# Video: Pack 1 β’ Paper 3 β’ Question 8

Pack 1 β’ Paper 3 β’ Question 8

03:12

### Video Transcript

In this diagram, π΄π΅, π΅πΆ, and πΆπ· are three sides of a regular polygon π. Show that polygon π is a nonagon, a regular nine-sided shape.

In any regular polygon, the sum of the interior and exterior angles is 180 degrees. And the exterior angles can be calculated by dividing 360 degrees by π, the number of sides. If we firstly consider the 18-sided polygon, the exterior angle can be calculated by dividing 360 by 18, the number of sides. This is equal to 20 degrees. As the sum of the exterior and interior angles equal 180, the interior angle can be calculated by subtracting 20 from 180. This gives us 160 degrees. The angle on the diagram is 160 degrees.

As all three sides of an equilateral triangle are equal, all three angles must be equal. The angles in a triangle add up to 180 degrees. Therefore, each interior angle will be 180 divided by three. This is equal to 60 degrees. We can now calculate the interior angle of the polygon π. Angles at a point or in a circle add up to 360 degrees. Therefore, the interior angle is 360 minus 160 minus 60. This gives us 140 degrees. The interior angle of polygon π is 140 degrees.

We have now worked out that the interior angle of polygon π is 140 degrees. This means that the exterior angle must be equal to 40 degrees, as the sum of the interior and exterior angles of regular polygons equals 180 degrees.

A nonagon has nine sides. This means that the exterior angle of a regular nonagon can be calculated by dividing 360 by nine. 360 divided by nine is equal to 40. Therefore, the exterior angle of a nonagon is 40 degrees. Since the angles are the same, exterior angle of 40 degrees and an interior angle of 140 degrees, then shape π has to be a nonagon.

As the 18-sided polygon, the equilateral triangle, and the nonagon fit together at a point, we can say that these three shapes tessellate.