We want to find the area of a regular triangle with an apothem that measures 2sqrt(3).
The area of a regular polygon is half the product of the apothem and the perimeter. We will use the Pythagorean Theorem to find the side length.
Finding the Perimeter
To find the perimeter, let's start by drawing the radii of the given triangle.
The radii divide the triangle into three 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 3 to obtain their measures.
360/3=120^(∘)
The vertex angles of the isosceles triangles measure 120^(∘) 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 longer leg we multiply 2sqrt(3) by sqrt(3).
Longer Leg: 2sqrt(3)(sqrt(3))= 6
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* 6= 12.
In a regular triangle all three sides have the same length. Therefore, we can obtain its perimeter by multiplying the length of a side by 3.
Perimeter: 12* 3=36
Finding the Area
Finally, we have that the apothem is 2sqrt(3) and that the perimeter is 36. To find the area, we will substitute these two values in the formula A= 12aP and simplify.
