Area of triangle with apothem of 6?

Answer 1

First, what is apothem?

Apothem: The length of the line joining the center of a regular polygon to the midpoint of one of its sides.

So, we have a regular triangle, or an equilateral triangle (very important fact that we need to use all the way of our work), where all three sides are equal, and all three angles are equal, each equals 60 degrees.

1. Draw the equilateral triangle ABC (label it in a counterclockwise direction starting with letter A, length BC is horizontal).

2. From A, and B draw the medians AD and BE, which we also know that are perpendicular to the sides BC and AC and bisect the vertex angles A and B, each part equals 30 degrees.

3. Label with O the point of intersection of these medians.

4. Look at the right triangles ODB and OEA. We know that angles B and A equal 30 degrees, and angles D and E equal 90 degrees. So that sides OD and OE, which equal 6 as they are apothems, are one half of the length measure of the hypotenuses OB and OA, as sides opposite to an angle of 30 degrees. Thus, these hypotenuses OB and OA equal 12. (or you can use the fact that the apothem is 1/3 of the length of the median)

5. So the height AD of the triangle ABC is 18 (6 + 12). In order to find the area of the triangle we need to know what length measure has the base BC.

6. In the right triangle BEC, the length of BC is approximately 20.78 (cos 30 degrees = 18/BC).

7. Since we know the length of the base BC, 20.78, and the length of the height, 18, we are able to find the area of the triangle ABC which is approximately 187 square unit ( A = (bh)/2, A = (20.78 x 18)/2 = 187.02)

8. Area of the triangle can be obtained by one of these two formulae:

A = 1/2 a*p where a is the apothem and p is the perimeter.

A = 3 a^2*sqrt(3)

9. Applying the first formula: A = 1/2*6 *62.34 = 187.02

10. Applying the second formula: A = 3*6^2*3^1/2 = 187.06