question 4 of 10

the diagram below shows a square inside a regular octagon. the apothem of the octagon is 18.11 units. to the nearest square unit, what is the area of the shaded region?

apothem length: 18.11

a. 1087 square units
b. 1948 square units
c. 862 square units
d. 1027 square units

Question

question 4 of 10

the diagram below shows a square inside a regular octagon. the apothem of the octagon is 18.11 units. to the nearest square unit, what is the area of the shaded region?

apothem length: 18.11

a. 1087 square units
b. 1948 square units
c. 862 square units
d. 1027 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", "Properties of a Square", "Apothem" ], "new_concepts": [], "current_concepts": [ "Regular Polygon Area", "Properties of a Square", "Apothem", "Decomposition of Polygons" ] } ## Calculate the area of the regular octagon $$ \begin{aligned} s &= 15 \ n &= 8 \ P &= n \cdot s = 8 \cdot 15 = 120 \ a &= 18.11 \ A_{\text{octagon}} &= \frac{1}{2} \cdot P \cdot a = \frac{1}{2} \cdot 120 \cdot 18.11 = 1086.6 \end{aligned} $$ ## Calculate the area of the inner square $$ \begin{aligned} s_{\text{square}} &= 15 \ A_{\text{square}} &= s_{\text{square}}^2 = 15^2 = 225 \end{aligned} $$ ## Calculate the area of the shaded region $$ \begin{aligned} A_{\text{shaded}} &= A_{\text{octagon}} - A_{\text{square}} \ A_{\text{shaded}} &= 1086.6 - 225 = 861.6 \approx 862 \end{aligned} $$

(A) 1087 square units

(B) 1948 square units

(C) 862 square units

(D) 1027 square units

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Regular Polygon Area" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Calculate the area of the regular octagon

Calculate the area of the inner square

Calculate the area of the shaded region

Answer: