question 5 of 10

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. 1039 square units
d. 373 square units

Question

question 5 of 10

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. 1039 square units
d. 373 square units

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Regular Polygon Area", "Apothem", "Rectangle Area Formula", "Decomposition of Polygons" ], "new_concepts": [], "current_concepts": [ "Regular Polygon Area", "Apothem", "Rectangle Area Formula", "Decomposition of Polygons" ] } ## Calculate the side length and area of the regular hexagon $$ \begin{aligned} a &= 17.32 \ s &= \frac{2a}{\sqrt{3}} \approx \frac{2 \times 17.32}{1.73205} \approx 20 \ P &= 6s = 6 \times 20 = 120 \ A_{\text{hexagon}} &= \frac{1}{2} a P = \frac{1}{2} \times 17.32 \times 120 = 1039.2 \end{aligned} $$ ## Calculate the area of the inner rectangle $$ \begin{aligned} w &= 20 \ h &= 16 \ A_{\text{rectangle}} &= w \times h = 20 \times 16 = 320 \end{aligned} $$ ## Calculate the area of the shaded region $$ \begin{aligned} A_{\text{shaded}} &= A_{\text{hexagon}} - A_{\text{rectangle}} \ A_{\text{shaded}} &= 1039.2 - 320 = 719.2 \approx 719 \end{aligned} $$

A. 1359 square units

B. 719 square units

C. 1039 square units

D. 373 square units

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Decomposition of Polygons" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the side length and area of the regular hexagon

Calculate the area of the inner rectangle

Calculate the area of the shaded region

Answer: