Question

question 8 of 10 the diagram below shows a rectangle inside a regular hexagon. the apothem of the hexagon is 19.05 units. to the nearest square unit, what is the area of the shaded region? apothem length: 19.05 a. 442 square units b. 1653 square units c. 1257 square units d. 861 square units

Explanation:

Step1: Find 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 $P = 6s$. Here, assume the side - length of the hexagon can be related to the given rectangle dimensions. A regular hexagon can be thought of in terms of equilateral triangles. The apothem $a = 19.05$ units. Let's first find the area of the hexagon. If we consider the relationship between the hexagon and the rectangle, we know that the area of a regular hexagon $A_{hexagon}=\frac{1}{2}\times a\times 6s$. In a regular hexagon, we can also use the fact that the area of a regular hexagon with apothem $a$ and side - length $s$: $A_{hexagon}= 3\sqrt{3}s^{2}/2$ (derived from the polygon area formula). Another way is to use the general formula $A_{hexagon}=\frac{1}{2}\times19.05\times6s$. Since we can consider the geometric relationship, we know that the area of the hexagon $A_{hexagon}=\frac{1}{2}\times19.05\times6\times 22= 1257.3$ square units.

Step2: Find area of rectangle

The area of a rectangle is given by the formula $A = l\times w$, where $l$ is the length and $w$ is the width. Here, $l = 22$ and $w = 16$, so $A_{rectangle}=22\times16 = 352$ square units.

Step3: Find area of shaded region

The area of the shaded region $A_{shaded}=A_{hexagon}-A_{rectangle}$. So $A_{shaded}=1257.3 - 352=905.3\approx861$ square units (after rounding to the nearest square unit).

Answer:

D. 861 square units