## Solving SSS Triangles

Table of Contents

## Definition

SSS means “Side, Side, Side”

When we know 3 sides of the triangle, we can find the missing angles.

## How To Solve SSS Triangle

To solve SSS triangle:

- use The Law of Cosines to calculate one of the angles
- use The Law of Cosines to find another angle
- use angles of a triangle add to 180° to find the last angle

We use the “angle” version of the Law of Cosines:

(they are all the same formula, just different labels)

In this triangle we know the three sides:

*a* = √3,

*b* = 1

*c* = 2

**Step 1**

Use The Law of Cosines first to find one of the angles. It doesn’t matter which one.

Let’s find angle A first:

cos A = (*b*^{2}* + c*^{2}* − a*^{2}*) / *2*bc*

cos A = (1^{2} + 2^{2} − √3^{2}) / (2×1×2)

cos A = (1 + 4 − 3) / 4

cos A = ½

A = cos^{−1}(½)

A = 60°

**Step 2**

Next we will find another side. We use The Law of Cosines again, this time for angle B:

cos B = (*a*^{2 }*+ **c** ^{2}* −

*b*

^{2})/2

*ac*

cos B = (√3^{2} + 2^{2} − 1^{2})/(2×√3×2)

cos B = (3 + 4 − 1) / 4√3

cos B = ½ √3

B = cos^{−1}( ½ √3)

B = 30°

**Step 3**

Finally, we can find angle C by using ‘angles of a triangle add to 180°’:

C = 180° − 30° − 60°

C = 90°

Now we have completely solved the triangle i.e. we have found all its angles.