Question

using the image below, find the area. 14 cm select one: a. 94.64 cm² b. 1379 cm² c. 946.4 cm² d. 137.9 cm²

Explanation:

Step1: Find the number of sides and side - length

The polygon is an octagon, $n = 8$, and side - length $s=14$ cm.

Step2: Find the apothem

For a regular octagon, we can consider the right - triangle formed. The central angle of a regular octagon is $\theta=\frac{360^{\circ}}{n}=\frac{360^{\circ}}{8} = 45^{\circ}$. If we consider the right - triangle formed by the apothem ($a$), half of the side - length ($\frac{s}{2}$), and the radius of the circum - circle. In a right - triangle related to a regular octagon, if the side - length is $s = 14$ cm, then half of the side - length $x=\frac{s}{2}=7$ cm. The apothem $a$ can be found using trigonometry. Since the central angle of the right - triangle is $22.5^{\circ}$, and $\tan22.5^{\circ}=\sqrt{2}-1\approx0.414$. Also, $\tan22.5^{\circ}=\frac{\frac{s}{2}}{a}$, so $a=\frac{\frac{s}{2}}{\tan22.5^{\circ}}=\frac{7}{ \sqrt{2}-1}=7(\sqrt{2} + 1)\approx7(1.414 + 1)=7\times2.414 = 16.898$ cm.

Step3: Use the area formula for a regular polygon

The area formula for a regular polygon is $A=\frac{1}{2}a\times P$, where $P$ is the perimeter. The perimeter $P=ns=8\times14 = 112$ cm. Then $A=\frac{1}{2}\times16.898\times112=946.288\approx946.4$ cm².

Answer:

C. $946.4$ cm²