select the correct answer.

a business award is in the shape of a regular hexagonal pyramid. the height of the award is 95 millimeters and the base edge is 44 millimeters.

what is the surface area of the pyramid to the nearest square millimeter?

$18,541\text{ mm}^2$
$10,060\text{ mm}^2$
$13,611\text{ mm}^2$
$17,616\text{ mm}^2$

Question

select the correct answer.

a business award is in the shape of a regular hexagonal pyramid. the height of the award is 95 millimeters and the base edge is 44 millimeters.

what is the surface area of the pyramid to the nearest square millimeter?

$18,541\text{ mm}^2$
$10,060\text{ mm}^2$
$13,611\text{ mm}^2$
$17,616\text{ mm}^2$

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": [ "Surface Area of Pyramids", "Regular Hexagonal Pyramid" ], "current_concepts": [ "Surface Area of Pyramids", "Regular Hexagonal Pyramid", "Area of Polygons" ] } ## Calculate the area of the regular hexagonal base $$ \begin{aligned} s &= 44\text{ mm} \ A_{\text{base}} &= \frac{3\sqrt{3}}{2} s^2 \ A_{\text{base}} &= \frac{3\sqrt{3}}{2} (44)^2 = 2904\sqrt{3} \approx 5029.876\text{ mm}^2 \end{aligned} $$ ## Calculate the slant height of the pyramid $$ \begin{aligned} h &= 95\text{ mm} \ a &= \frac{s}{2\tan(30^\circ)} = \frac{s\sqrt{3}}{2} = 22\sqrt{3} \approx 38.105\text{ mm} \ l &= \sqrt{h^2 + a^2} = \sqrt{95^2 + (22\sqrt{3})^2} = \sqrt{9025 + 1452} = \sqrt{10477} \approx 102.357\text{ mm} \end{aligned} $$ ## Calculate the total surface area $$ \begin{aligned} A_{\text{lateral}} &= \frac{1}{2} \cdot P \cdot l = \frac{1}{2} \cdot (6 \cdot 44) \cdot \sqrt{10477} = 132 \cdot \sqrt{10477} \approx 13511.151\text{ mm}^2 \ A_{\text{total}} &= A_{\text{base}} + A_{\text{lateral}} \approx 5029.876 + 13511.151 = 18541.027\text{ mm}^2 \approx 18541\text{ mm}^2 \end{aligned} $$

(A) 18,541 mm²

(B) 10,060 mm²

(C) 13,611 mm²

(D) 17,616 mm²

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

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QUESTION IMAGE

Question

Explanation:

Calculate the area of the regular hexagonal base

Calculate the slant height of the pyramid

Calculate the total surface area

Answer: