Find the length of the third side calculator

This calculator displays 2 scenarios
1) Enter 3 sides of a triangle, and it will determine if the side lengths satisfy the properties of the triangle inequality and form a triangle
2) Enter 2 sides of a triangle, and this will determine an acceptable range for the length of the 3rd side of a triangle so that the 3rd side respects the Triangle Inequality.
This calculator has 3 inputs.

This calculator calculates for the length of one side of a right triangle given the length of the other two sides. A right triangle has two sides perpendicular to each other. Sides "a" and "b" are the perpendicular sides and side "c" is the hypothenuse. Enter the length of any two sides and leave the side to be calculated blank. Please check out also the Regular Triangle Calculator and the Irregular Triangle Calculator.

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\alphaα, β\betaβ,
γ\gammaγ. The sides of the triangle are related to each other with the cosine law:

cos⁡α=b2+c2−a22×b×ccos⁡β=−b2+c2+a22×a×ccos⁡γ=b2−c2+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*}​cosα=2×b×cb2+c2−a2​cosβ=2×a×c−b2+c2+a2​cosγ=2×b×ab2−c2+a2​​

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°γ=90°.
2. Enter the length of side, a=3a = 3a=3.
3. Input the length of side, b=4b = 4b=4.
4. Using the triangle length calculator:

cos⁡γ=b2−c2+a22×b×acos⁡90°=42−c2+322×4×30=16−c2+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*}cosγcos90°0c2c​=2×b×ab2−c2+a2​=2×4×342−c2+32​=1216−c2+9​=16+9=16+9​=5​

The third side of the triangle is b0. Considering the third angle as b1, the third side is obtainable using the cosine law. Since cos(90°) = 0, the cosine law now translates to a Pythagoras theorem, i.e., b2.

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