Question

b if an interior angle of a regular polygon measures 140°, how many sides does the polygon have? e the sum of the measures of the interior angles of polygon is 2,880°. how many sides does the polygon have?

Explanation:

Problem B:

Step1: Recall the formula for the measure of an interior angle of a regular polygon.

The formula for the measure of an interior angle \( I \) of a regular polygon with \( n \) sides is \( I=\frac{(n - 2)\times180^{\circ}}{n} \). We know that \( I = 140^{\circ} \), so we set up the equation:
\( 140=\frac{(n - 2)\times180}{n} \)

Step2: Solve the equation for \( n \).

First, multiply both sides of the equation by \( n \):
\( 140n=(n - 2)\times180 \)
Expand the right - hand side:
\( 140n = 180n-360 \)
Subtract \( 140n \) from both sides:
\( 0 = 180n-140n - 360 \)
\( 0 = 40n-360 \)
Add 360 to both sides:
\( 40n=360 \)
Divide both sides by 40:
\( n=\frac{360}{40}=9 \)

Problem E:

Step1: Recall the formula for the sum of interior angles of a polygon.

The formula for the sum of the interior angles \( S \) of a polygon with \( n \) sides is \( S=(n - 2)\times180^{\circ} \). We know that \( S = 2880^{\circ} \), so we set up the equation:
\( 2880=(n - 2)\times180 \)

Step2: Solve the equation for \( n \).

Divide both sides of the equation by \( 180 \):
\( \frac{2880}{180}=n - 2 \)
\( 16=n - 2 \)
Add 2 to both sides:
\( n=16 + 2=18 \)

Answer:

• For problem B: The polygon has 9 sides.
• For problem E: The polygon has 18 sides.