Hypotenuse calculator: find the longest side of a right triangle

The hypotenuse is the longest side of a right triangle, opposite the right angle. From the two legs a and b, it is c = √(a² + b²), so legs of 3 and 4 give a hypotenuse of 5. From one side and an angle, divide: c = opposite ÷ sin A, or c = adjacent ÷ cos A. This hypotenuse calculator is part of our Trigonometry Calculator, which works it out either way and gives the angles too.

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How to find the hypotenuse with the calculator

1. Open the Trigonometry Calculator. It starts on the unit circle, so at the top of the sidebar choose Standard triangle.
2. Under Triangle, find the panel that says "Type any three values, including a side, then Solve". It has boxes for the sides a, b and c and the angles A, B and C. Each side is opposite the angle with the same letter.
3. Press Right angle at C. This fills in C = 90°, so c is the hypotenuse.
4. Type what you know: the two legs in a and b, or one leg and one of the other angles. Angles are in degrees while Degrees is on.
5. Press Solve. Measurements shows c, the hypotenuse, with every other side and angle, and the triangle is redrawn to fit the view.

A line under the buttons names the method it used, such as the cosine rule for two legs or the sine rule for a side and an angle. It gives the results rather than written working, so the methods below show how to get the same answers by hand. Measurements rounds lengths to 2 decimal places.

Hypotenuse formula from two sides

Square each leg, add, and take the square root. This is the Pythagorean theorem, a² + b² = c², rearranged for c. The Pythagorean theorem calculator guide covers the theorem in full, including finding a missing leg and checking whether a triangle has a right angle.

Example 1: two legs with decimals

The legs are 2.4 m and 3.2 m.

The hypotenuse is 4 m. These are the 3, 4, 5 triangle scaled by 0.8, which is why the answer comes out whole. In the calculator, type a = 2.4 and b = 3.2, press Right angle at C, then Solve.

Example 2: a straight-line distance

You walk 300 m east and then 400 m north. How far are you from where you started, in a straight line? East and north are at right angles, so the two walks are the legs and the shortcut is the hypotenuse: √(300² + 400²) = √250,000 = 500 m, compared with 700 m walked.

Example 3: the length of a ramp

A loading ramp climbs 0.8 m and covers 5 m of ground. How long is its sloping surface?

The ramp is 5.06 m long. Solve also gives its angle: with the climb typed as a and the ground as b, angle A is 9.09°, the slope of the ramp.

Hypotenuse from an angle and a side

If you know one acute angle and one leg, use sine or cosine. Name the leg first: it is opposite the angle if it doesn't touch it, and adjacent if it does. Then:

Tangent doesn't involve the hypotenuse, so it can't give it directly. In the calculator, side a is opposite angle A and side b is next to it, so type the leg in a or b to match.

Example 4: the side opposite the angle

In a right triangle, angle A is 35° and the side opposite it is 10 cm.

The hypotenuse is 17.43 cm. In the calculator: Right angle at C, a = 10, A = 35, then Solve.

Example 5: the side next to the angle

Now the 10 cm side is adjacent to the 35° angle instead.

The hypotenuse is 12.21 cm. In the calculator, type b = 10 instead of a. The same 10 cm and 35° give two different answers, which is why naming the side comes first. Your calculator must be in degree mode for these.

Practise finding a missing side from an angle, then press Mark to check your answer in the course.

Special right triangles: hypotenuse shortcuts

A 45-45-90 triangle with legs of 6 has hypotenuse 6√2 ≈ 8.49, and the calculator's badge calls it an isosceles right triangle. In a 30-60-90 triangle the shortest leg is opposite the 30° angle, so a shortest leg of 5 gives a hypotenuse of exactly 10, because sin 30° = 1/2. These ratios come from the fact that right triangles with the same angles are similar, which is also why sine and cosine work at all.

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Checking a hypotenuse answer

• It must be the longest side. If your hypotenuse is shorter than a leg, something went wrong.
• It must be shorter than the two legs added together, because a straight line is the shortest route. In example 2, 500 m is less than 700 m.
• Check the angles add to 180°. With the right angle, the other two must add to 90°. Solve shows all three.
• Square it back. c² should equal a² + b². For example 1, 4² = 16 = 5.76 + 10.24.

Common mistakes

• Adding the legs. The hypotenuse of 3 and 4 is 5, not 7. Square, add, then square root.
• Forgetting the square root. a² + b² is c², not c.
• Multiplying by sin or cos instead of dividing. Multiplying the hypotenuse by sin A gives a leg. To get the hypotenuse from a leg, divide.
• Using sin when the side is adjacent, or cos when it is opposite. Name the side relative to the angle first.
• Radians mode. sin 35 in radians is a very different number. Check your calculator says DEG.
• Using it on a triangle without a right angle. Only right triangles have a hypotenuse. For other triangles, use the sine or cosine rule, which the calculator's solver also does from any three values.

Read next

Maths · GeometryPythagorean theorem calculator: find any side with stepsFind the hypotenuse or a missing leg of a right triangle with the Pythagorean theorem, worked examples you can check, and a right triangle test.