Answer:
45.14 square feet
Explanation:
To find the area of a regular octagon with a radius of 4 feet, we can divide the octagon into eight congruent triangles, each with a central angle of 45 degrees.
The apothem, or the distance from the center of the octagon to the midpoint of a side, can be found using the formula:
apothem = radius * cos(22.5 degrees)
where 22.5 degrees is half of the central angle of 45 degrees.
apothem = 4 feet * cos(22.5 degrees)
apothem = 4 feet * 0.9239 (rounded to four decimal places)
apothem = 3.6955 feet (rounded to four decimal places)
The area of each triangle can be found using the formula:
area of triangle = (1/2) * base * height
where the base is the length of one side of the octagon, and the height is the apothem.
The length of one side of the octagon can be found using the formula:
length of side = 2 * radius * sin(22.5 degrees)
length of side = 2 * 4 feet * sin(22.5 degrees)
length of side = 2 * 4 feet * 0.3827 (rounded to four decimal places)
length of side = 3.0607 feet (rounded to four decimal places)
Now, we can find the area of each triangle:
area of triangle = (1/2) * base * height
area of triangle = (1/2) * 3.0607 feet * 3.6955 feet
area of triangle = 5.6428 square feet (rounded to four decimal places)
Since there are eight congruent triangles in the octagon, the total area of the octagon can be found by multiplying the area of one triangle by 8:
area of octagon = 8 * area of triangle
area of octagon = 8 * 5.6428 square feet
area of octagon = 45.1424 square feet (rounded to four decimal places)
Therefore, the area of a regular octagon with a radius of 4 feet is approximately 45.14 square feet.