Question

Derive the formula for the area of a sector, and then use it to choose all that are correct.

Answer 1

Answer:

Part A) $A_s=\frac{\pi r^{2}}{360^o}{\theta}$

Part B) option 1,option 4

Step-by-step explanation:

Part A) Derive the formula for the area of a sector

we know that

The area of circle is equal to

$A=\pi r^{2}$

The area of circle subtends a central angle of 360 degrees

so

using proportion

Find out the area of a sector  $A_s$  by a central angle of ∅ degrees

$\frac{\pi r^{2}}{360^o}=\frac{A_s}{\theta}$

$A_s=\frac{\pi r^{2}}{360^o}{\theta}$

Part B) Verify each case

case 1) we have

radius = 5 cm

angle = 120°

area = 26.2 cm 2

Find the area of the sector and then compare with the value of the given area

assume

$\pi=3.14$

substitute the given values

$A_s=\frac{(3.14)(5)^{2}}{360^o}{120^o}$

$A_s=26.2\ cm^2$

so

The given value of area is correct

case 2) we have

radius = 4 cm

angle = 105°

area = 16.7 cm 2

Find the area of the sector and then compare with the value of the given area

assume

$\pi=3.14$

substitute the given values

$A_s=\frac{(3.14)(4)^{2}}{360^o}{105^o}$

$A_s=14.7\ cm^2$

so

The given value of area is not correct

case 3) we have

radius = 6 cm

angle = 85°

area = 23.7 cm 2

Find the area of the sector and then compare with the value of the given area

assume

$\pi=3.14$

substitute the given values

$A_s=\frac{(3.14)(6)^{2}}{360^o}{85^o}$

$A_s=26.7\ cm^2$

so

The given value of area is not correct

case 4) we have

radius = 7

angle = 75°

area = 32.1 cm 2

Find the area of the sector and then compare with the value of the given area

assume

$\pi=3.14$

substitute the given values

$A_s=\frac{(3.14)(7)^{2}}{360^o}{75^o}$

$A_s=32.1\ cm^2$

so

The given value of area is correct