If the central angle of a circle has measure 100° and makes a minor arc with length 25, what is the radius?

Question

If the central angle of a circle has measure 100° and makes a minor arc with length 25, what is the radius?

Answer 1

The applicable formula here is: s = r • θ

-- Here, s represents the arc length; r represents the radius length; and θ represents the central angle measured in radians.

-- We know that the arc length is 25 and that the angle measure is 100 degrees, which we must first convert to radians with the formula: angle measure in radians = angle measure in degrees • (π / 180).

-- 100 • (π / 180) = 100π / 180 = 5π / 9

-- We can now use the measure of the angle in radians in the formula, plugging in the other information that we know.

-- s = r • θ --› 25 = r • 5π/9

-- Multiply both sides by the reciprocal of 5π/9, which is 9/5π.

-- (9/5π)(25) = r • (5π/9)(9/5π)

-- 45/π = r

-- r = 14.32

Answer 2

Use a proportion to find the circumference C: 25 / C = 100^° / 360^°, because arclength is to C as central angle is to full circle. This gives us C = 90, and we know that C = 2 π r in general. The radius r is then r = 45 / π.