How do you find the hypotenuse of a right triangle?

Square both legs, add them, and take the square root: c equals the square root of (a squared plus b squared). For legs of 6 and 8 that is the square root of 100, which is 10.

Right Triangle Calculator

Q: How do you find a leg when you know the hypotenuse and the other leg?

Rearrange the Pythagorean theorem to subtract: a equals the square root of (c squared minus b squared). The leg is always shorter than the hypotenuse, so the value under the root stays positive.

Q: What does SOH-CAH-TOA mean?

It is a memory aid for the three trig ratios. Sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. Choose the ratio that uses the two sides you have.

Q: How do you find the angles of a right triangle?

One angle is always 90 degrees. For an acute angle, use an inverse trig ratio such as A equals arctangent of (opposite divided by adjacent). The other acute angle is 90 degrees minus the first, since the two add to 90.

Q: Which side is the hypotenuse?

The hypotenuse is the side opposite the right angle and is always the longest side. The other two sides, which meet at the right angle, are the legs.

Enter any two values — the two legs, the hypotenuse, or a side and an acute angle — and get the rest using the Pythagorean theorem and trigonometry, drawn to scale. Click a value on the drawing to change it.

Known values

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Enter any two values, including at least one side.

Leg a

Leg b

Hypotenuse c

Angle A

Angle B

The right angle is fixed: C = 90°

Units Decimals

Click an amber value to edit it · not to printed scale

Results

The Pythagorean theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. It's the fastest way to find a missing side when you know the other two.

a² + b² = c²

Using an angle instead

If you know a side and one acute angle, trigonometry finds the rest: the opposite leg is c·sin, the adjacent leg is c·cos, and the two acute angles always add to 90°.

a = c·sin A · b = c·cos A · B = 90° − A

What it calculates

Both legs, the hypotenuse, both acute angles, area, perimeter, the altitude to the hypotenuse, and the inscribed- and circumscribed-circle radii.

Solving a right triangle

A right triangle has one 90° angle, and the side opposite it — the longest side — is the hypotenuse. That single right angle makes these triangles much quicker to solve than ordinary ones. There are two ways in, depending on what you start with: the Pythagorean theorem when you know two sides, and trigonometry when you know a side and one of the acute angles.

Two ways in

Two sides → use Pythagoras. The relationship a² + b² = c² ties the three sides together, so knowing any two gives the third.
A side and an acute angle → use trigonometry. Sine, cosine and tangent connect an angle to the ratio of two sides, so one side plus one angle unlocks the rest.

Worked example: two legs (Pythagoras)

Suppose the two legs are a = 6 and b = 8, and you need the hypotenuse, both angles and the area.

Step 1 — hypotenuse, by the Pythagorean theorem

c = √(a² + b²) = √(6² + 8²) = √(36 + 64) = √100 = 10

Step 2 — an acute angle, by the tangent ratio

Angle A sits opposite side a, so tan A = opposite ÷ adjacent = a ÷ b:

A = arctan(6 ÷ 8) = arctan(0.75) = 36.87°

Step 3 — the other angle, since the acute angles add to 90°

B = 90° − A = 90° − 36.87° = 53.13°

Step 4 — area, half the product of the legs

Area = ½ · a · b = ½ · 6 · 8 = 24

Result

Legs 6 and 8, hypotenuse 10, angles 36.87° and 53.13°, area 24, perimeter 24 — the classic 6-8-10 right triangle.

Worked example: a side and an angle (trigonometry)

Now suppose you know the hypotenuse c = 10 and one acute angle A = 30°. SOH-CAH-TOA gives the legs directly.

Opposite leg, by sine (SOH)

a = c · sin A = 10 · sin 30° = 10 · 0.5 = 5

Adjacent leg, by cosine (CAH)

b = c · cos A = 10 · cos 30° = 10 · 0.8660 = 8.6603

Other angle

B = 90° − 30° = 60°

Formulas this calculator uses

Pythagoras · a² + b² = c²

SOH-CAH-TOA · sin = opp/hyp · cos = adj/hyp · tan = opp/adj

Acute angles · A + B = 90°

Area · ½ · a · b

Frequently asked questions

What is the Pythagorean theorem?

In any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². It only works for right triangles, and the hypotenuse (c) is always the side opposite the 90° angle.

How do you find the hypotenuse?

Square both legs, add them, and take the square root: c = √(a² + b²). For legs of 6 and 8, that's √(36 + 64) = √100 = 10.

How do you find a leg when you know the hypotenuse and the other leg?

Rearrange Pythagoras to subtract instead of add: a = √(c² − b²). The leg is always shorter than the hypotenuse, so the value under the square root stays positive.

What does SOH-CAH-TOA mean?

It's a memory aid for the three trig ratios in a right triangle. SOH: Sine = Opposite ÷ Hypotenuse. CAH: Cosine = Adjacent ÷ Hypotenuse. TOA: Tangent = Opposite ÷ Adjacent. Pick the one that uses the two sides you have.

How do you find the angles of a right triangle?

One angle is always 90°. For an acute angle, use an inverse trig ratio — for example A = arctan(opposite ÷ adjacent). Once you have one acute angle, the other is simply 90° minus it, because the two always add to 90°.

Which side is the hypotenuse?

The hypotenuse is the side directly opposite the right angle, and it's always the longest of the three sides. The other two sides, which meet at the right angle, are called the legs.