A sector is cut from a circle of radius 21 cm. The angle of the sector is 150 degrees.
Find the length of the arc.
Find the perimeter of the sector (take π).
Answer (2)
The length of the arc of the sector is approximately 54.95 cm and the perimeter of the sector is approximately 96.95 cm.
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To find the length of the arc and the perimeter of the sector, we start by understanding the relationship between the circle and the sector.
1. Length of the Arc:
The arc length L of a sector is given by the formula: L = 360 θ × 2 π r where θ is the angle of the sector in degrees, and r is the radius of the circle.
Given that the radius r is 21 cm and the angle θ is 150 degrees: L = 360 150 × 2 × π × 21
First, simplify 360 150 :
360 150 = 12 5
Then calculate L :
L = 12 5 × 2 × π × 21 L = 12 5 × 42 π L = 17.5 π
Therefore, the length of the arc is approximately 55 cm when you approximate π ≈ 3.14 .
2. Perimeter of the Sector:
The perimeter P of the sector includes the arc length plus the two radii: P = L + 2 r P = 17.5 π + 2 × 21 P = 17.5 π + 42
Substituting π ≈ 3.14 for an approximate value: P = 17.5 × 3.14 + 42 P = 54.95 + 42 P ≈ 96.95
Thus, the perimeter of the sector is approximately 96.95 cm.
