Triangle Length Calculator

Created by Rahul Dhari
Reviewed by Kenneth Alambra
Last updated: Jan 19, 2023

The triangle length calculator tells you the length of the third side if you enter two sides and an angle. A triangle has three sides and three angles. While we know by courtesy of the angle sum property that the sum of interior angles is 180°, the length of sides can be anything. To this end, you need to employ a sine law or the cosine law to relate them to each other.

Sine or cosine forms the crux of trigonometry functions that have numerous applications. One is about finding the third side or any angle for a triangle: the calculator and the accompanying text do the same. Read on to understand more about triangle length and cosine law.

Relationship to calculate triangle lengths

Let's consider a triangle whose sides are a, b, and c and angles α\alpha, β\beta,
γ\gamma. The sides of the triangle are related to each other with the cosine law:

cosα=b2+c2a22×b×ccosβ=b2+c2+a22×a×ccosγ=b2c2+a22×b×a\begin{align*} &\cos{\alpha} = \frac{b^2 + c^2 - a^2}{2\times b\times c}\\\\ &\cos{\beta} = \frac{-b^2 + c^2 + a^2}{2\times a\times c}\\\\ &\cos{\gamma} = \frac{b^2 - c^2 + a^2}{2\times b\times a} \end{align*}

Using the triangle length calculator

Let ⊿ABC be a right-angled triangle having sides, a and b, forming the right angle, equal to 3 and 4, respectively.

To find the missing side length:

  1. Fill in the angle, γ=90°\gamma = 90°.
  2. Enter the length of side, a=3a = 3.
  3. Input the length of side, b=4b = 4.
  4. Using the triangle length calculator:
cosγ=b2c2+a22×b×acos90°=42c2+322×4×30=16c2+912c2=16+9c=16+9=5\scriptsize \begin{align*} \qquad \cos{\gamma} &= \frac{b^2 - c^2 + a^2}{2 \times b\times a} \\\\ \qquad \cos{90°} &= \frac{4^2 - c^2 + 3^2}{2\times 4 \times 3} \\\\ 0 &= \frac{16 - c^2 + 9}{12} \\\\ c^2 &= 16 + 9 \\ c &= \sqrt{16 + 9} = 5 \end{align*}

The third side of the triangle is 5.


How do I find an angle of the triangle using the side?

To find the angle of the triangle opposite one of its sides, say side "a":

  1. Square the first side, a.
  2. Add the square of the second side, b to it.
  3. Subtract the square of the third side, c from the sum.
  4. Divide the difference by the length of second side.
  5. Divide the quotient by the length of first side.
  6. Divide the quotient by 2.
  7. Find the cosine inverse of the final value to obtain the angle. Mathematically, α = arccos ((a² + b² - c²)/(2ab))

What is the third side of the right triangle having two sides, 9 and 16?

The third side of the triangle is 18.36. Considering the third angle as 90°, the third side is obtainable using the cosine law. Since cos(90°) = 0, the cosine law now translates to a Pythagoras theorem, i.e., c = √(9² + 16²) = 18.36.

Rahul Dhari
Triangle with sides a,b,c and angles α, β, γ

Enter at least 1 angle
Angle α
Angle β
Angle γ
Side (a)
Side (b)
Side (c)
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