Created by Hanna Pamuła, PhD

Reviewed by

Bogna Szyk and Jack Bowater

Last updated:

Jan 18, 2024

- How to find the sides of a right triangle
- How to find the angle of a right triangle
- How do you solve a right angle triangle with only one side?
- How to find the missing side of a right triangle? How to find the angle? Example
- FAQ

Finding the missing side or angle couldn't be easier than with our great tool – right triangle side and angle calculator. Choose two given values, type them into the calculator, and the calculator will determine the remaining unknowns in a blink of an eye! If you are wondering how to find the missing side of a right triangle, keep scrolling, and you'll find the formulas behind our calculator.

## How to find the sides of a right triangle

There are a few methods of obtaining right triangle side lengths. Depending on what is given, you can use different relationships or laws to find the missing side:

**1. Given two sides**

If you know two other sides of the right triangle, it's the easiest option; all you need to do is apply the Pythagorean theorem:

`a² + b² = c²`

If leg

`a`

is the missing side, then transform the equation to the form where`a`

is on one side and take a square root:`a = √(c² - b²)`

If leg

`b`

is unknown, then:`b = √(c² - a²)`

For hypotenuse c missing, the formula is:

`c = √(a² + b²)`

🙋 Our Pythagorean theorem calculator will help you if you have any doubts at this point.

**2. Given an angle and the hypotenuse**

Apply the law of sines or trigonometry to find the right triangle side lengths:

`a = c × sin(α)`

or`a = c × cos(β)`

`b = c × sin(β)`

or`b = c × cos(α)`

🙋 Refresh your knowledge with Omni's law of sines calculator!

**3. Given an angle and one leg**

Find the missing leg using trigonometric functions:

`a = b × tan(α)`

`b = a × tan(β)`

**4. Given the area and one leg**

As we remember from basic triangle area formula, we can calculate the area by multiplying the triangle height and base and dividing the result by two. A right triangle is a special case of a scalene triangle, in which one leg is the height when the second leg is the base, so the equation gets simplified to:

`area = a × b / 2`

For example, if we know only the right triangle area and the length of the leg `a`

, we can derive the equation for the other sides:

`b = 2 × area / a`

`c = √(a² + (2 × area / a)²)`

🙋 For this type of problem, see also our area of a right triangle calculator.

## How to find the angle of a right triangle

If you know one angle apart from the right angle, the calculation of the third one is a piece of cake:

Given `β`

: `α = 90 - β`

Given `α`

: `β = 90 - α`

However, if only two sides of a triangle are given, finding the angles of a right triangle requires applying some basic trigonometric functions:

for `α`

:

`sin(α) = a / c`

so`α = arcsin(a / c)`

(inverse sine);`cos(α) = b / c`

so`α = arccos(b / c)`

(inverse cosine);`tan(α) = a / b`

so`α = arctan(a / b)`

(inverse tangent);`cot(α) = b / a`

so`α = arccot(b / a)`

(inverse cotangent);

and for `β`

:

`sin(β) = b / c`

so`β = arcsin(b / c)`

(inverse sine);`cos(β) = a / c`

so`β = arccos(a / c)`

(inverse cosine);`tan(β) = b / a`

so`β = arctan(b / a)`

(inverse tangent);`cot(β) = a / b`

so`β = arccot(a / b)`

(inverse cotangent).

## How do you solve a right angle triangle with only one side?

To solve a triangle with one side, you also need **one of the non-right angled angles**. If not, it is impossible:

If you have the

**hypotenuse**, multiply it by**sin(θ)**to get the length of the side**opposite**to the angle.Alternatively, multiply the hypotenuse by cos(θ) to get the side adjacent to the angle.

If you have the non-hypotenuse side

**adjacent**to the angle, divide it by**cos(θ)**to get the length of the**hypotenuse**.Alternatively,

**multiply**this length by tan(θ) to get the length of the side opposite to the angle.If you have an angle and the side

**opposite**to it, you can divide the side length by**sin(θ)**to get the**hypotenuse**.Alternatively, divide the length by tan(θ) to get the length of the side adjacent to the angle.

## How to find the missing side of a right triangle? How to find the angle? Example

Let's show how to find the sides of a right triangle with this tool:

Assume we want to find the missing side given area and one side.

**Select the proper option from a drop-down list**. It's the third one.**Type in the given values**. For example, the area of a right triangle is equal to 28 in² and b = 9 in.**Our right triangle side and angle calculator displays missing sides and angles!**Now we know that:- a = 6.222 in
- c = 10.941 in
- α = 34.66°
- β = 55.34°

Now, let's check how finding the angles of a right triangle works:

Refresh the calculator.

**Pick the option you need**. Assume that we have two sides, and we want to find all angles. The default option is the right one.**Enter the side lengths**. Our right triangle has a hypotenuse equal to 13 in and a leg a = 5 in.**Missing side and angles appear**. In our example, b = 12 in, α = 22.62° and β = 67.38°.

## FAQ

### How many lines of symmetry does a right triangle have?

If a right triangle is isosceles (i.e., its two non-hypotenuse sides are the same length), it has **one line of symmetry**. Otherwise, the triangle will have **no lines of symmetry**.

### Can a right angled triangle have equal sides?

**No, a right triangle cannot have all 3 sides equal**, as all three angles cannot also be equal. **One has to be 90°** by definition. A right triangle can, however, have its two non-hypotenuse sides equal in length. This would also mean the two other angles are equal to 45°.

### Are all right triangles similar?

**Not all right-angled triangles are similar**, although some can be. They are similar if all their angles are the same length, or if the ratio of two of their sides is the same.

Hanna Pamuła, PhD