Find the apothem and the side length of the regular polygon. Then, use the formula A= 12ap to find its area.
59 in^2
Practice makes perfect
The area of a regular polygon is half the product of the apothem and the perimeter. We will first find the apothem and then the side length to obtain the perimeter. Finally, we will use this information to find the area.
Let's do it!
Apothem
By drawing the five radii, we can divide the pentagon into five isosceles triangles. Since the triangles are congruent and a full turn measures 360^(∘), the central angles of the isosceles triangles formed by the radii measure 3605=72^(∘).
Now, let's consider just one of these isosceles triangles. We will also draw the apothem a of the hexagon, which is perpendicular to the side.
Remember that an apothem bisects the central angle and the side of the regular polygon. Therefore, we obtain a right triangle with an acute angle that measures 36^(∘).
Let's use the cosine ratio to find an expression for a.
cos(θ)=Adj./Hyp. ⇒ cos(36)=a/5
Let's solve this equation for a.
The apothem of the regular pentagon is 5cos(36) inches.
Perimeter
Consider the right triangle one more time.
We now want to find the length of the opposite side. To do this we will use the sine ratio.
sin(θ)=Opp./Hyp. ⇒ sin(36)=s/5
Let's solve this equation for s.
As previously mentioned, the apothem bisects the side of the regular hexagon. Therefore, the length of the side of the given polygon is twice the length of the side of the above triangle.
Consequently, the side length of the regular hexagon is 10sin(36) inches. Since this polygon has five congruent sides, to find its perimeter we will multiply the side length by 5.
Perimeter [0.8em] 5* 10sin(36)=50sin(36)in.
Area
Now we know that the apothem of the figure is 5cos(36) inches and that the perimeter is 50sin(36) inches. To find its area, we will substitute these values in the formula A= 12ap. Let's do it!
