Three sectors of a circle of radius 7 cm make angles of 60, 80 and 40 degrees at the centre, are shaded. Find the area of the shaded region. Plz reply fast.?

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Solution:

Let's break down the problem step by step to find the area of the shaded region.

Step 1: Finding the Area of the Circle
The area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius. In this case, the radius is given as 7 cm. So, the area of the entire circle is:

A_circle = π(7 cm)^2
A_circle = 49π cm^2

Step 2: Finding the Area of the Sectors
We are given that three sectors of the circle are shaded, and the central angles of these sectors are 60°, 80°, and 40°. To find the area of each sector, we can use the formula:

A_sector = (θ/360°) × A_circle

For the first sector with a central angle of 60°:
A_sector1 = (60°/360°) × 49π cm^2
A_sector1 = (1/6) × 49π cm^2
A_sector1 = 8.17π cm^2

For the second sector with a central angle of 80°:
A_sector2 = (80°/360°) × 49π cm^2
A_sector2 = (2/9) × 49π cm^2
A_sector2 = 10.89π cm^2

For the third sector with a central angle of 40°:
A_sector3 = (40°/360°) × 49π cm^2
A_sector3 = (1/9) × 49π cm^2
A_sector3 = 5.44π cm^2

Step 3: Finding the Area of the Shaded Region
To find the area of the shaded region, we need to subtract the sum of the areas of the three sectors from the area of the entire circle. Mathematically,

A_shaded = A_circle - (A_sector1 + A_sector2 + A_sector3)
A_shaded = 49π cm^2 - (8.17π cm^2 + 10.89π cm^2 + 5.44π cm^2)
A_shaded = 24.50π cm^2

Therefore, the area of the shaded region is 24.50π cm^2.

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