We are already aware of some methods of finding perimeters and areas of simple plane figures such as rectangles, squares, parallelograms, triangles, etc. The measure of the surface enclosed by a figure is its area. In our daily life, we come across many objects that relate to a circular shape in some form or the other. Cycle wheels, wheelbarrow, dartboard, round cake, drain cover, bangles, brooches, circular paths, etc. are some examples of such objects. In this chapter, we shall begin our discussion about the circle, its area and apply this knowledge in identifying the area of circle formula. We will also, discuss two special parts of a circular region known as sector and segment. Let us start with our learning!

**Area of Circle Formula**

**What is a Circle?**

In our daily life, we see many objects around us, which are round in shape, such as wheels of a vehicle, bangles, dials of many clocks, coins of denominations 50 p, Re 1, Rs 2, Rs 5 and Rs 10, key rings, buttons of shirts, etc. In a clock, we observe that the second’s hand goes around the dial of the clock rapidly and its tip moves in a round path. This path traversed by the tip of the second’s hand forms a circle.

Let us take another example. Let us take a compass and fix a pencil in it. Put its point leg on a point on a sheet of a paper. Open the other leg to some distance. Keep the point leg on the same point, rotate the other leg through one revolution. We get a closed figure traversing through the pencil on paper.

As we know, it is a circle. We get the circle by keeping one-point static and drawing all the points that are at a fixed distance. This gives us the definition of a circle as, the collection of all the points in a plane, which are at a static distance from a static point in the plane.

**Formula**

**Area** = \( \pi r^{2} \)

Where,

\(\pi\) | Read as pi. This is the constant ratio whose value is equal to 3.1416 or 22/7 |

r | the radius of the circle |

**d** = 2 x r

Where,

d | The diameter of the circle |

r | The radius of the circle |

**Area** =\( (\pi d^{2})/4 \)

Where,

\( \pi\) | Read as pi. This is the constant ratio whose value is equal to 3.1416 or 22/7 |

d | The diameter of the circle |

**Solved Examples**

Now that we have some clarity about the concept and meaning of the area of a circle, let us try some examples to deepen our understanding of the subject.

Q: The diameter of a circular garden is 9.8 m. Find its area.

Ans: As we already have a formula for calculating the area of a square. Let us substitute the values

Diameter, d = 9.8 m.

Therefore, radius r = d/2 = 9.8/2 = 4.9 m

Area of the circle = \(\pi r^{2} \)

Area of the circle = \(\frac{22}{7}\) x (4.9)^2, considering pi = \(\frac{22}{7}\)

= \(\frac{22}{7}\) x \(\frac{49}{10}\) x \(\frac{49}{10}\) = 75.46 sq m.

Q: What will be the cost of polishing a circular table-top of radius 2 m at the rate of Rs 10 per square meter.

Ans: To identify the total cost of polishing, we first need to find the total area of the circular table-top.

Let us first identify the area of table-top.

The radius of the circular table-top is given as 2m.

Hence, its area will be given by,

Area = \(\pi r^{2} \)

= 3.14 x 2 x 2 sq metres, considering pi = 3.14

= 12.56 sq m

Now, the cost of polishing per square metre is given as Rs 10,

Therefore, the cost of polishing the total area of table-top = area x 10 = 12.56 x 10 = 125.60

Hence, the total cost of polishing the circular table-top is Rs 125.60

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