Question

the diagram below shows a rectangle inside a regular hexagon. the apothem of the hexagon is 17.32 units. to the nearest square unit, what is the area of the shaded region? apothem length: 17.32 a. 1359 square units b. 719 square units c. 373 square units d. 1039 square units

Explanation:

Step1: Find hexagon side length

For a regular hexagon, apothem $a = \frac{s\sqrt{3}}{2}$. Rearrange to solve for $s$:
$$s = \frac{2a}{\sqrt{3}} = \frac{2\times17.32}{\sqrt{3}} \approx 20$$

Step2: Calculate hexagon perimeter

Perimeter $P = 6s = 6\times20 = 120$

Step3: Compute hexagon area

Area formula: $A_{hex} = \frac{1}{2} \times a \times P$
$$A_{hex} = \frac{1}{2} \times 17.32 \times 120 = 1039.2$$

Step4: Calculate rectangle area

Area formula: $A_{rect} = l \times w$
$$A_{rect} = 20 \times 16 = 320$$

Step5: Find shaded region area

Subtract rectangle area from hexagon area:
$$A_{shaded} = 1039.2 - 320 = 719.2 \approx 719$$

Answer:

B. 719 square units