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Question

$ AB$ and $ CD$ are arcs of two concentric circles of radii $ \text{21 cm}$ and $ \text{7 cm}$ respectively and centre at $ O$. If $ \text{∠AOB = 30°}$, find the area of the shaded region.


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Solution

Step 1: Finding the areas of sectors OCD and OAB:

Let A1 and A2 be the areas of sectors OCD and OAB respectively.

Angle made by both sectors at the centre of circle =30°.

A1= Area of sector OCD whose radius is 7cm, is

A1=30360×227×72(UsingA=θ360×πr2)=776cm2

A2= Area of sector OAB whose radius is 21cm, is

A2=30360×227×212(UsingA=θ360×πr2)=4624cm2

Step 2: Finding the shaded area:

Areaofshadedregion=AreaofsectorOAB-AreaofsectorOCD=A2-A1=4624-776=462×3-77×212=1386-15412=123212=102.67cm2

Hence, the area of shaded region is 102.67cm2.


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