Hypotenuse Calculator

Find the hypotenuse c of a right triangle from the two legs, from one leg and an acute angle, or from the area and one leg, with the working, the exact radical and a scaled drawing.

By the PythagoreanTheorem.net teamUpdated Free · no sign-up

c = 3√13 ≈ 10.8167

Exact value in simplest radical form, then the decimal rounded to 4 decimal places.

  • Area27
  • Perimeter25.8167
  • Angle A33.69°
  • Angle B56.31°
a = 6b = 9c = 10.8167

Step-by-step solution

  1. Write the theorem: a2 + b2 = c2
  2. Substitute the two legs: 62 + 92 = c2
  3. Square each leg: 36 + 81 = c2
  4. Add: c2 = 117
  5. Take the square root (a length is never negative): c = √117 = 3√13 ≈ 10.8167
Decimals (2.5), fractions (3/4) and roots (2√3, sqrt(8)) all work.

To find the hypotenuse, square both legs, add the results and take the square root: c = √a2 + b2. For legs 6 and 8 that is √36 + 64 = √100 = 10. The hypotenuse is the side across from the right angle, and it is always the longest of the three sides.

How to use the hypotenuse calculator

  1. Choose what you know. Legs, with both legs given, is the usual case. Pick Leg + angle when a problem gives one leg and an acute angle, or Area + leg when it gives the triangle's area.
  2. Enter the numbers. The legs are the two sides that form the right angle. In leg + angle mode, say whether the angle is opposite the leg or next to it; the drawing marks the angle so you can check you picked the right one.
  3. Read the answer. You get c as an exact value when there is a tidy one (such as 3√13), the rounded decimal, every step of the working, the other measurements and a drawing to scale.

Units are labels only, so both lengths must already be in the same unit. The page address updates as you type, which lets you bookmark or share a calculation, for example leg 5 with a 30° angle opposite it.

Hypotenuse formulas for each mode

abc = ?ac = ?AAreaac = ?Two legsLeg + angleArea + legc = √(a² + b²)c = a ÷ sin Ab = 2 · Area ÷ a,then c = √(a² + b²)
The three modes of the calculator. Whatever you start with, the last step is always the Pythagorean theorem or a trig ratio that leads to the same c.

From two legs

This is the Pythagorean theorem formula solved for c:

c = √a2 + b2

Square first, add second, take the root last. The order matters: √a2 + b2 is not the same as a + b, one of the slips listed on the common mistakes page.

From one leg and an acute angle

If you know leg a and the angle A across from it, the sine ratio links them: sinA = a/c. If the known angle B sits next to leg a, use cosine instead: cosB = a/c. Either way, rearrange for c:

c = a over sinA  c = a over cosB

The calculator then finds the other leg with the tangent and checks all three sides with a2 + b2 = c2. For when to reach for trigonometry instead of the theorem, see Pythagorean theorem or trigonometry?

From the area and one leg

In a right triangle the two legs are a base and its height, so the area is 1 over 2ab. Solve that for the missing leg, then use the theorem:

b= 2 · Area over ac= √a2 + b2

Worked examples

Example 1: two whole-number legs

The legs are 9 cm and 12 cm.

c2= 92 + 122 = 81 + 144 = 225c= √225 = 15

The hypotenuse is 15 cm. This is the 3-4-5 triangle scaled up by 3.

Example 2: an answer that is a radical

The legs are 4 and 7.

c2= 42 + 72 = 16 + 49 = 65c= √65 ≈ 8.0623

65 has no square factor except 1, so √65 is already in simplest form. Rounded to the nearest hundredth, c ≈ 8.06.

Example 3: a leg and the angle opposite it

Leg a = 5, and the angle across from it is A = 30°.

c = 5 over sin30° = 5 over 0.5 = 10

The hypotenuse is 10, twice the short leg, as in every 30-60-90 triangle. The other leg is 5√3 ≈ 8.6603.

Example 4: a leg and the angle next to it

Leg a = 8, and the angle beside it is B = 40°. Since cos40° ≈ 0.7660:

c = 8 over cos40° ≈ 10.4433

c ≈ 10.44. Check: the other leg is 8tan40° ≈ 6.7128, and 82 + 6.71282 ≈ 109.06, which matches 10.44332 ≈ 109.06.

Example 5: the area and one leg

A right triangle has an area of 30 square units and one leg of 5 units.

b= 2 · 30 over 5 = 12c= √52 + 122 = √169 = 13

The hypotenuse is 13 units: the 5-12-13 triangle.

Common leg pairs and their hypotenuses

These pairs come up again and again in homework. The whole-number rows are Pythagorean triples; the others have radical answers, shown in simplest form and rounded to four decimal places.

Leg aLeg bExact cDecimal c
11√21.4142
12√52.2361
13√103.1623
23√133.6056
333√24.2426
242√54.4721
3455
555√27.0711
47√658.0623
681010
693√1310.8167
5121313
101010√214.1421
9121515
8151717
7242525
20212929
9404141

To simplify a radical like √117 = 3√13 yourself, pull out the largest square factor (here 117 = 9 · 13). The square root simplifier shows the factor steps.

Check the answer: the shortcut rule

a = 3b = 4c = 5Along the legs: 3 + 4 = 7Straight across: c = 54 < 5 < 7longer than either leg,shorter than both together
The hypotenuse is a shortcut across the corner. That gives a quick check on any answer: c must be longer than the longer leg and shorter than a + b.

The hypotenuse is the straight path across the corner, so it can never be longer than walking along both legs, and it can never be shorter than the longer leg:

max(a, b) < c < a + b

Two quick estimates catch most typing slips:

  • Equal legs: c = a√2 ≈ 1.414a, so legs of 10 give about 14.14. This is the 45-45-90 triangle.
  • One leg much longer than the other: c is only a little more than the long leg, roughly b + a2 over 2b. For legs 3 and 40 the estimate is 40 + 9 over 80 = 40.1125; the exact answer is √1609 ≈ 40.1123.

Degrees, radians and special angles

In leg + angle mode, this calculator always works in degrees. A handheld calculator may not: in radian mode, sin30 means the sine of 30 radians, about −0.988, so 5 ÷ sin30 comes out near −5.06. A negative length is the giveaway. Switch the calculator to degree mode (DEG) and you get 10.

Three angles give exact answers you can find without a calculator. With leg a = 1 and the angle A opposite it:

Angle AExact cDecimal cWhy
30°22sin30° = 1 over 2: the hypotenuse is twice the short leg
45°√21.4142sin45° = √2 over 2: both legs are equal
60°2√3 over 31.1547sin60° = √3 over 2: here a is the long leg

For any other leg length, multiply the row by that length. The special right triangle calculator handles these cases from any starting side.

Frequently asked questions

What is the hypotenuse of a right triangle?

It is the side opposite the 90° angle. It is always the longest side, and it is the c in a2 + b2 = c2. The other two sides, which meet at the right angle, are the legs.

Can I find the hypotenuse if I only know one side?

Not from one length alone, because infinitely many right triangles share a leg of 5. You need one more fact: the other leg, an acute angle, the area or the perimeter. This calculator handles the first three.

How do I find the hypotenuse of a 45-45-90 triangle?

Multiply a leg by √2. With legs of 7, c = 7√2 ≈ 9.8995, the same as √72 + 72 = √98.

Can the legs be decimals or square roots?

Yes. Type 2.5, 3/4 or sqrt(5) directly. Legs of √5 and 2 give c = √5 + 4 = 3, a whole number, and legs of 1.5 and 2 give exactly 2.5.

Is the hypotenuse always longer than both legs?

Always. Since c2 = a2 + b2 and both squares are positive, c2 is bigger than a2 and bigger than b2, so c is longer than either leg. An answer shorter than a leg means a number went into the wrong box.

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