# Solving SSS Triangles

## 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:

1. use The Law of Cosines to calculate one of the angles
2. use The Law of Cosines to find another angle
3. use angles of a triangle add to 180° to find the last angle

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

Examples

(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 = (b2 + c2 − a2) / 2bc

cos A = (12 + 22 − √32) / (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 = (a2 + c2 − b2)/2ac

cos B = (√32 + 22 − 12)/(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.