Triangle Side Calculator

Created by Purnima Singh, PhD
Reviewed by Madhu Raman
Last updated: Aug 30, 2022

Omni's triangle side calculator allows you to calculate the length of the sides of a triangle.

Continue reading to find out how to calculate the sides of a triangle for three different cases:

  • If two sides and one angle are known;
  • If two angles and one side are known; and
  • If two sides and the perimeter are known.

You will also find an example of using the calculator to find the missing sides of a triangle.

How to find the missing side of a triangle

There are several methods to find the missing side of a triangle. Depending upon the type of given information, you can choose one of the following:

Triangle with sides a,b,c and angles α, β, γ
Fig 1: Triangle with sides a,b,c and angles α, β, γ.

If two sides and one angle are known

If we know, the measurements of the two sides and the opposite angle to one of them, we can use the Law of Sines to find the missing side.

asin(α)=bsin(β)=csin(γ)\footnotesize \frac{a}{\text{sin} (\alpha)} = \frac{b}{\text{sin} (\beta)}=\frac{c}{\text{sin} (\gamma)} \\

We can use any pair of the above ratios to solve for the third side.

On the other hand, if we know two sides and the angle between them, we can use the Law of Cosines to find the remaining side.

a2=b2+c22bc cos(α)b2=a2+c22ac cos(β)c2=a2+b22ab cos(γ)\footnotesize a^2 = b^2 + c^2 - 2bc\ \text{cos} (\alpha)\\ \\ b^2 = a^2 + c^2 - 2ac\ \text{cos} (\beta)\\ \\ c^2 = a^2 + b^2 - 2ab\ \text{cos} (\gamma)

If two angles and one side are known

We know that the sum of the three angles of a triangle is 180°180\degree. Hence if we know the measurement of the two angles, we can find the third angle using the equation:

α+β+γ=180°\footnotesize \alpha + \beta + \gamma = 180\degree \\

Once we know the third angle, we can easily calculate the missing side using the Law of Sines mentioned above.

If two sides and the perimeter are known

This is the most simple scenario. As we know that the perimeter of a triangle is just the sum of its three sides, i.e.,

Perimeter=a+b+c\scriptsize \text{Perimeter} = a + b +c

This means that if we know the measurement of two sides of a triangle (say aa and bb), we can calculate the length of the third side as:

c=Perimeter(a+b)\scriptsize c = \text{Perimeter} - (a + b)

In the next section, we will see an example of how to calculate the sides of a triangle.

Finding the missing sides of a triangle - An example

Let us consider a triangle where β=45°\beta = 45\degree, γ=30°\gamma = 30 \degree and b=10b=10. To calculate the triangle side lengths, we will proceed as follows:

  1. Using the angle sum property of a triangle, we will solve for the third angle:
α=180°(α+γ)=180°(45°+30°)=105°\quad \footnotesize \quad \begin{align*} \alpha & = 180\degree - (\alpha + \gamma)\\ & = 180\degree - (45\degree + 30 \degree)\\ & = 105\degree \end{align*}
  1. Now, we will use the Law of Sines to find the missing sides
asin(α)=bsin(β)a=bsin(β)×sin(α)=10sin(45°)×sin(105°)=13.66\quad \footnotesize \quad \begin{align*} \frac{a}{\text{sin} (\alpha)} &= \frac{b}{\text{sin} (\beta)} \\ \implies a & = \frac{b}{\text{sin} (\beta)} \times \text{sin} (\alpha) \\ & = \frac{10}{\text{sin} (45\degree)} \times \text{sin} (105\degree) \\ & = 13.66 \end{align*}
bsin(β)=csin(γ)c=bsin(β)×sin(γ)=10sin(45°)×sin(30°)=7.071\quad \footnotesize \quad \begin{align*} \frac{b}{\text{sin} (\beta)} &= \frac{c}{\text{sin} (\gamma)} \\ \implies c & = \frac{b}{\text{sin} (\beta)} \times \text{sin} (\gamma) \\ & = \frac{10}{\text{sin} (45\degree)} \times \text{sin} (30\degree) \\ & = 7.071 \end{align*}

How to use the triangle side calculator

Now let us see how we can use our calculator to find the missing side of a triangle. We will use the same example as in the previous section.

  1. Using the drop-down menu against Given, choose the option side and 2 angles.

  2. Now choose the option two angles + adjacent side (AAS).

  3. Enter the dimension of angles (β=45°\beta = 45\degree and γ=30°\gamma = 30 \degree) and side (b=10b=10).

  4. The triangle side calculator will display the third angle (α=105°\alpha = 105\degree) and the two sides (a=13.66a = 13.66 and c=7.071c = 7.071).

FAQ

How to find the third side of a triangle if its two side and perimeter are given?

To find the third side of a triangle, if its two sides and perimeter are given, follow the given instructions:

  1. Find the sum of the two sides, a and b.

  2. Subtract the sum a + b from the perimeter P of the triangle.

  3. You will get the third side, c as c = P - (a + b).

Calculate the triangle side lengths if two of its angles are 60° each and one of the sides is 10 cm.

The length of each side is 10 cm. Since two of the angles are 60° each, the third angle will be 180° - (60° + 60°) = 60°.

As all the three angles are equal, the triangle is an equilateral triangle. Hence, all the sides will be of the same length, i.e., 10 cm.

Purnima Singh, PhD
Given
side and 2 angles
Enter the selected two angles and one side.
Option
two angles + side between (ASA)
Triangle with sides a,b,c and angles α, β, γ
ASA inputs
α
deg
c
in
ß
deg
ASA results
γ
deg
a
in
b
in
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