Question

the diagram below shows a rectangle inside a regular hexagon. the apothem of the hexagon is 15.59 units. to the nearest square unit, what is the area of the shaded region? apothem length: 15.59 a. 842 square units b. 183 square units c. 1220 square units d. 464 square units

Explanation:

Step1: Calculate area of hexagon

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 hexagon with side - length $s$, the perimeter $P = 6s$. In a regular hexagon, we can find the side - length using the right - triangle formed by the apothem and half of the side. However, we can also note that if we assume the side - length of the hexagon is related to the rectangle's dimensions. Let's use the formula directly with the given apothem. First, we need to find the perimeter. For a regular hexagon, we know that the relationship between the apothem $a$ and the side - length $s$ is $a=\frac{\sqrt{3}}{2}s$. But we can also calculate the area using the fact that the area of a regular hexagon with apothem $a$ and side - length $s$: $A_{hexagon}=\frac{1}{2}\times a\times6s$. Since we know $a = 15.59$, and assume the side - length of the hexagon can be related to the rectangle's dimensions. The area of the hexagon $A_{hexagon}=\frac{1}{2}\times15.59\times(6\times18)= \frac{1}{2}\times15.59\times108 = 15.59\times54=841.86$.

Step2: Calculate area of rectangle

The area of a rectangle is given by $A = l\times w$, where $l$ is the length and $w$ is the width. Here, $l = 21$ and $w = 18$, so $A_{rectangle}=21\times18 = 378$.

Step3: Calculate area of shaded region

The area of the shaded region $A_{shaded}=A_{hexagon}-A_{rectangle}$. So $A_{shaded}=841.86 - 378=463.86\approx464$.

Answer:

D. 464 square units