Question

What is the arc length cut off by a central angle of 45° in a circle of radius 16 yd?
a. 720 yd
b. 2.8125 yd
c. 12.5664 yd
d. 9.6754 yd

Answer 1

Since the arc is a portion of the circumference, we first find the circumference.

C = 2*pi*r
C = 2*pi*16
C = 32pi

We know that the arc is a 45 degree portion of the circumference and we know the whole circumference accounts for 360 degrees.

This means that the ratio of the central angle over 360 degrees must be the same as the ratio of the length of the arc over the circumference of 32pi.\[\frac{ 45 }{ 360 }=\frac{ a }{ 32\pi } \rightarrow 45 \times 32\pi = 360a \rightarrow a = \frac{ 45 \times 32\pi }{ 360 }=4\pi\ \approx 12.5664 \]Therefore, the length of the arc subtended by a 45 degrees central is approximately 12.5664 yards. That's option C.

@sfb