You can calculate the measure of a central angle of an octagon by dividing 360^(∘) by 8. Use the formula for the area of a regular polygon.
≈ 120 square inches
Practice makes perfect
It is given that a platter is in the shape of a regular octagon. Using this information, we can create a diagram.
Let's recall that the area of a regular polygon is equal to one-half the product of the perimeter P and the apothem a.
A=1/2aP
We know that the length of the apothem of the octagon is 6 inches. Hence, we only need to find the perimeter of the octagon. Let's consider the following diagram.
First, we need to find the measure of the central angle ∠ AOB. An octagon has 8 central angles and their sum is 360^(∘). Hence, by dividing 360 by 8, we can calculate the measure of one central angle.
m∠ AOB=360/8=45^(∘)
Since △ AOB is an isosceles triangle, height OC is also a median of AB and a bisector of ∠ AOB. Thus, the measure of ∠ COB is half the measure of ∠ AOB.
m∠ COB=45/2=22.5^(∘)
Now we can use the tangent function to find the length of CB.
The tangent of m∠ COB is equal to the ratio of the opposite side CB and the adjacent side OC.
tg (m∠ COB) &= CB/OC
&⇓
tg (22.5^(∘))&= CB/6
Let's solve this equation for CB.
Now that we know the length of CB, we can find AB. Since OC is a median of AB, segment AB is twice as long as CB.
AB=2.5* 2=5 in
This way we found the side length of the octagon. There are 8 sides in every octagon, so by multiplying its length by 8 we can calculate the perimeter of the octagon.
P=8* 5=40in
Finally, we have enough information to find the area of the octagon. Let's substitute a with 6 and P with 40 into the formula.
