**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 = πr^{2}
or πd^{2}/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

= πr^{2}

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} = πd^{2}/4

Therefore, **Area of Circle
Formula** is given by,

**Area of circle = πr ^{2}
**

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

**Area of circle = πd ^{2}/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,

πr^{2} = A (Area of the circle)

i.e. r^{2} =
A/π

or, r = √(A/π)

Therefore, the **r****adius 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,

πd^{2}/4= A (Area of the circle)

i.e. πd^{2} =
4A

or, d^{2} =
4A/π

or, d = √(4A/π)

Therefore, the **d****iameter 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
= πr ^{2}/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
= πr ^{2}θ/360°**

**Area of major sector
= πr ^{2}(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 = πr^{2}

= 22/7 × (14cm)^{2}

= 22/7 × 196 cm^{2}

= 616cm^{2}

∴ Area of circle is 616cm^{2}. 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 = πr^{2}

= 22/7 × 5^{2}

= 22/7 × 25

= 78.6cm^{2}

∴ Area of
circle is 78.6cm^{2}. 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 = πr^{2}

= 22/7 × 7^{2}

= 22/7 × 49

= 154cm^{2}

∴ The area of the circle is 154cm^{2}. Ans.

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

**Solution:** Here,

Area of circle (A) = 300
cm^{2}

i.e. πr^{2} = 300

or, 22/7 × r^{2} = 300

or, r^{2} = 300 × 7/22

or, r^{2} = 95.45

or, r^{2} = 7^{2}

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 = πr^{2}

Perimeter and area of a circle are numerically equal,

∴ πr^{2} = 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 (A_{1}) = l^{2}

=
(8 cm)^{2}

=
64 cm^{2}

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

Area of 4
quarter circles (A_{2})

= Area of one full
circle

= πr^{2}

= 22/7 × 4^{2}

= 50.28 cm^{2}

∴ Area of
shaded region (A) = A_{1} – A_{2}

= 64 cm^{2} – 50.28 cm^{2}

= 13.72 cm^{2}

∴ Area of the shaded region is13.72 cm^{2}. 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
(A_{1}) = l × b

=
14 cm × 7 cm

=
98 cm^{2}

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

Area of 4
half circles (A_{2})

= Area of two full
circle

= 2 × πr^{2}

= 2 × 22/7 × (3.5)^{2}

= 77 cm^{2}

∴ Area of
shaded region (A) = A_{1} – A_{2}

= 98 cm^{2} – 77 cm^{2}

= 21 cm^{2}

∴ The area of the shaded region is 21cm^{2}.
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 = πr^{2}

= 22/7 × (14cm)^{2}

= 22/7 × 196 cm^{2}

= 616cm^{2}

∴ Area of quadrant circle = ¼ of Area of circle

= ¼ × 616cm^{2}

= 154cm^{2} 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 d^{2}/2. Therefore,

Area of the square inscribed in a circle = d^{2}/2

=
(2r)^{2}/2

=
4r^{2}/2

=
2r^{2}

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

Area of square = 2r^{2} 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 = 2r^{2}

=
2 × (12cm)^{2}

=
2 × 144cm^{2}

=
288cm^{2} 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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