A circle has a circumference of 20\pi. A sector of this circle is defined by a central ang… — GRE General Test

A circle has a circumference of $20\pi$ . A sector of this circle is defined by a central angle that intercepts an arc of length $5\pi$ . What is the area of this sector?

A
$10\pi$
$25\pi$ Cevap
C
$50\pi$
D
$100\pi$
E
$5\pi$

Cevap

The area of the sector is

25\pi

First, the radius is found using

2\pi r = 20\pi

, which gives

r = 10

. The total area of the circle is

\pi (10)^2 = 100\pi

. The fraction of the circle defined by the sector is

\frac{5\pi}{20\pi} = \frac{1}{4}

. Multiplying the total area by this fraction gives

\frac{1}{4} \times 100\pi = 25\pi

Adım Adım Çözüm

Find the radius of the circle from the circumference.

Since

C = 2\pi r = 20\pi

, solving for

r

gives

r = 10

The radius is required to calculate the total circle area.

Determine the fraction of the circle that the sector represents.

\text{Fraction} = \frac{\text{Arc Length}}{\text{Circumference}} = \frac{5\pi}{20\pi} = \frac{1}{4}.

The ratio of arc length to circumference gives the proportion of the total circle occupied by the sector.

Calculate the total area of the circle.

A = \pi r^2 = \pi (10)^2 = 100\pi.

The area of a circle with radius 10 is

100\pi

Multiply the total area by the sector fraction.

\text{Sector Area} = \frac{1}{4} \times 100\pi = 25\pi.

Applying the proportional fraction yields the sector area.

Anahtar Kavram

The ratio of a sector's arc length to the full circumference is equal to the ratio of the sector's area to the full circle's area.

Soru

Cevap

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