Find the area of each regular polygon to the nearest tenth. Octagon with side length of 10 kilometers.
Heres what I tried; but I don't know if im on the right track.
Area=1/2 * Perimeter* Apothem
Perimeter=base of octagon* number of sides
Perimeter=10*8=80 km.
SO A=40a
(1/2 of 80 times apothem)
How do I find the apothem of the octagon?
Work on one of the 8 triangles, since they are all equal
Such a triangle is isosceles with a 45º formed by the equal sides and the other side is 10 km.
the other two angles are 67.5º each.
The apothem, which is simply the perpendicular distance from the centre of your regular polygon to one of the sides. It becomes the height of the triangle
Let the height be h, and use trig.
tan 67.5 = x/5
x = 5 tan 67.5 = 12.071
so the area of one triangle = 1/2 * 10 * 12.071 = 60.355....
the area of the octogon is 8 times that
= 482.84 km^2
To find the apothem of the octagon, you correctly identified that you can work on one of the 8 triangles since they are all equal. The triangle formed by one side of the octagon, the apothem, and the center of the octagon is an isosceles triangle.
In this triangle, one angle is 45º (formed by the equal sides) and the other two angles are 67.5º each. Let's find the apothem using trigonometry.
We have the following equation:
tan(67.5º) = apothem / (side length/2)
Simplifying, we have:
apothem = (side length/2) * tan(67.5º)
apothem = (10 km / 2) * tan(67.5º)
apothem = 5 km * tan(67.5º)
apothem ≈ 12.071 km
Now that we have the apothem, we can calculate the area of each triangle.
Area of one triangle = (1/2) * (side length) * (apothem)
Area of one triangle = (1/2) * 10 km * 12.071 km
Area of one triangle ≈ 60.355 km²
Finally, to find the area of the octagon, we multiply the area of one triangle by 8 (since there are 8 triangles in the octagon).
Area of octagon = 8 * 60.355 km²
Area of octagon ≈ 482.84 km² (rounded to the nearest tenth)
To find the apothem of the octagon, we can work with one of the eight triangles that form the octagon since they are all equal.
Each triangle is an isosceles triangle with a 45-degree angle formed by the equal sides and the other side being the side length of the octagon, which is 10 kilometers in this case. The other two angles are each 67.5 degrees.
To find the apothem, we can use trigonometry. Let's call the apothem "h" and the base of the triangle "x." We want to find the value of "x."
Using the tangent function (tan), we can set up the equation tan(67.5 degrees) = x / 5 (since the adjacent side to the angle is "x" and the opposite side is the apothem "h" which is equal to the radius of the circumscribed circle).
Solving for "x," we have x = 5 * tan(67.5 degrees) ≈ 12.071 kilometers.
Now that we have the base of the triangle (x), we can calculate the area of one triangle using the formula A = (1/2) * base * height. In this case, the base is 10 kilometers, and the height is the apothem we just calculated (12.071 kilometers).
So, the area of one triangle is A = (1/2) * 10 * 12.071 ≈ 60.355 square kilometers.
Since the octagon is made up of eight of these triangles, the total area of the octagon can be found by multiplying the area of one triangle by 8:
Total area of the octagon = 8 * 60.355 ≈ 483.64 square kilometers.
Therefore, the area of the octagon is approximately 483.64 square kilometers, rounded to the nearest tenth.