Area of Circle

What is Area of Circle?

 

The Area of Circle Definition: “Total surface on the plane covered by the circumference of a circle is called the Area of Circle.”

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How to find area of circle?

Area of circle can be calculated by using the formula Area = πr² or πd²/4, where r is the radius of circle and d is the diameter of circle. This formula can be derived from the following activities:

1.    Take a circular-shaped paper.

2.    Cut the circle into two equal halves.

3.    Cut each half further into small pieces equally as you can see in the figure.

4.    Arrange the pieces of two halves in one place to form like a rectangle as shown in the figure.

5.    As we go on increasing the number of small pieces it will tend to be a rectangle PQRS of length = half of circumference (c/2) and breadth = radius (r).

 

We know, 

Circumference of the circle (c) = 2πr

∴ Length of rectangle (l) = C/2 = 2πr/2 = πr

   Breadth of rectangle (b) = r

Now,

Area of circle,    

= Area of rectangle PQRS

                   = length × breadth

                   = πr × r

                   = πr² 

Since, the radius is half of the diameter, the area of a circle in terms of diameter can be found by replacing r = d/2, i.e. 

Area of circle = π(d/2)² = πd²/4

 

Therefore, Area of Circle Formula is given by, 

 Area of circle = πr²

 

And, area of a circle with diameter is given by, 

Area of circle = πd²/4

 

 

How to find the area of a circle with the circumference?

 

If the circumference of a circle is given, we have to find first the radius or diameter of the circle from the given circumference. Then, we can calculate the area of a circle by using the formula in terms of radius or diameter.

 

 

How to find the radius of a circle with the area?

 

We have,

πr² = A (Area of the circle)

i.e.    r² = A/π

or,     r = √(A/π)

 

Therefore, the radius of circle formula with the area is given by,

r = √(A/π)

 

 

How to find the diameter of a circle with the area?

 

We have,

πd²/4= A (Area of the circle)

i.e.    πd² = 4A

or,     d² = 4A/π

or,     d = √(4A/π)

 

Therefore, the diameter of the circle formula with the area is given by,

d = √(4A/π)

 

 

Semicircle | What is a Semicircle?

Semicircle is half of a circle. The diameter of a circle divides a circle into two equal parts, and each part is called a semicircle.

 

 

Area of Semicircle

 

Since semicircle is exactly the half of a full circle. Its area is given by the half of the area of full circle. Therefore, the area of semicircle formula is given by,

Area of Semicircle =  πr²/2

 

 

Sector | What is sector?

The region enclosed by any two radii and the arc of a circle is called a Sector of the circle. In the figure, the shaded region POQ is an example of a sector. There are minor sector and major sector. The shaded region is a minor sector and the unshaded region is a major sector.

 

 

Area of sector | How to find the area of a sector?

The area of a sector can be calculated by the area of the full circle and the central angle made by the corresponding sector. Since a complete central angle in a circle is 360°, each angle represents the 1/360 part of the area of a full circle. If r is the radius of circle and θ° is the central angle made by a minor sector, then the area of sector formula is given by the following formula,

Area of minor sector = πr²θ/360°

Area of major sector = πr²(1 - θ/360°)

 

 

Worked Out Examples

 

Example 1: find the area of a circle of radius 14 cm?

 

Solution: Here,

Radius (r) = 14cm

 

We know,

Area of circle = πr²

          = 22/7 × (14cm)²

          = 22/7 × 196 cm²

          = 616cm²

 

∴ Area of circle is 616cm². Ans.

 

Example 2: what is the area of a circle with a diameter of 10cm?

 

Solution: Here,

Diameter (d) = 10 cm

∴ Radius (r) = d/2 = 10/2 = 5cm

 

We know,

Area of circle = πr²

                 = 22/7 × 5²

                 = 22/7 × 25

                 = 78.6cm²

 

∴ Area of circle is 78.6cm². Ans.

 

 

Example 3: Find the area of circle whose circumference is 44 cm.

 

Solution: Here,

Circumference (C) = 44 cm

i.e.    2πr = 44

or,     2 × 22/7 × r = 44

or,     44/7 × r = 44

or,     r = 44 × 7/44

or,     r = 7 cm

 

We know,

Area of circle = πr²

                 = 22/7 × 7²

                 = 22/7 × 49

                 = 154cm²

 

∴ The area of the circle is 154cm². Ans.

 

 

Example 4: Find the circumference of a circle if the area is 300 square centimeters.

 

Solution: Here,

Area of circle (A) = 300 cm²

i.e.    πr² = 300

or,     22/7 × r² = 300

or,     r² = 300 × 7/22

or,     r² = 95.45

or,     r² = 7²

or,     r = 9.77 cm

 

Now,

Circumference of circle = 2πr

                            = 2 × 22/7 × 9.77

                            = 61.41 cm

 

∴ The circumference of the circle is 61.41cm. Ans.

 

 

Example 5: If the perimeter and the area of a circle are numerically equal then find the radius of the circle.

 

Solution: Here,

Perimeter of circle = 2πr

Area of circle = πr²

 

Perimeter and area of a circle are numerically equal,

∴       πr² = 2πr

Or,    r = 2 units. Ans.

 

 

Example 6: Find the area of the shaded region of the figure.

Solution: Here,

Side of square (l) = 8 cm

∴ Area of square (A₁) = l²

                             = (8 cm)²

                             = 64 cm²

 

Radius of quarter circle (r) = 8/2 = 4 cm

Area of 4 quarter circles (A₂)

= Area of one full circle

                             = πr²

                             = 22/7 × 4²

                              = 50.28 cm²

 

∴ Area of shaded region (A) =  A₁ – A₂

                             = 64 cm² – 50.28 cm²

                             = 13.72 cm²

 

∴ Area of the shaded region is13.72 cm². Ans.

 

Example 7: Find the area of the shaded region of the figure.

Solution: Here,

Length of rectangle (l) = 14 cm

Breadth of rectangle (b) = 7 cm

∴ Area of rectangle (A₁) = l × b

                             = 14 cm × 7 cm

                             = 98 cm²             

 

Radius of each half circle (r) = 7/2 = 3.5 cm

Area of 4 half circles (A₂)

= Area of two full circle

                             = 2 × πr²

                             = 2 × 22/7 × (3.5)²

                             = 77 cm²

∴ Area of shaded region (A) =  A₁ – A₂

                             = 98 cm² – 77 cm²

                             = 21 cm²

 

∴ The area of the shaded region is 21cm². Ans.

 

 

Example 8: ABC is a quadrant of a circle of radius 14cm. Find its area.

Solution: Here,

Radius (r) = 14cm

 

We know,

Area of circle = πr²

                    = 22/7 × (14cm)²

                    = 22/7 × 196 cm²

                    = 616cm²

∴ Area of quadrant circle = ¼ of Area of circle

= ¼ × 616cm²

= 154cm² Ans.

 

Example 9: Find the area of a square inscribed in a circle with radius r.

 

Solution: Here,

Radius of circle = r

Diameter = 2r

 

The diameter is the diagonal of the square inscribed in a circle. And, the area of square in terms of diagonal is given by d²/2. Therefore,

 

Area of the square inscribed in a circle = d²/2

                             = (2r)²/2

                             = 4r²/2

                             = 2r²

Therefore, the area of a square inscribed in a circle is given by the formula,

Area of square = 2r² Ans.

 

 

Example 10: What is the area of the largest square that can be inscribed in a circle of radius 12cm.

 

Solution: Here,

Radius of circle = 12cm

 

We know,

Area of the square inscribed in a circle = 2r²

                             = 2 × (12cm)²

                             = 2 × 144cm²

                             = 288cm² Ans.

 

 

If you have any questions or problems regarding the Area of Circle, you can ask here, in the comment section below.

 

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