 cm

The right triangle’s area is 0.6 m² and its perimeter is 4 m. The complete rectangle’s area is 1.2 m².

Your answer does not have to be a whole number

With legs of 2 cm and 3 cm, the hypotenuse is √13 cm ≈ 3.6056 cm. Keep √13 for an exact mathematical answer, or round the decimal to the precision your task requires. Try your own leg lengths to follow the calculation.

UNDERSTAND THE FORMULA

Hypotenuse Formula and Pythagorean Theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. Take the positive square root to turn that squared value into a length.

a b c 90°
Find the side opposite the right angle c = √(a² + b²). The hypotenuse is longer than either individual leg when both leg lengths are positive.
a b c 90°
A rectangle’s diagonal uses the same formula Length and width form the perpendicular legs. The diagonal is the hypotenuse of either triangle.
a and b are perpendicular legs; c is the hypotenuse. Convert all lengths to the same unit before substituting. Diagrams are illustrative.
Measurement Formula Units
Hypotenuse c = √(a² + b²) Length, e.g. cm
Squared hypotenuse c² = a² + b² Square, e.g. cm²
Triangle area A = ab ÷ 2 Square, e.g. cm²
Triangle perimeter P = a + b + c Length, e.g. cm
Equal legs of length a c = a√2 Length, e.g. cm
Missing leg, if c and b are known a = √(c² − b²) Length; requires c > b

Why not simply add the two legs?

Adding a and b gives the distance along two sides. The hypotenuse connects their endpoints directly. For positive legs, it is shorter than a + b but longer than either a or b on its own.

Keep length and squared length separate

Squaring a length in centimeters produces a value in cm². Adding the squares gives c², not c. Taking the square root returns the answer to centimeters. The area result remains in square units.

Calculator scope: Enter both legs to calculate the hypotenuse automatically. The missing-leg formula above is a manual reference. For a triangle without a right angle, the third-side formula also needs the included angle: c² = a² + b² − 2ab cos(θ).

COMMON QUESTIONS

Frequently Asked Questions

Answers about identifying the hypotenuse, choosing measurements, converting units and interpreting the results.

How do I identify the hypotenuse?

Find the 90° angle. The side opposite it is the hypotenuse, and it is the longest side of the right triangle. The two sides meeting at that angle are the legs. These names stay the same when the triangle is rotated; the hypotenuse does not have to look like a sloping line on the page.

What formula calculates the hypotenuse?

Use c = √(a² + b²), where a and b are the perpendicular legs and c is the hypotenuse. Square both lengths, add the squares and take the positive square root. You can explore a visual explanation of the Pythagorean theorem on Math Is Fun.

Can I find the hypotenuse from only one leg?

Not for a general right triangle. You need the other leg or additional information, such as an acute angle. If an angle θ is opposite your known leg a, then c = a ÷ sin(θ). If θ is adjacent to that leg, use c = a ÷ cos(θ). This calculator’s input fields require both legs.

Can the two legs use different units?

Yes. Select the matching unit beside each input. The calculator converts both legs to consistent units before calculating. For example, legs of 30 cm and 0.4 m are equivalent to 0.3 m and 0.4 m, giving a hypotenuse of 0.5 m.

Can I calculate a rectangle’s diagonal with this tool?

Yes. Enter the rectangle’s length and width as the two legs. Its diagonal is the hypotenuse. The area and perimeter shown by this tool belong to the right triangle formed by that diagonal. They are not the complete rectangle’s area and perimeter.

What happens when the two legs are equal?

Equal perpendicular legs form a 45°–45°–90° triangle. If each leg has length a, then c = a√2. For example, two 10 cm legs give a hypotenuse of 10√2 cm ≈ 14.1421 cm.

Can I enter a hypotenuse and find a missing leg?

This tool calculates the hypotenuse from two legs. To find a missing leg manually, rearrange the theorem to a = √(c² − b²). Use the same units for c and b, and make sure the hypotenuse is longer than the known leg. For c = 13 cm and b = 12 cm, a = 5 cm.

Does this formula work when the angle is not 90°?

No. For two sides a and b with included angle θ, use the law of cosines: c = √(a² + b² − 2ab cos(θ)). It reduces to the hypotenuse formula at 90° because cos(90°) = 0. This calculator assumes a right angle and cannot verify it from two entered lengths alone.

HAVE BOTH LEG LENGTHS?

Find Your Triangle’s Hypotenuse

Enter the two perpendicular sides to find the longest side of your right triangle. Choose your units and follow the calculation from formula to answer.

Use Hypotenuse Calculator