Answer :

The area of the heptagon with an apothem of 40 feet and a side length of 38.5 feet is 5390 square feet.

To find the area of a regular heptagon (a polygon with seven equal sides and angles), we can use the formula for the area of a regular polygon:

[tex]\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \][/tex]where:

- The **Perimeter** is the sum of the lengths of all sides. - The **Apothem** is the line from the center of the polygon to the midpoint of one of its sides, which is perpendicular to that side.

First, let's calculate the perimeter of the heptagon by multiplying the length of one side by the number of sides.

Since the heptagon has seven sides and each side is 38.5 feet:

[tex]\[ \text{Perimeter} = 38.5 \, \text{ft} \times 7 \][/tex]

[tex]\[ \text{Perimeter} = 269.5 \, \text{ft} \][/tex]

Now that we have the perimeter, we can calculate the area.

The apothem is given as 40 feet. We can use this information in the formula:

[tex]\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \][/tex]

[tex]\[ \text{Area} = \frac{1}{2} \times 269.5 \, \text{ft} \times 40 \, \text{ft} \][/tex]

[tex]\[ \text{Area} = 134.75 \, \text{ft} \times 40 \, \text{ft} \][/tex]

[tex]\[ \text{Area} = 5390 \, \text{ft}^2 \][/tex]

So, the area of the heptagon with an apothem of 40 feet and a side length of 38.5 feet is 5390 square feet.

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Rewritten by : Barada