Question

The sum of the measures of the interior angles of a polygon is known to be between 2100 and 2200. How many sides does the polygon have?

Short answer

The polygon has 14 sides.

Step-by-step solution

Step 1. Write the theorem for the sum of interior angles for a polygon.

The theorem for the sum of interior angles for a polygon states that the sum of the measures of the interior angles for a polygon with sides is $\left(n-2\right)180$.

Step 2. Description of step.

Consider the number of sides of the polygon be n.

Step 3. Find the value of n.

As, the sum of measures of the interior angles of a polygon is between 2100 and 2200.

$$\begin{gathered} 2100<\text{sumofinteriorangles}<\text{2200} \\ \text{2100}<\left(n-2\right)180<2200 \\ \frac{2100}{180}<n-2<\frac{2200}{180} \\ 11.67<n-2<12.22 \\ 11.67+2<n<12.22+2 \\ 13.67<n<14.22 \end{gathered}$$

Therefore, the number of sides of the polygon must be between 13.67 and 14.22.

But the number of sides of the polygon must be an integer, therefore the integer which is lying between 13.67 and 14.22 is 14.

Therefore, the value of is 14.

Therefore, the number of the sides of polygon is 14.