A circle has a central angle of 60^\circ that subtends an arc of length 4\pi. What is the … — SAT

A circle has a central angle of $60^\circ$ that subtends an arc of length $4\pi$ . What is the circumference of the circle?

The circumference of the circle is

24\pi

The correct answer is

24\pi

. A full circle is

360^\circ

. A central angle of

60^\circ

represents

\frac{60^\circ}{360^\circ} = \frac{1}{6}

of the entire circle. Therefore, the length of the arc subtended by this angle is

\frac{1}{6}

of the total circumference. To find the circumference, we multiply the arc length by

6

4\pi \times 6 = 24\pi

Find the fraction of the circle represented by the

60^\circ

central angle.

\frac{60^\circ}{360^\circ} = \frac{1}{6}

A full circle has a central angle of

360^\circ

, so the fraction of the circle is the ratio of the given central angle to

360^\circ

Set up the proportion relating the arc length to the total circumference (

C

\frac{1}{6} = \frac{4\pi}{C}

The ratio of the arc length to the total circumference equals the ratio of the central angle to the total degree measure of a circle.

Solve for the circumference (

C

) by multiplying both sides by

6

C = 6 \times 4\pi = 24\pi

Multiplying the arc length of the sector by the reciprocal of the fraction yields the total circumference of the circle.

Finding the circumference of a circle given the central angle and the subtended arc length using proportional reasoning.

Soru