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. 1463 square units b. 764 square units c. 816 square units d. 647 square units

Explanation:

Step1: Find octagon side length

Let regular octagon side = $s$. For a regular octagon formed by cutting 13-unit squares from a square, $s = 13\sqrt{2} \approx 18.38$ units.

Step2: Calculate octagon perimeter

$P = 8s = 8 \times 18.38 \approx 147.04$ units

Step3: Compute octagon area

Use formula $A_{octagon} = \frac{1}{2} \times \text{apothem} \times P$
$A_{octagon} = \frac{1}{2} \times 15.69 \times 147.04 \approx 1165$ square units

Step4: Calculate inner square area

$A_{square} = 13^2 = 169$ square units

Step5: Find shaded area

$A_{shaded} = A_{octagon} - A_{square} = 1165 - 169 = 816$ square units

Answer:

C. 816 square units