an arc of length 110cm subtends an angle of 150° at the centre of a circle. find the perimeter of the sector
since the arc length is s=rθ, the perimeter is p = 2r + s
2(110/(5π/6)) + 110 = 110 + 264/π
To find the perimeter of the sector, we need to find the length of the arc and the two radii that form the sector.
Given:
- Length of the arc = 110 cm
- Angle at the center = 150°
To find the length of the arc, we can use the formula:
Length of the arc = (angle / 360°) * (2 * π * r), where r is the radius.
Let's calculate the radius:
1. We know that the angle at the center is 150°.
2. The angle at the center is twice the angle at the circumference subtended by the same arc.
So, the angle at the circumference = 150° / 2 = 75°.
3. We can use the formula: circumference = 2 * π * r, to find the radius.
circumference = 360°, r = ? (unknown)
2 * π * r = 360°
r = 360° / (2 * π)
r ≈ 57.29 cm (rounded to two decimal places)
Now, let's find the length of the arc:
Length of the arc = (angle / 360°) * (2 * π * r)
Length of the arc = (150° / 360°) * (2 * π * 57.29 cm)
Length of the arc ≈ (0.4167) * (2 * π * 57.29 cm)
Length of the arc ≈ 0.4167 * 359.9 cm
Length of the arc ≈ 149.9 cm (rounded to one decimal place)
The length of the arc is approximately 149.9 cm.
Finally, let's find the perimeter of the sector:
Perimeter of the sector = Length of the arc + 2 * (radius)
Perimeter of the sector = 149.9 cm + 2 * 57.29 cm
Perimeter of the sector = 149.9 cm + 114.58 cm
Perimeter of the sector ≈ 264.5 cm (rounded to one decimal place)
Therefore, the perimeter of the sector is approximately 264.5 cm.
To find the perimeter of the sector, we need to determine the length of the arc and the length of the two radii that form the sector.
1. Calculate the length of the arc:
The length of an arc is given by the formula: arc length = (θ/360) * 2πr, where θ is the central angle in degrees and r is the radius of the circle.
In this case, the arc length is 110 cm and the central angle is 150°:
arc length = (150/360) * 2πr
110 = (5/12) * 2πr
22/5 = πr
2. Find the radius of the circle:
Divide both sides of the equation by π:
22/5π = r
3. Calculate the perimeter of the sector:
The perimeter of the sector is equal to the sum of the arc length and the lengths of the two radii:
perimeter = arc length + 2r
perimeter = 110 + 2(22/5π)
perimeter = 110 + (44/5)π
Therefore, the perimeter of the sector is 110 + (44/5)π cm, or approximately 143.92 cm.