The angle of a sector in a given circle is 200 and the area of the sector is equal to 800 sq. cm. Find the arc length of the sector.
To find the arc length of the sector, you will need to use the formula:
Arc Length = (Angle / 360) × Circumference
First, let's find the circumference of the circle using the area of the sector. The formula to find the area of the sector is:
Area = (Angle / 360) × πr²
We are given that the area of the sector is 800 sq. cm. So, we can rearrange the equation to solve for the radius (r):
800 = (200/360) × πr²
800 = (5/9) × πr²
800 × 9/5 = πr²
(7200/5) = πr²
r² = 1440/π
r ≈ √(1440/π)
r ≈ √(1440/3.14)
r ≈ √(457.96)
r ≈ 21.4 cm (rounded to one decimal place)
Now that we have found the radius of the circle, we can find the circumference using the formula:
Circumference = 2πr
Circumference = 2 × 3.14 × 21.4
Circumference ≈ 134.8 cm (rounded to one decimal place)
Finally, we can find the arc length by substituting the angle (200) and the circumference (134.8) into our arc length formula:
Arc Length = (Angle / 360) × Circumference
Arc Length = (200 / 360) × 134.8
Arc Length ≈ 74.9 cm (rounded to one decimal place)
Therefore, the arc length of the sector is approximately 74.9 cm.
1/2 r^2 θ = 800
arc length s = rθ
A = pi r^2 (200 / 360) = 200
solve for r
then
circumference = 2 pi r
arc length = (200/360) * circumference
assuming that is 200°,
200/360 = θ/2π
θ = 2π(200/360) = 10π/9 radians
area of sector = r^2 θ
800 = r^2(10π/9)
r^2 = 9/10π (800) = 720/π
r = 15.139
arc length = r θ = 15.139(10π/9) = 52.844 cm