Question

Consider circle O, in which arc XY measures 16π cm. The length of a radius of the circle is 32 cm.

What is the circumference of the circle?
π units
What is the ratio of the arc length to the circumference?


What is the measure of central angle XOY?

Answer 1

Circumference: 64π

Ratio: 1 : 4

Measure of ∠xoy: π/2

Explanation:

We are given an arc length of 16π. Since it's in terms of pi, we use the formula

S = rФ where r is the radius, and Ф is the measure of the angle in radians (in terms of pi)

We are given S = 16π and r = 32, plug those in and find Ф

16π = 32Ф

16π/32 = Ф

π/2 = Ф

This is the measure of the central angle.

The angle is π/2 radians. There are 2π radians in the circumference, so the circumference is 4 times the arc length created by the central angle. (There are 4 halves in 2) so the ratio of the arc length tothe circumference is 1 : 4

The formula for circumference is C = 2πr, where r is the radius, so we hace

C = 2π(32) = 64π

Answer 2

The answer is

64

1/4

90

Hope this helps.