select the correct answer.

an engineering firm designs a custom hexagonal screw for a computer board. a sketch of the top of the screw is shown. what is the area of the screw head?

(a) 62 \text{ mm}^2
(b) 187 \text{ mm}^2
(c) 36 \text{ mm}^2
(d) 108 \text{ mm}^2

Question

select the correct answer.

an engineering firm designs a custom hexagonal screw for a computer board. a sketch of the top of the screw is shown. what is the area of the screw head?

(a) 62 \text{ mm}^2
(b) 187 \text{ mm}^2
(c) 36 \text{ mm}^2
(d) 108 \text{ mm}^2

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Area of Polygons", "Coordinate Geometry" ], "current_concepts": [ "Area of Polygons", "Coordinate Geometry" ] } ## Divide the hexagon into simpler geometric shapes $$ \begin{aligned} &\text{Vertices of the hexagon: } (0, 6), (6, 3), (6, -3), (0, -6), (-6, -3), (-6, 3)\ &\text{Divide the hexagon vertically along the } y\text{-axis into two symmetric pentagons, or horizontally into:}\ &\text{1. A central rectangle with vertices } (-6, 3), (6, 3), (6, -3), (-6, -3)\ &\text{2. A top triangle with vertices } (-6, 3), (6, 3), (0, 6)\ &\text{3. A bottom triangle with vertices } (-6, -3), (6, -3), (0, -6) \end{aligned} $$ ## Calculate the area of each component $$ \begin{aligned} &\text{Area of central rectangle: } A_{\text{rect}} = \text{width} \times \text{height} = (6 - (-6)) \times (3 - (-3)) = 12 \times 6 = 72\text{ mm}^2\ &\text{Area of top triangle: } A_{\text{top}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times (6 - 3) = \frac{1}{2} \times 12 \times 3 = 18\text{ mm}^2\ &\text{Area of bottom triangle: } A_{\text{bottom}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times (-3 - (-6)) = \frac{1}{2} \times 12 \times 3 = 18\text{ mm}^2 \end{aligned} $$ ## Sum the areas to find the total area $$ \begin{aligned} &A_{\text{total}} = A_{\text{rect}} + A_{\text{top}} + A_{\text{bottom}} = 72 + 18 + 18 = 108\text{ mm}^2 \end{aligned} $$

$62\text{ mm}^2$

$187\text{ mm}^2$

$36\text{ mm}^2$

$108\text{ mm}^2$

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

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QUESTION IMAGE

Question

Explanation:

Divide the hexagon into simpler geometric shapes

Calculate the area of each component

Sum the areas to find the total area

Answer: