Question

What is the measure of the exterior angle of a decagon?

Answer 1

Hint: In this problem, we have to find the measure of the exterior angle of a decagon. We should know that the general formula to find the sum of the interior angle of any polygon is \[180\left( n-2 \right)\], where n is the number of sides of the polygon. Here we have to find the sum of the measures of the interior angles of a decagon as we know that a decagon has 12 sides. We can now substitute the value of n as 12, we can then find the interior angle of each side and subtract it from \[{{180}^{\circ }}\], to get the supplementary exterior angle.

Complete step by step solution:
Here we have to find the measure of the exterior angle of a decagon
We know that the general formula to find the sum of the interior angle of any polygon is \[180\left( n-2 \right)\], where n is the number of sides of the polygon.
Here we have to find the sum of the measures of the interior angles of a decagon as we know that a decagon has 12 sides.
We can now substitute n = 12 in \[180\left( n-2 \right)\], we get
\[\Rightarrow 180\left( 12-2 \right)\]
We can now simplify the above step, we get
\[\Rightarrow 180\left( 10 \right)={{1800}^{\circ }}\]
We can now find the interior angle of each side, we get
\[\Rightarrow \dfrac{{{1800}^{\circ }}}{12}={{150}^{\circ }}\]
We can now subtract the above value from \[{{180}^{\circ }}\], we get
\[\Rightarrow {{180}^{\circ }}-{{150}^{\circ }}={{30}^{\circ }}\]
Therefore, the measure of the exterior angle of a decagon is \[{{30}^{\circ }}\].

Note: We should always remember that an octagon has 8 sides. We should also remember that the general formula to find the sum of the interior angle of any polygon is \[180\left( n-1 \right)\], where n is the number of sides of the polygon. We can divide the sum of the interior measures of the angle by the number of sides to get the measure of angle in each side.