A sector of a circle has an area of 15\pi square units and a perimeter of 10 + 6\pi units.… — GRE General Test

A sector of a circle has an area of $15\pi$ square units and a perimeter of $10 + 6\pi$ units. What is the measure of the central angle of the sector, in degrees?

A
$108^\circ$
B
$135^\circ$
C
$144^\circ$
$216^\circ$ Cevap
E
$270^\circ$

Cevap

The correct answer is

216^\circ

, corresponding to the option stating

216^\circ

The perimeter of a sector is defined as

2r + s = 10 + 6\pi

, which yields a radius of

r = 5

and an arc length

s = 6\pi

. Checking with the sector area formula

\frac{1}{2}rs = \frac{1}{2}(5)(6\pi) = 15\pi

confirms these measurements. The total area of the circle is

\pi (5)^2 = 25\pi

. The sector thus constitutes

\frac{15\pi}{25\pi} = \frac{3}{5}

of the circle. Multiplying this fraction by

360^\circ

gives a central angle of

216^\circ

Adım Adım Çözüm

Express the sector perimeter formula and solve for radius and arc length.

Radius

r = 5

and arc length

s = 6\pi

The perimeter of a sector equals two radii plus its arc length:

\text{Perimeter} = 2r + s = 10 + 6\pi

. Equating standard and

\pi

-termed components gives

2r = 10 \implies r = 5

and arc length

s = 6\pi

Verify consistency using the sector area formula.

\text{Area} = 15\pi

, matching the given information.

The area of a sector can also be calculated as

\frac{1}{2} r s = \frac{1}{2}(5)(6\pi) = 15\pi

Find the total area of the circle and the fraction of the circle occupied by the sector.

Total area

= 25\pi

, area fraction

= \frac{3}{5}

The full circle area is

\pi r^2 = \pi (5^2) = 25\pi

. The sector represents

\frac{15\pi}{25\pi} = \frac{3}{5}

of the entire circle.

Calculate the central angle

\theta

in degrees.

\theta = 216^\circ

Multiply the fraction by

360^\circ

\theta = \frac{3}{5} \times 360^\circ = 216^\circ

Anahtar Kavram

Perimeter, Arc Length, and Area of a Circle Sector

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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