the area of a sector PQR is a @rad.and the radius of the of the circle is rcm,the area of the sector is 20cm^2 and the perimeter is 26cm.find r?
If the angle is x radians,
1/2 r^2 x = 20
2r + rx = 26
now just solve for r and x.
r= 10.5 , 2.5
To find the radius of the circle (r), we can use the given information about the area and perimeter of the sector.
1. Area of the sector (A) = 20 cm^2
We are given that the area of sector PQR is 20 cm^2.
The formula to calculate the area of a sector is:
A = (θ/360) * π * r^2
Here, θ is the angle in degrees of the sector.
Since the angle is not given, let's assume that the entire circle is 360 degrees, as it is usually the case.
2. Perimeter of the sector (P) = 26 cm
The perimeter of a sector is the sum of the lengths of the two radii and the arc length between them.
The formula to calculate the perimeter of the sector is:
P = 2r + 2πr(θ/360)
Now, we can use these two equations to find the value of r.
Substituting the given values into the formulas:
20 cm^2 = (θ/360) * π * r^2 (equation 1)
26 cm = 2r + 2πr(θ/360) (equation 2)
Since the value of θ is not given, we need to find it first.
Divide equation 2 by equation 1 to eliminate θ:
26 cm / 20 cm^2 = (2r + 2πr(θ/360)) / ((θ/360) * π * r^2)
Simplify:
1.3 cm^-1 = 2r + 2πr(θ/360) / (r^2 * θ)
Now, rearrange the equation to isolate θ:
1.3 cm^-1 * r^2 = 2r + 2πr(θ/360)
Multiply both sides by (360 / 2πr):
(1.3 cm^-1 * r^2) * (360 / 2πr) = (2r + 2πr(θ/360)) * (360 / 2πr)
Simplify:
234 cm = 720 + 360θ
Subtract 720 from both sides:
360θ = -486 cm
Divide both sides by 360:
θ = -1.35 degrees
Since the angle cannot be negative, there might be an error in the problem statement or calculations done before.
Please double-check the given information or provide additional details.