Question

each exterior angle in a regular polygon has a measure of 18. how many sides does the polygon have? what is the sum of the measures of the interior angles? a 10; 1,440 b 18; 2,880 c 20; 3,240 d 36; 6,120

Explanation:

Step1: Find the number of sides (n) of the regular polygon.

The sum of the exterior angles of any polygon is \( 360^\circ \). For a regular polygon, each exterior angle \( \theta \) is equal, so \( n=\frac{360^\circ}{\theta} \). Given \( \theta = 18^\circ \), we have \( n=\frac{360}{18}=20 \). Wait, no, wait—wait, the options have 20? Wait, no, wait, let's check again. Wait, the problem says each exterior angle is 18? Wait, no, wait, maybe I miscalculated. Wait, no, the sum of exterior angles is 360, so number of sides \( n=\frac{360}{\text{exterior angle}} \). If exterior angle is 18, then \( n = 360/18 = 20 \)? But the options: let's check the options. Wait, the options are A:10;1,440, B:18;2,880, C:20;3,240, D:36;6,120. Wait, maybe I made a mistake. Wait, no, the formula for the sum of interior angles is \( (n - 2)\times180^\circ \). Let's check each option.

Wait, let's re-express the problem: Each exterior angle in a regular polygon has a measure of 18. How many sides? Sum of interior angles?

First, number of sides \( n \): sum of exterior angles is \( 360^\circ \), so \( n=\frac{360}{\text{exterior angle}}=\frac{360}{18}=20 \). Then sum of interior angles: \( (n - 2)\times180=(20 - 2)\times180 = 18\times180 = 3240 \). So that's option C: 20; 3,240. Wait, but let's check again. Wait, maybe the exterior angle is 18, so \( n = 360/18 = 20 \). Then sum of interior angles: \( (20 - 2)\times180 = 18\times180 = 3240 \). So that's option C.

Wait, but let's confirm the steps:

Step1: Calculate the number of sides (n) of the regular polygon.

The sum of the exterior angles of any polygon is \( 360^\circ \). For a regular polygon, each exterior angle \( \theta \) is equal, so \( n=\frac{360^\circ}{\theta} \). Given \( \theta = 18^\circ \), we have \( n=\frac{360}{18}=20 \).

Step2: Calculate the sum of the interior angles.

The formula for the sum of the interior angles of a polygon with \( n \) sides is \( (n - 2)\times180^\circ \). Substituting \( n = 20 \), we get \( (20 - 2)\times180 = 18\times180 = 3240 \).

Answer:

C. 20; 3,240