Here we will discuss about the area of a circular ring along with some example problems.
The area of a circular ring bounded by two concentric circle of radii R and r (R > r)
= area of the bigger circle – area of the smaller circle
= πR\(^{2}\) - πr\(^{2}\)
= π(R\(^{2}\) - r\(^{2}\))
= π(R + r) (R - r)
Therefore, the area of a circular ring = π(R + r) (R - r), where R and r are the radii of the outer circle and the inner circle respectively.
Solved example problems on finding the area of a circular ring:
1. The outer diameter and the inner diameter of a circular path are 728 m and 700 m respectively. Find the breadth and the area of the circular path. (Use π = \(\frac{22}{7}\)).
Solution:
The outer radius of a circular path R = \(\frac{728 m}{2}\) = 364 m.
The inner radius of a circular path r = \(\frac{700 m}{2}\) = 350 m.
Therefore, breadth of the circular path = R - r = 364 m -
350 m = 14 m.
Area of the circular path = π(R + r)(R - r)
= \(\frac{22}{7}\)(364 + 350) (364 - 350) m\(^{2}\)
= \(\frac{22}{7}\) × 714 × 14 m\(^{2}\)
= 22 × 714 × 2 m\(^{2}\)
= 31,416 m\(^{2}\)
Therefore, the area of the circular path = 31416 m\(^{2}\)
2. The inner diameter and the outer diameter of a circular path are 630 m and 658 m respectively. Find the area of the circular path. (Use π = \(\frac{22}{7}\)).
Solution:
The inner radius of a circular path r = \(\frac{630 m}{2}\) = 315 m.
The outer radius of a circular path R = \(\frac{658 m}{2}\) = 329 m.
Area of the circular path = π(R + r)(R - r)
= \(\frac{22}{7}\) (329 + 315)(329 - 315) m\(^{2}\)
= \(\frac{22}{7}\) × 644 × 14 m\(^{2}\)
= 22 × 644 × 2 m\(^{2}\)
= 28,336 m\(^{2}\)
Therefore, the area of the circular path = 28,336 m\(^{2}\)
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