In this diagram, 𝐴𝐵, 𝐵𝐶, and
𝐶𝐷 are three sides of a regular polygon 𝑃. Show that polygon 𝑃 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 𝑛, 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
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 𝑃. 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 𝑃 is
We have now worked out that the
interior angle of polygon 𝑃 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 𝑃 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.