Question

the diagram below shows a square inside a regular octagon. the apothem of the octagon is 15.69 units. to the nearest square unit, what is the area of the shaded region? apothem length: 15.69 a. 816 square units b. 647 square units c. 1463 square units d. 764 square units

Explanation:

Step1: Find area of the octagon

The formula for the area of a regular polygon is $A = \frac{1}{2}aP$, where $a$ is the apothem and $P$ is the perimeter. For a regular octagon with side - length $s$, the perimeter $P = 8s$. Here, we can also use the fact that the area formula $A=\frac{1}{2}aP$. Given $a = 15.69$. Let's assume the side - length of the octagon is related to the square's side. The area of the octagon $A_{octagon}=\frac{1}{2}\times15.69\times(8\times13)=815.88$.

Step2: Find area of the square

The area of a square with side - length $s = 13$ is $A_{square}=s^{2}=13^{2}=169$.

Step3: Find area of the shaded region

The area of the shaded region $A = A_{octagon}-A_{square}=815.88 - 169=646.88\approx647$.

Answer:

B. 647 square units