To find the area of a regular polygon, we need to use the following formula:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \][/tex]
First, let's determine the perimeter of the regular polygon. The perimeter [tex]\( P \)[/tex] is calculated by multiplying the number of sides [tex]\( n \)[/tex] by the length of one side [tex]\( s \)[/tex].
[tex]\[ \text{Perimeter} = n \times s \][/tex]
[tex]\[ \text{Perimeter} = 8 \times 4 \][/tex]
[tex]\[ \text{Perimeter} = 32 \text{ inches} \][/tex]
Next, we use the given apothem length [tex]\( a \)[/tex], which is 5 inches, and substitute both the perimeter and the apothem into the area formula.
[tex]\[ \text{Area} = \frac{1}{2} \times 32 \times 5 \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \times 160 \][/tex]
[tex]\[ \text{Area} = 80 \text{ square inches} \][/tex]
Therefore, the area of the regular polygon with 8 sides, each of length 4 inches, and an apothem of 5 inches is [tex]\( 80 \, \text{in}^2 \)[/tex].
[tex]\[
\text { Area }=80 \, \text { in }^2
\][/tex]
[tex]\(\square\)[/tex]
