The sum of the measures of the interior angles of a polygon is (n-2)180, where n represents the number of sides.
We will start by finding the sum of the interior angles of each polygon. In the diagram, the circled vertex is the common vertex of a square, a regular hexagon, and a regular 12-gon, or a dodecagon. Therefore, we will substitute 4, 6, and 12 for n in the expression for the sum to find the sum of the interior measures in each polygon.
Polygon
n
(n-2)180^(∘)
Simplify
square
4
( 4-2)180^(∘)
360^(∘)
regular hexagon
6
( 6-2)180^(∘)
720^(∘)
regular 12-gon
12
( 12-2)180^(∘)
1800^(∘)
Now, recall that in a regular polygon, all the angles have the same measure. To find the measure of one interior angle of each regular polygon, we will divide the sum of the angles by the number of angles in the polygon. Notice that in regular polygons the number of sides and the number of interior angles are equal. This means that we need to divide the sum of the angles by n. Let's do it!
Polygon
n
Sum of Angles
One Angle
square
4
360^(∘)
360^(∘)/4= 90^(∘)
regular hexagon
6
720^(∘)
720^(∘)/6= 120^(∘)
regular 12-gon
12
1800^(∘)
1800^(∘)/12= 150^(∘)
We found the measure of each angle at the circled vertex.
Now we can calculate the sum of the angles that we got.
90^(∘)+ 120^(∘)+ 150^(∘)=360^(∘)
The sum of the angles is 360^(∘).
