A circle has a radius of 12\text{ centimeters}. What is the length, in centimeters, of the… — ACT

A circle has a radius of $12\text{ centimeters}$ . What is the length, in centimeters, of the arc intercepted by a central angle of $30^\circ$ ?

2\pi

centimeters

To find the arc length, multiply the total circumference of the circle,

2\pi r

, by the fraction of the circle represented by the central angle,

\frac{\theta}{360^\circ}

. With a radius of

12

centimeters and a central angle of

30^\circ

, this calculation yields

2\pi (12) \times \frac{30}{360} = 24\pi \times \frac{1}{12} = 2\pi

centimeters.

Identify the formula for arc length:

s = 2\pi r \left(\frac{\theta}{360^\circ}\right)

, where

r

is the radius and

\theta

is the central angle in degrees.

Formula established:

s = 2\pi r \left(\frac{\theta}{360^\circ}\right)

Arc length is the fraction of the total circumference determined by the central angle.

Substitute the given radius

r = 12\text{ cm}

and central angle

\theta = 30^\circ

into the formula.

Equation set up:

s = 2\pi (12) \left(\frac{30}{360}\right)

This sets up the specific calculation for the given circle.

Simplify the expression to find the final arc length.

s = 24\pi \left(\frac{1}{12}\right) = 2\pi

Simplification yields the exact length of the arc in terms of

\pi

The length of an arc is proportional to the fraction of the circle's circumference represented by the central angle.

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