Given
two sides a
in
b
in
c
in
Angle α
deg
Angle β
deg

# Right Triangle Side and Angle Calculator

By Hanna Pamuła, PhD candidate

Finding out 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 remaining unknowns will be determined 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 when 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)`

1. Given angle and 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)`

1. Given angle and one leg

Find the missing leg using trigonometric functions:

• `a = b * tan(α)`

• `b = a * tan(β)`

1. Given area and one leg

As we remember from basic triangle area formula, we can calculate the area by multiplying 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 other sides:

• `b = 2 * area / a`
• `c = √(a² + (2 * area / a)²)`

## How to find the angle of a right triangle

If you know one angle apart from the right angle, 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 = arctan(b / c)` (inverse cosine)
• `tan(α) = a / b` so `α = arctan(a / b)` (inverse tangent)
• `cot(α) = b / a` so `α = arctan(b / a)` (inverse cotangent)

and for `β`

• `sin(β) = b / c` so `β = arcsin(b / c)` (inverse sine)
• `cos(β) = a / c` so `β = arctan(a / c)` (inverse cosine)
• `tan(β) = b / a` so `β = arctan(b / a)` (inverse tangent)
• `cot(β) = a / b` so `β = arctan(a / b)` (inverse cotangent)

## 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:

1. 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.
2. Type in the given values. For example, an area of a right triangle is equal to 28 in² and b = 9 in.
3. 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°
Hanna Pamuła, PhD candidate

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