Answer
(a.) $26.2\; m$
(b.) $327 \;m^2$
(c.) $56 \;m^2$.
Work Step by Step
Given values are:-
Radius $r=25.0\; m$.
Central angle $\theta = 60.0^{\circ}$.
All sides of the triangle are equal. (equilateral triangle)
Therefore, the length of the cord $c= r=25.0 \; m$.
By using conversion factor $\frac{\pi \; rad}{180^{\circ}}$ Central angle in radian measure is $\theta = 60.0^{\circ}\times \frac{\pi \; rad}{180^{\circ}}$
Simplify.
$\theta = \frac{\pi }{3}\; rad$
(a.) Formula for the length of the arc is
$s=r\theta$
Substitute all values.
$s=(25.0\;m)(\frac{\pi }{3})$
Simplify.
$s=26.2\; m$ correct to one decimal place.
(b.) Formula for the area of the sector is
$A=\frac{1}{2} r^2 \theta $
Substitute all values
$A=\frac{1}{2} (25.0 \; m)^2 (\frac{\pi }{3}) $
Simplify.
$A=327 \;m^2$ (Rounded value).
(c.) Formula for the area of the segment is
$A=\frac{1}{2}r^2\theta-\frac{c\sqrt{4r^2-c^2}}{4}$
From the part (b.) $\frac{1}{2} r^2 \theta = 327\; m^2 $
Substitute all values into the formula.
$A=327\;m^2-\frac{(25.0\; m)\sqrt{4(25.0\;m)^2-(25.0\; m)^2}}{4}$
Simplify.
$A=327\;m^2-271 \; m^2$
$A=56 \;m^2$. (Rounded value.)
