Triangle Ratio Calculator

Created by Anna Szczepanek, PhD
Reviewed by Davide Borchia
Last updated: Oct 11, 2022

We've built this triangle ratio calculator so that you can easily find missing angles in triangles using ratios. We'll discuss the math formula for triangle ratio as well as lead you through some examples of solving the angle ratio of a triangle.

Triangle ratio formula

If we deal with the angle ratio of a triangle, the formula we need to remember is the one saying that the sum of angles in a triangle is equal to the straight angle, i.e., to 180°:

α + β + γ = 180°.

How to make use of this formula? If you know your angles are in the ratio a:b:c, you can write them as ax, bx, cx, where x is unknown. And, according to the above formula, we obtain:

ax + bx + cx = 180°

From this you can determine first the unknown x, and then the angles: ax, bx, cx. In the next section, we translate these considerations into a step-by-step guide on how to find missing angles in triangles using ratios.

How do I find missing angles in triangles using ratios?

If you know the angles are in the ratio a:b:c and want to determine angles:

  1. Write the unknown angles as ax, bx, cx.
  2. Use the fact that they add up to the straight angle: ax + bx + cx = 180°.
  3. Simplify the equation: (a + b + c)x = 180°.
  4. Compute x = 180°/(a + b + c).
  5. Use x to determine the missing angles as ax, bx, cx.
  6. If you need the ratio of sides as well, use the law of sines.

Let's go together through examples to see how it all works in practice:

Example 1

The angle ratio now reads 2:3:4.

Write the angles as 2x, 3x, 4x, and write the equation for their sum as 2x + 3x + 4x = 180°. After simplification, we obtain 9x = 180° and then x = 20°. As a result, 2x = 40°, 3x = 60°, 4x = 80°. This means our missing angles are 40°, 60°, 80°.

Example 2

Let's say the angles are in the ratio of 1:1:2.

We write the angles as x, x, 2x. Since their sum is 180°, we get: x+x+2x = 180°. Hence, 4x = 180°, which means that x = 45°. Very easily we get 2x = 90°, and so the missing angles are 45°, 45°, 90°. In particular, this is a right triangle!

How to use this triangle ratio calculator?

Our triangle ratio calculator can work in two ways:

  • enter the angle ratio and get the angles; or
  • enter the angles and get the angle ratio.

Make sure to test both possibilities!

💡 To make your life easier, if you enter two angles, our triangle ratio calculator will determine the third one automatically.

As a bonus, we'll also tell you the ratio of side lengths. If you need to convert the side length to angles, go to SSS triangle calculator.

There are two famous angle ratios for triangles: 1:2:3 and 1:1:2. Both these triangles are right triangles and the latter is also isosceles (and we've met it already!). Read more about them here:

FAQ

How do I find the ratio of angles in a triangle?

To find the ratio of angles in a triangle:

  1. Take the triangle's angles: α, β, and γ.
  2. Write them down as α:β:γ. This is your ratio!. But you may want to simplify it.
  3. Divide all three numbers by their greatest common divisor.
  4. For instance, if your ratio is 30:60:90, divide all three numbers by thirty: 1:2:3.

What are the angles of a triangle with ratio 3 : 4 : 5?

  1. Write your angles using the given ratio as 3x, 4x, 5x.
  2. The sum of angles in a triangle is the straight angle: 3x + 4x + 5x = 180°.
  3. Solve this equation for x. We get x = 15°.
  4. The missing angles are 3x = 45°, 4x =60°, 5x = 75°.
Anna Szczepanek, PhD
Angle ratio x : y : z
x
y
z
Angles
α
deg
β
deg
γ
deg
Side length ratio
Input data to get the results!
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