Question

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 ​

Answer 1

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

Answer 2

Answer:

C.  647 square units

Step-by-step explanation:

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

Area of the octagon

$\textsf{Area of a regular polygon}=\dfrac{n\:l\:a}{2}$

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:

$\implies \textsf{Area}=\sf \dfrac{8 \cdot 13 \cdot 15.69}{2}$

$\implies \textsf{Area}=\sf \dfrac{1631.76}{2}$

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

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)