1/2 r^2 θ = 800
arc length s = rθ
arc length s = rθ
solve for r
then
circumference = 2 pi r
arc length = (200/360) * circumference
200/360 = θ/2π
θ = 2π(200/360) = 10π/9 radians
area of sector = r^2 θ
800 = r^2(10Ï€/9)
r^2 = 9/10Ï€ (800) = 720/Ï€
r = 15.139
arc length = r θ = 15.139(10π/9) = 52.844 cm
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 = 2Ï€r
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.