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14 votes
14 votes
The diagram below shows a square inside a regular octagon. The apothem of the octagon is 15.69 units. To the nearest square unit, what is the area of the shaded region? 13 U 13 U Apothem length: 15.69 O A. 1463 square units B. 816 square units C. 647 square units OD. 764 square units ​

The diagram below shows a square inside a regular octagon. The apothem of the octagon-example-1
User Tage
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2 Answers

20 votes
20 votes

Answer:

C. 647 square units

Explanation:

To find the shaded area, subtract the area of the unshaded square from the area of the octagon.

Area of the octagon


\textsf{Area of a regular polygon}=(n\:l\:a)/(2)

where:

  • n = number of sides
  • l = length of one side
  • a = apothem

Given:

  • n = 8
  • l = 13
  • a = 15.69

Substitute the given values into the formula and solve for A:


\implies \textsf{Area}=\sf (8 \cdot 13 \cdot 15.69)/(2)


\implies \textsf{Area}=\sf (1631.76)/(2)


\implies \textsf{Area}=\sf 815.88\:\:square \:units

Area of the square


\implies \textsf{Area}=\sf 13^2=169 \:\:square \:units

Area of the shaded region

= area of the octagon - area of the square

= 815.88 - 169

= 646.88

= 647 square units (nearest square unit)

17 votes
17 votes

perimeter of octagon

  • 8(13)
  • 104units

Now area

  • perimeter×apothem /2
  • 104(15.69)/2
  • 52(15.69)
  • 815.88units²

Area of square

  • 13²
  • 169units²

Area of shaded region

  • 815.88-169
  • 646.88units²
  • 647units²
User Trosendal
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