Question

1. Find the measure, to the nearest degree, of the acute central angle in the figure below.

(The radius is 21 and the arc length is 18.)

2 Explain what would happen to the measure of the central angle in the figure aboce if the radius were doubled (and the arc length remained unchanged).

Answer 1

(arc length) ÷ circumference = (central angle) ÷ 360°

arc length - L=18

Radius r=21

Circumference - C.

$D=2r=C/\pi$

$C=2r\pi=2(21)3.14=131.88$

$L/C=\alpha/360$

$18/131.88=\alpha/360$

$\alpha=360L/C=360L/2r\pi$

$\alpha=49.1 °$

if the radius were doubled (and the arc length remained unchanged C will be doubled, then central angle will become smaller for two times.

If R=2r, then C=2x2rπ=4rπ, then $\alpha_{new}=360L/4r\pi$

$\alpha/\alpha_{new}=360L(4r\pi)/(2r\pi(360L))=2$