# Circles

A Circle is defined as a closed shape with a set of points in a plane at a specific distance from the center. In general the equation of a circle in x,y quadrant is given as

$(x-h)^{2} + (y-k)^{2} = r^{2}$

where (x,y) are the coordinate points
(h,k) is the coordinate of the centre of a circle
and r is the radius of a circle.

In other words, Locus of all points at a fixed distance from a reference central point is called a Circle.

The distance between any of the points and the center is known as the radius of a circle.

To understand what circles are in simple terms try the following exercise-

1. Take an empty sheet of paper and just mark a single point on the sheet, somewhere in the middle of the sheet, and name it to point O.
2. Select a random length for radius, for example, 3 cm.
3. Using a ruler, keep the reference zero mark on point O and randomly mark 3cm away from point O in all the direction.
4. Mark as many points as u want away from point O, but all of them should be exactly 3 cm away from point O.

If you’ve selected sufficient points, you may notice that the shape is starting to resemble a circle and this is exactly what the definition of a circle is.

“A line segment connecting the centre of a circle to any point on the circle itself .”

Diameter (d) –

“ A line segment having both the endpoints on the circle.” It is twice the length of radius i.e. d = 2r

Circumference (C) –

“It is the distance around the circle.” The word ‘perimeter’ is also sometimes used, although this usually refers to the distance around polygons, figures made up of the straight line segment.

Given by –

$C = \pi d = 2 \pi r$

Where, $\pi = 3.1415$

Area (A) –

Area of a circle is the amount of space occupied by the circle.

Given By-

Area = $\pi r^{2}$

Where, $\pi = 3.1415$

 Example- Find the Area and the circumference of a circle whose radius is 10 cm. (Take the value of $\pi$ to be 3.14) Solution- Given – Radius = 10 cm. Area = $\pi r^{2}$, $= 3.14 \times (10)^{2} \;\; cm^{2}$ $= 314 \;\; cm^{2}$ Circumference $C = 2 \pi r$ $= 2 \times 3.14 \times 10$ $= 62.8 cm^{2}$

Terminology used-

(i) Annulus-
The region bounded by two concentric circles. It is basically a ring-shaped object.

(ii) Arc – It is basically the connected curve of a circle.

(iii) Sector – A region bounded by two radii and an arc.

(iv) Segment- A region bounded by a chord and an arc lying between the chord’s endpoints. It is to be noted that segments do not contain the centre.

(v) Centre – It is the midpoint of a circle.

(vi) Chord- A line segment whose endpoints lie on the circle.

(vii) Diameter- A line segment having both the endpoints on the circle

(viii) Radius- A line segment connecting centre of a circle to any point on the circle itself.

(ix) Secant- A straight line cutting the circle at two points. It is also called as an extended chord.

(x)Tangent- A coplanar straight line touching the circle at a single point.

(xi) Semicircle- An arc which connects to both the ends of a diameter.

Proof for Area of a circle-

We know that Area is the space occupied by the circle.

Consider a concentric circle having external circle radius to be ‘r.’

Open all the concentric circle to form a right-angled triangle.

The outer circle would form a line having length $2 \pi r$ forming the base.

The height would be ‘r’

Therefore the area of the right-angled triangle formed would be equal to the area of a circle.

Area of a circle = Area of triangle = $\frac{1}{2} \times b \times h$

= $\frac{1}{2} \times 2 \pi r \times r$

= $\pi r^{2}$

#### Practise This Question

In the below figure the shaded region I and II are respectively