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Difficulty: HardPerimeter and Area of 2D Figures

A circle is perfectly inscribed within a square that has a perimeter of 4848 centimeters. What is the area of the region located inside the square but outside the circle, in square centimeters?

  1. A
    144π−144144\pi - 144
  2. B
    48−36π48 - 36\pi
  3. 144−36π144 - 36\piAnswer
  4. D
    144−12π144 - 12\pi

Answer

The correct answer is 144−36π144 - 36\pi.
To find the area of the specified region, the area of the inscribed circle must be subtracted from the area of the square. A perimeter of 48 gives the square a side length of 12 and an area of 144. The inscribed circle's diameter matches the square's side length (12), meaning its radius is 6. Its area is calculated as pi times the radius squared, which is 36pi. Subtracting these gives the final area of 144 - 36pi.

Step-by-Step Solution

1
Find the side length of the square.
The side length is 48÷4=1248 \div 4 = 12 cm.
The perimeter of a square is equal to 44 times its side length.
2
Calculate the area of the square.
The area is 12×12=14412 \times 12 = 144 cm 2^2.
The area of a square is the side length squared.
3
Determine the radius of the inscribed circle.
The radius is 12÷2=612 \div 2 = 6 cm.
Because the circle is perfectly inscribed, its diameter equals the side length of the square (1212 cm). The radius is half of the diameter.
4
Calculate the area of the circle.
The area is π×62=36π\pi \times 6^2 = 36\pi cm 2^2.
The formula for the area of a circle is A=πr2A = \pi r^2.
5
Find the area of the region outside the circle but inside the square.
The area is 144−36π144 - 36\pi cm 2^2.
Subtracting the circle's area from the total square's area leaves the remaining corner regions.

Key Concept

Area of composite figures and inscribed shapes
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Topics

Perimeter and Area of 2D Figures

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