What is the area of the circular ring included between two concentric circles of radius 14 cm and 10.5 cm?

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150 Questions 100 Marks 180 Mins

Given:

As shown above in the figure, two concentric circles with centre At point O. Radius of the bigger circle = R cm, of the smaller circle = r cm. Area of between the two concentric circles = 286 cm2.Difference between the radii of the two circles = 7 cm = R – r ____ (1)

Formula Used:

Area of a circle = πR2 (Where R = Radius of the circle)

Calculation:

⇒ πR2 – πr2 = 286, 22/7 × (R2 – r2) = 286

⇒ (R – r)(R + r) = 13× 7, (R + r)× 7 = 13× 7, (R + r) = 13 cm ____(2)

From (1) & (2), we get

∴ R = 10 cm, r = 3 cm

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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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