The following solution is arrived at by using the formula for the area of a regular octagon in relation to its Apotem which in this case is 18. Hence the area of the Octagon given is 1,446.
What is an Apothem?
An apothem is defined as a line from the center of a regular polygon at right angles to any of its sides.
What is the area of a Regular Octagon?
The calculation for this shape using only it's apothem has some curved bends. In order to derive the area, we have to divide the octagon into triangles.
When we find the area of the triangles we can then multiply by the total number to get the area of the Octagon.
Hence we can say that the area of the Octagon (A) = (1/2b*h)n
Where b = base
h = height
n = number of triangles.
Recall that the total angle in a circle is 360°, hence given that all the triangles are equal, we must divide 360° by 8 triangles to get the angle in the vertex of each triangle.
Thus, 360°/8 = 45°
So we have The angle of our triangle which is opposite the base.
Recall that the triangle is split in two so that each of the triangles are right-angled triangle,
Hence, the angle opposite the base for each right angle triangle is given as:
45°/2 = 22.5°
So to get the length of the opposite side (x), we use the law of tangents:
Which means:
x = 18 Tan 22.5°
= 18 * 0.55785173935
≈ 10.04
So if x is ≈ 10.04, then area of that triangle is
= 1/2 * 10.04 * 18 (that is 1/2bh)
= 90.3792
From the above, we can state that the Area of the triangle with the Apothem is
= 90.3792 * 2
= 180.74
Recall our formula for finding the area of the octagon:
A = (1/2bh)n
A = (108.74) *8
A = 1,445.95
≈ 1,446
Learn more about Areas of Octagons at:
https://brainly.com/question/358118
