Question

the diagram below shows a square inside a regular octagon. the apothem of the octagon is 13.28 units. to the nearest square unit, what is the area of the shaded region? apothem length: 13.28 a. 584 square units b. 540 square units c. 463 square units d. 1048 square units

Explanation:

Step1: Find area of 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 $s = 11$, the perimeter $P=8s = 8\times11 = 88$. The apothem $a = 13.28$. So the area of the octagon $A_{octagon}=\frac{1}{2}\times13.28\times88=13.28\times44 = 584.32$.

Step2: Find area of square

The side - length of the square is $s = 11$, so the area of the square $A_{square}=s^{2}=11^{2}=121$.

Step3: Find area of shaded region

The area of the shaded region $A = A_{octagon}-A_{square}$. So $A=584.32 - 121=463.32\approx463$.

Answer:

C. 463 square units