Question

Find the area and arc length of a sector with central angle 140.

Answer 1

a Remember that

The area of complete circle is equal to

$A=\pi\cdot r^2$

The area of a complete circle subtends a central angle of 360 degrees

so

Applying proportion

Find out the area of the sector with a central angle of 240 degrees

$\frac{\pi\cdot r^2}{360}=\frac{x}{240}$

Solve for x

$x=\frac{240\cdot\pi\cdot r^2}{360}$

Simplify

$x=\frac{2\cdot\pi\cdot r^2}{3}$

that is the area of the sector with a central angle of 240 degrees

Part b

The circumference of the complete circle is equal to

$C=2\pi r$

the arc length of the complete circle subtends a central angle of 360 degrees

so

Applying proportion

Find out the arc length by a central angle of 240 degrees

$\frac{2\pi r}{360}=\frac{x}{240}$

solve for x

$x=\frac{4\pi r}{3}$

that is the arc length for a central angle of 240 degrees