The area of a regular polygon is half the product of the apothem and the perimeter. Note that we are given the apothem but are missing the perimeter. Let's first find the perimeter and use it to find the area.
Finding the Perimeter
To find the perimeter, let's start by drawing the radii of the given polygon.
The radii divide the regular hexagon into six congruent isosceles triangles. Since corresponding angles of congruent figures are congruent, the vertex angles of the isosceles triangles formed by the radii are congruent. Moreover, since a full turn measures 360^(∘), we can divide 360 by 6 to obtain their measures.
360/6=60^(∘)
The vertex angles of the isosceles triangles measure 60^(∘) each.
Next, recall that the apothem bisects the vertex angle of the isosceles triangle formed by the radii. As a result, 30^(∘)-60^(∘)-90^(∘) triangles are obtained. Let's consider one of them.
In this type of special triangle the length of the longer leg is sqrt(3) times the length of the shorter leg. Therefore, to obtain the length of the shorter leg we divide 6 in. by sqrt(3).
Shorter Leg: 6/sqrt(3)
Let's simplify the expression.
Not only does the apothem bisect the vertex angle of the isosceles triangle but it also bisects its opposite side, which is a side of the hexagon. Therefore, the length of one side of the given regular polygon is 2* 2sqrt(3)= 4 sqrt(3).
In a regular hexagon all six sides have the same length. Therefore, we can obtain its perimeter by multiplying the length of a side by 6.
Perimeter: 4sqrt(3)* 6=24sqrt(3) in.
Finding the Area
Finally, we have that the apothem is 6 in. and that the perimeter is 24sqrt(3) inches. To find the area, we will substitute these two values in the formula A= 12ap and simplify.
