find the perimeter of a regular octagon with an area of 80 m and an apothem of 5m
Look at one of the 8 congruent triangles.
The height of one of those is 5 m and let the base be b
each triangle has an area of 80/8 = 10 m^2
(1/2)(b)(5) = 10
5b = 20
b = 4
so we 8 sides of 4 m, for a perimeter of 32 m
To find the perimeter of a regular octagon, we can use the formula: Perimeter = 8 × side length.
However, we are not given the side length directly. Instead, we are given the area and apothem of the octagon. So, we need to find a way to determine the side length using these given measurements.
First, let's understand what an apothem is. The apothem is the distance from the center of the octagon to any of its sides, forming a right angle with the side. In this case, the given apothem is 5m.
We can form a right triangle using half of one side length, the apothem, and the radius. Since the octagon is regular, the radius is equal to the apothem.
Now, let's calculate the length of half of one side using the Pythagorean theorem:
(side length/2)^2 = (apothem)^2 + (radius)^2
(side length/2)^2 = 5^2 + 5^2
(side length/2)^2 = 50 + 25
(side length/2)^2 = 75
Next, we need to find the square root of both sides to get the value of half the side length:
side length/2 = √75
side length/2 = √(25 × 3)
side length/2 = √25 × √3
side length/2 = 5√3
Now, to find the complete side length, we multiply both sides by 2:
side length = 2 × 5√3
side length = 10√3 m
Finally, we can find the perimeter of the octagon by multiplying the side length by 8:
Perimeter = 8 × side length
Perimeter = 8 × 10√3
Perimeter = 80√3 m
Therefore, the perimeter of the regular octagon with an area of 80 m² and an apothem of 5 m is 80√3 m.