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Difficulty: MediumCircle Geometry: Arc Length and Sector Area

A circular metal disk has a radius of 1212 centimeters. A wedge-shaped sector with a central angle of 75∘75^\circ is cut out and removed from the disk. What is the perimeter, in centimeters, of the remaining portion of the disk?

  1. A
    5π5\pi
  2. B
    5π+245\pi + 24
  3. C
    19π19\pi
  4. D
    19π+1219\pi + 12
  5. 19π+2419\pi + 24Answer

Answer

19π+2419\pi + 24
The central angle of the remaining portion is 360∘−75∘=285∘360^\circ - 75^\circ = 285^\circ. The arc length of this remaining sector is 285∘360∘×2π(12)=19π\frac{285^\circ}{360^\circ} \times 2\pi(12) = 19\pi cm. Because cutting out the wedge exposes two straight sides equal to the radius (1212 cm each), the total perimeter is the sum of the curved arc and the two radii: 19π+12+12=19π+2419\pi + 12 + 12 = 19\pi + 24 cm.

Step-by-Step Solution

1
Find the central angle of the remaining major sector
360∘−75∘=285∘360^\circ - 75^\circ = 285^\circ
Removing a 75∘75^\circ wedge from a full 360∘360^\circ circle leaves a central angle of 285∘285^\circ.
2
Calculate the arc length of the remaining major sector
Arc length =285∘360∘×2π(12)=1924×24π=19π= \frac{285^\circ}{360^\circ} \times 2\pi(12) = \frac{19}{24} \times 24\pi = 19\pi cm
Arc length is given by the formula s=θ360∘×2πrs = \frac{\theta}{360^\circ} \times 2\pi r.
3
Calculate the total perimeter of the remaining shape
Perimeter =19π+12+12=19π+24= 19\pi + 12 + 12 = 19\pi + 24 cm
The total boundary of the remaining piece consists of the curved arc plus the two straight straight edges formed by radii where the cut occurred.

Key Concept

Perimeter of a Sector and Arc Length Formula
Estimated Time:1m 0s
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Topics

Circle Geometry: Arc Length and Sector Area

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