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Zorluk: OrtaArea of Two-Dimensional Shapes

A circular tabletop is designed to have a square glass inlay. The square glass inlay is inscribed in the circle and has an area of 6464 square inches. What is the area, in square inches, of the circular tabletop?

  1. A
    16π16\pi
  2. 32π32\piCevap
  3. C
    64π64\pi
  4. D
    128π128\pi

Cevap

The area of the circular tabletop is 32π32\pi square inches.
The correct answer is 32π32\pi because the inscribed square has a side length of 88 inches. The diagonal of the square serves as the diameter of the circle, which is 828\sqrt{2} inches. The radius is therefore 424\sqrt{2} inches, and the area of the circle is π(42)2=32π\pi (4\sqrt{2})^2 = 32\pi square inches.

Adım Adım Çözüm

1
Find the side length of the inscribed square from its area.
The side length of the square is 88 inches.
The area of a square is given by s2s^2, where ss is the side length. Since the area is 6464, we solve s2=64s^2 = 64 to find s=8s = 8.
2
Find the length of the diagonal of the square, which represents the diameter of the circle.
The diagonal length is 828\sqrt{2} inches.
In a square with side length ss, the diagonal is s2s\sqrt{2}. Since the square is inscribed in the circle, the diagonal of the square is equal to the diameter of the circle.
3
Calculate the radius of the circle and then the area.
The radius is 424\sqrt{2} inches and the area is 32π32\pi square inches.
The radius rr is half of the diameter, so r=822=42r = \frac{8\sqrt{2}}{2} = 4\sqrt{2}. The area of the circle is given by A=πr2=π(42)2=32πA = \pi r^2 = \pi (4\sqrt{2})^2 = 32\pi.

Anahtar Kavram

Relating the area of an inscribed polygon to the circumscribed circle
Tahmini Süre:1m 30s
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Konular

Area of Two-Dimensional Shapes

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