Question

The interior angle measure of a regular polygon is calculated with the following formula, where
represents the number of sides or interior angles of the regular polygon.
interior angle = 180( n - 2)/n


A. Determine the measure of the interior angles of a regular decagon. In your answer, include all calculations.

B. Use your result from part A to identify whether or not a regular decagon can be used as the only shape in a regular tessellation.

WRITER

Answer 1

A. Use n = 10 in the formula interior angle = 180(n − 2)/n.

interior angle = 180(10 − 2)/10
= 180(8)/10
= 144 degrees.

So each interior angle of a regular decagon measures 144°.

B. For a regular tessellation with one regular polygon, a whole number of those interior angles must meet at a point and sum to 360°. That requires 360 ÷ 144 = 2.5, which is not an integer. Equivalently, 144° × 2 = 288° and 144° × 3 = 432°, so you cannot get exactly 360° using only regular decagons.

Therefore a regular decagon cannot be the only shape in a regular tessellation. (The only regular polygons that do tessellate by themselves are equilateral triangles, squares, and regular hexagons.)