An arc of a circle is of length 5p cm and the sector is bounds has an area of

20p cm^2
find the radius of the circle.

thank u @reiny

let the radius be r

circumference = 2πr
area = πr^2

πr^2/20π = 2πr/5π
r^2/20 = 2r/5
5r^2 = 40r
r^2 - 8r = 0
r(r-8) = 0
r = 8 or r = 0, of course we would want r > 0 to any area at all

r = 8

check:
if r = 8
circumf= 16π
area= 64π

20π/64π = 5/16
5π/16π = 5/16
All looks good

To find the radius of the circle, we can use the formulas for the length of an arc and the area of a sector of a circle.

1. Length of an Arc (l):
The length of an arc can be calculated using the formula:
l = (θ / 360) * (2πr)
where θ is the central angle (in degrees) of the sector, and r is the radius of the circle.

2. Area of a Sector (A):
The area of a sector can be calculated using the formula:
A = (θ / 360) * πr^2
where θ is the central angle (in degrees) of the sector, and r is the radius of the circle.

Let's solve the problem step by step:

Given:
Length of Arc (l) = 5π cm
Area of Sector (A) = 20π cm^2

Step 1: Finding the central angle (θ)
We have the formula for the length of an arc:
l = (θ / 360) * (2πr)
Rearranging the formula to solve for θ:
θ = (l * 360) / (2πr)

Substituting the given values:
θ = (5π * 360) / (2πr)
θ = 900 / r

Step 2: Find the radius (r) using the area of the sector (A)
We have the formula for the area of a sector:
A = (θ / 360) * πr^2

Substituting the values and solving for r:
20π = (900 / r) * (πr^2)
20 = (900 / r) * r^2
20r = 900
r = 900 / 20
r = 45 cm

Therefore, the radius of the circle is 45 cm.