Question

select the correct answer.
this is a regular pentagon. the apothem is shown. what is perimeter of the pentagon?
a. 43.8 cm
b. 30.7 cm
c. 36.3 cm
d. 18.2 cm

Explanation:

Step1: Recall the formula for the area of a regular polygon

The area formula for a regular polygon is \(A=\frac{1}{2}ap\), where \(a\) is the apothem and \(p\) is the perimeter. But we can also use the relationship between the side - length and the apothem. For a regular pentagon, the central angle \(\theta=\frac{360^{\circ}}{n}=\frac{360^{\circ}}{5} = 72^{\circ}\). The angle between the apothem and the radius is \(\alpha=\frac{\theta}{2}=36^{\circ}\). If we assume the side - length is \(s\), and the apothem \(a = 5\) cm. We know that \(\tan\alpha=\frac{\frac{s}{2}}{a}\).

Step2: Calculate the side - length

Since \(\alpha = 36^{\circ}\) and \(a = 5\) cm, and \(\tan36^{\circ}\approx0.727\). From \(\tan\alpha=\frac{\frac{s}{2}}{a}\), we have \(s = 2a\tan\alpha\). Substituting \(a = 5\) cm and \(\tan36^{\circ}\approx0.727\), we get \(s\approx2\times5\times0.727 = 7.27\) cm.

Step3: Calculate the perimeter

The perimeter \(p\) of a regular pentagon with \(n = 5\) sides is \(p=ns\). Substituting \(n = 5\) and \(s\approx7.27\) cm, we get \(p = 5s\approx5\times7.27=36.35\approx36.3\) cm.

Another way:

Step1: Use the formula for the perimeter of a regular polygon related to the apothem

The formula for the area of a regular polygon \(A=\frac{1}{2}ap\). But if we assume we know the apothem \(a = 5\) cm. For a regular pentagon, we can also use the fact that if we consider one of the 5 isosceles triangles that make up the pentagon. The base of the isosceles triangle is the side - length \(s\) of the pentagon and the height is the apothem \(a\).
The central angle of a regular pentagon \(C=\frac{360^{\circ}}{5}=72^{\circ}\). If we split the isosceles triangle (formed by two radii and a side of the pentagon) into two right - triangles (by the apothem), the angle at the center of the pentagon for the right - triangle is \(\frac{72^{\circ}}{2}=36^{\circ}\).
We know that \(\tan36^{\circ}=\frac{\frac{s}{2}}{a}\), so \(s = 2a\tan36^{\circ}\). Then \(p=5s\).
Since \(\tan36^{\circ}\approx0.727\), \(s = 2\times5\times0.727\) and \(p=5\times(2\times5\times0.727)=50\times0.727 = 36.35\approx36.3\) cm.

Answer:

B. \(30.7\) cm