**Area of a circle**Â is the region occupied by the circle in a two-dimensional plane. ItÂ can be determined easily using a formula,Â **Ï€r ^{2}**, where r is the radius of the circle. Apart from the area, circles have a radius, diameter and circumference, as its property. This circumference is similar to the perimeter, where the total distance covered by the boundary of the circle is evaluated.Â Â If the area of a given circle with radius r units is Ï€r

^{2}Â square units, then an

**area of semi-circle**and quarter-circle with the same radius will be equal to Ï€r

^{2}/ 2Â andÂ Ï€r

^{2Â }/4Â square units respectively.

## Area of a Circle Formula

Let us take a circle with radius r.

Then the area for this circle, A, is given by;**Â **

AÂ = Ï€r^{2} |

Here, the value of pi,Â **Ï€ = 22/7 or 3.14**

Area of a circle can be visualized & proved using two methods, namely

- Determining the circle’s area using rectangles
- Determining the circle’s area using triangles

Let us understand both the methods one-by-one-

### Using Areas of rectangles

The circle is divided into 16 equal sectors and the sectors are arranged as shown in the fig. 3. The area of the circle will be equal to that of the parallelogram-shaped figure formed by the sectors cut out from the circle. Since the sectors have equal area, each sector will have equal arc length. The green coloured sectors will contribute to half of the circumference and blue coloured sectors will contribute to the other half. If the number of sectors cut from the circle is increased, the parallelogram will eventually look like a rectangle with length equal to Ï€r and breadth equal to r.

The area of a rectangle (A) will also be the area of a circle. So, we have

- AÂ = Ï€rÃ—r
- AÂ = Ï€r
^{2}

### Using Areas of triangles

Fill the circle with radius r with concentric circles. After cutting the circle along the indicated line in fig. 4 and spreading the lines, the result will be a triangle. The base of the triangle will be equal to the circumference of the circle and its height will be equal to the radius of the circle.

So, the area of the triangle (A) will be equal to the area of the circle. We have

AÂ = 1/2Ã—baseÃ—height

AÂ = 1/2Ã—(2Ï€r)Ã—r

AÂ = Ï€r^{2}

**Circumference of Circle**

A perimeter of closed figures is defined as the length of its boundary. When it comes to circles, the perimeter of a circle is given a different name. It is called Circumference of the circle. To define the circumference of the circle, knowledge of a term known as â€˜piâ€™ is required. Consider the circle shown in the fig. 1, with centre at O and radius r.

The circumference/Perimeter of the circle is equal to the length of its boundary. The length of rope which wraps around its boundary perfectly will be equal to its circumference, which can be measured by using the formula **C = 2Ï€r**, where r is the radius of the circle.

Ï€, read as â€˜piâ€™is defined as the ratio of the circumference of a circle to its diameter. This ratio is the same for every circle. Consider a circle with radius â€˜râ€™ and circumference â€˜Câ€™. For this circle

- Ï€Â = Circumference/Diameter
- Ï€Â = C/2r
- C = 2Ï€r

The same is shown in fig. 2.

### Examples

**Question: If radius of a circle is 15cm. Then find its area.**

**Solution:** Given, radius of circle = 15cm

The, area will be;

A =Â Ï€r^{2}

A =Â Ï€.15^{2}

A = 706.5 sq.cm.

**Question: If the diameter of a circle is 10cm. Then find its area.**

**Solution:** Given, diameter = 10cm

So, radius will be = 10/2 = 5cm

Hence, area A =Â Ï€r^{2}

A =Â Ï€.5^{2}

A= 78.5 sq.cm

**Question: If the circumference of a given circle is 30cm. Then what will be its area.**

**Solution:** Given, the circumference of a circle = 30cm

We know, from the formula of circumference, C =2Ï€r

So, we can write,

2Ï€r = 30

or r = 30/2Ï€ = 15/Ï€

As we found the value of r, now we can find the area;

A =Â Ï€r^{2}

A =Â Ï€(15/Ï€)^{2}

On solving we get,

A = 71.65 sq.cm.

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