SOLUTION: I want to find the area of a regular pentagon and I only have the apothem. Can I do this? The Apothem is 35'

Question 872015: I want to find the area of a regular pentagon and I only have the apothem. Can I do this? The Apothem is 35'
Answer by htmentor(1343) (Show Source): You can put this solution on YOUR website!

The pentagon can be broken up into 5 triangles.
To find the area, we need to find the area of one of the triangles and multiply by 5.
From the figure, we see that the apothem (in red) is the radius of the in-circle, which touches the pentagon at the midpoints of the sides.
It is also clear that the apothem is the height of one of these triangles.
The angles whose vertex is the center of the circle must sum to 360 degrees, which means each one is 72 degrees.
The measure of the half-angle is 36 degrees
From the figure, we see that tan(36) = BG/KG -> BG = KG*tan(36)
The area of the shaded triangle is BG*KG, and the area of the pentagon is 5*BG*KG
Substituting, we get Area = 5*KG*KG*tan(36) = 5*35^2*tan(36) = 4450.1

RELATED QUESTIONS
How do you find the area of a regular polygon if you only have the... (answered by shree840)

How do I find the Apothem of a regular polygon? The equation i am trying to use is to... (answered by solver91311)

What is the area of a regular pentagon with perimeter 10x and apothem... (answered by fcabanski)

How do I find the area, radius and apothem of a regular hexagon with the side length of... (answered by Earlsdon)

What is the area of a regular hexagon whose radius is 24? I have that the apothem is 12... (answered by Alan3354)

Find the area of a regular pentagon with an apothem length of 6cm. PLEASE... (answered by jim_thompson5910)

Find the area of a regular pentagon with apothem 15 meters. Show all work to receive... (answered by edjones)

Area of a regular pentagon with the apothem length of 5 inches and the side is 12... (answered by ikleyn)

I need help with this question. If the apothem is 12cm, and the perimeter is 81.6,... (answered by ikleyn,greenestamps)