The sector of an image has an area of 30 cm^2 and the length of the arc is s=7 cm. Determine the angle, theta. Enter the answer in exact form.

Question

Also, given the area of a sector, A, and the length or the arc of the sector, s, determine the general formula for the angle in radians as a function of A and s.

This is one of a few extra credit problems that I don't understand (extra credit problems in my class aren't previously covered so I don't know where to begin to solve this problem. Any help is greatly appreciated!

Answer 1

A sector is a slice of a circle. The arc length, "s" is the distance around the curve of the sector. So the arc length (s) is related to circumference, (c)  in the same way sector area (A_s ) relates to the circle area (A_c).

 

A_s / A_c = s/c

 

A_s / (pi)r² = s / 2(pi)r

 

2A_s / s  = r

 

Arc length, s = rΘ

where Θ = the central angle that forms the sector in radians

 

since we know r, the circle radius

 

s = 2A_s * Θ / s

 

s² / 2A_s = Θ

 

 

 

Answer 2

Hi Daniel,

area of sector = pi*r^2*(theta/360)
length of arc = 2*pi*r*(theta/360)
30 = pi*r^2*(theta/360)
7 = 2*pi*r*(theta/360)

Set (theta/360) from the two equations equal to each other.
30/pi*r^2 =7/( 2*pi*r)
r = 60/7 cm

plug this in one of the equations to solve for theta. I will use
7 = 2*pi*r*(theta/360)
theta = 46.79 degrees

 

Given the area of a sector, A, and the length or the arc of the sector, s, the general formula for the angle in radians as a function of A and s will be as follows

A = pi*r^2(theta/2*pi)

if you rearrange this, you will get

theta = 2A/r^2

 

s = 2*pi*r*(theta/2*pi)

theta = s/r