We want to find the areas of the pentagons. To do so, we first need to find their apothems. We know that an apothem bisects the central angle and the side opposite to the angle. The formula for the measure of the central angle of a regular polygon is 360n, where n is the number of sides in the polygon. Let's find the central angle of a pentagon.
360/n ⇔ 360/5=72
The measure of the central angle of a regular pentagon is 72^(∘). Since the apothem bisects this angle, the measure of the angle between the apothem and the radius of the pentagon is 36^(∘). Let's show the apothem of each pentagon.
Using the tangent ratio, we can find the apothems, a and b.
Tangent Ratio
Rewrite
Pentagon 1
tan 36^(∘)=3/a
a= 3/tan36 ^(∘)
Pentagon 2
tan 36^(∘)=4/b
b= 4/tan36 ^(∘)
Next, we will find the perimeters of the pentagons.
ccc
& Pentagon1 & Pentagon2 [0.5em]
Perimeter: & 5* 6 = 30 & 5* 8 = 40
We can now use the area formula for a regular polygon, A= 12ap. We can find the area A_1 of Pentagon 1 by substituting 3tan36^(∘) for a and 30 for p.
