Question

A circle has a radius of 17 feet. find the radian measure of the central angle that intercepts an arc of length16 feet?

Answer 1

The central angle is `16/17` radians (approx. 0.9412).

Explanation:

An angle in radians is, by definition, the ratio of an arc length from a circle to that circle's radius:

`"angle (in radians)" = "length of arc"/"length of radius"`

As an easy example, for an arc that was the whole circumference, the central angle would be

`"angle" = "circumference"/"radius" = (2pi xx "17 feet")/"17 feet"" = 2pi`

Notice something interesting: since every circle's circumference is `2pi` times longer than its radius, the angle doesn't depend on the size of the circle.

(A circle that's twice as big would have both a double-sized circumference and a double-sized radius, so these "doublings" would cancel out.)

Ultimately, if you know the length of the arc, and you know the length of the radius, the central angle (in radians) is just how many radii long the arc is.

For the given question, the arc length is 16 feet, the radius is 17 feet, and so the central angle (in radians) is

`"angle" = "arc length"/"radius length"="16 feet"/"17 feet" ~~ 0.9412.`

The arc is 0.9412 times the length of the radius, so the central angle is 0.9412 radians.

Note: since this formula is the ratio of two lengths, their units will always cancel, making a radian angle "unitless".