Circle

Circle - A shape with infinite symmetry, no beginning or end. It's a symbol of unity, wholeness, and completeness. From ancient times to modern Mathematics, the circle holds a special place. What about circles interests you?

The circle, a perfect shape, boasts infinite symmetry and boundless continuity. Its every point equidistant from the center, symbolizing unity and completeness. Ancient cultures revered it for its divine qualities, while Mathematics and Science found it fundamental. A symbol of eternity, cycles, and harmony, the circle's simplicity belies its profound significance in art, nature, and human understanding.

1.0Circle Definition

A Circle is a fundamental geometry in a two-dimensional geometric shape defined by a set of points that are all equidistant from a fixed center point. It consists of all the points on a plane that are at a fixed distance, known as the radius, away from the center. In essence, it is the locus of points equidistant from a single point, forming a closed curve.

2.0Circumference of a Circle

The circumference of a circle is the total distance around its outer edge. It's calculated using the formula:

Circumference = 2 × π × radius

or equivalently,

Circumference = π × diameter

where π (pi) is a mathematical constant around 3.14159, and the radius, the distance from the circle's center to any point on its edge. The diameter is the distance across the circle through its center, equal to twice the radius.

Certainly! While the term "perimeter" is traditionally used for polygons, such as squares, rectangles, or triangles, it is commonly interchanged with the term "circumference" when discussing circles. The perimeter of a polygon represents the combined length of its sides, whereas the circumference of a circle denotes the total distance around its outer boundary.

3.0Area of Circle Formula

The formula to calculate the area of a circle is:

Area = π × radius²

In this formula:

• π (pi) is a mathematical constant around 3.14159.
• The radius represents the measurement from the center of a circle to any point along its circumference.

You square the radius and multiply it by π to find the area of the circle. This formula gives you the amount of space enclosed by the circle's perimeter.

4.0Equation of Circle

In a coordinate plane, the equation that defines a circle is formulated as: 

(x − h)² + (y − k) ² = r²

where:

• The center of the circle is specified by the coordinates (h, k),
• r is the radius of the circle.

If you expand and simplify the left side of the equation, you'll see that the squared distance from any point (x, y) on the circle to its center (h, k) is equivalent to the square of the radius.

5.0General Equation of Circle

In equation of circle, coefficient of x² = Coefficient of y² and coefficient of xy = 0,

x²  + y² 2gx + 2fy + c = 0

⇒ Center = ( –g, –f) 

Radius = √g² + f² – c

6.0Properties of Circle 

Some properties of circles are listed below:

Property 1: If a straight line drawn from the center of a circle bisects a chord, not passing through the centre, then it cuts the chord at right angles. Conversely, if it cuts the chord at right angles, then it bisects the chord.

Property 2: Equal chords of a circle are equidistant from the center. AB = CD, OP ⊥ AB and OQ ⊥ CD ⇒ OP = OQ

Conversely, Chords which are equidistant from the centre are equal. OP = OQ, OP ⊥ AB and OQ ⊥ CD ⇒ AB = CD

Property 3: Of any two chords of a circle, that which is nearer to the center is greater than one lying remote. Conversely, the greater of two chords is nearer to the center.

OP > OQ ⇒ CD < AB

Property 4: If P be any point (interior, exterior or on the circle) then greatest distance between point P and circumference of the circle is PO + r and the smallest distance between point P and circumference of the circle is |PO – r| (where r is radius and O is centre)

7.0Area of a Sector of a Circle

The area of a sector of a circle is a fraction of the total area enclosed by the circle, defined by two radii and the connecting arc. It's calculated using the formula:

Area of Sector = $\frac{\text{ Central Angle }}{360^{\circ }}\times \pi r^2$ 

Where:

• Central Angle is the angle subtended by the arc at the center of the circle, measured in degrees.
• r is the radius of the circle.

This formula finds the fraction of the circle represented by the central angle and multiplies it by the total area of the circle πr² to get the area of the sector.

8.0Area of a Segment of a Circle

A segment of a circle is the part of the circle that lies between a chord (a line segment connecting two points on the circle) and the arc (the curved part of the circle between those two points).

There are two types of segments:

1. Minor Segment: The region bounded by a chord and the smaller of the two arcs determined by the chord.
2. Major Segment: The region bounded by a chord and the larger of the two arcs determined by the chord.

The area of a segment of a circle can be calculated using the formula:

$AreaofSegment=\frac{\theta }{360^{\circ }}\times \pi r^2-\frac{1}{2}{r}^2\sin \theta$ 

Where:

• A is the area of the segment,
• r is the radius of the circle,
• θ is the central angle of the segment in radians.

$A=\frac{\theta }{360^{\circ }}\times \pi r^2-\frac{1}{2}{r}^2\sin \theta$

Also Read: Area of the Sector of the Circle

9.0Solved Question of Circle

Question 1: If the circumference of a circle is tripled, then the ratio of the area of the new circle to the area of the original circle is:

(A) 1: 3 (B) 1:9 (C) 9:1 (D) 1:18

Solution: Let the original radius of the circle be r.

The original circumference of the circle is C = 2 × π × r.

When the circumference is tripled, the new circumference becomes 3C = 6 × π × r.

We know that the area of a circle is given by A = π × r².

So, the area of the original circle is A₁ = π × r², and the area of the new circle is A₂ = π × (3r)² = 9πr².

Therefore, the ratio of the area of the new circle to the area of the original circle is:

$\frac{{\mathrm{A}}_2}{{\mathrm{A}}_1}=\frac{9\pi {\mathrm{r}}^2}{\pi {\mathrm{r}}^2}=9$

which is 9 : 1

Hence, the correct answer is: (C) 9: 1

Question 2: The circle passing through the points (−1,0) and touching the y-axis at (0,2) also passes through the point:

1. (−3/2, 0) (B) (−5/2, 2) (C) (−3/2, 5/2) (D) (−4, 0)

Solution: Let (h, k) be center of circle.

The circle touches the y-axis.

∴ Radius of circle = –h

Equation of circle-

(x − h)² + (y − k) ² = h² .... (1)

Since the circle passes through (0, 2)

Therefore,

h² + (2 − k) ² = h²

⇒ (k − 2) ² = 0

⇒ k = 2

Given that the circle also passes through (−1, 0),

Therefore,

(−1 − h)² + 2² = h²

h² + 1 + 2h + 4 = h²

h = −5/2

Substituting the value of h and k in eqⁿ (1), we get

(x +5/2) ² + (y − 2) ² = (5/2) ² 

Now, we can find the point from which the circle passes.

As the point ( –4, 0) satisfy the equation of circle.

Thus, the circle will also pass through the point (−4, 0).

Hence the correct answer is (D) (−4, 0).

Question 3: Find the equation of the circle that passes through the point (2, 1) and just touches the coordinate axes. 

1. x² + y² – 2x – 2y – 1 = 0 (C) x² + y² + 2x + 2y + 1 = 0
2. x² + y² – 2x – 2y + 1 = 0 (D) x² + y² + 2x + 2y – 1 = 0

Solution: Given, the circle touching the coordinate axes.

The equation of circle touching the coordinate axes is

(x – a)² + (y – a)² = a², with (a, a) is center and a is radius of circle.

Since, the circle is passing through the point (2, 1).

⇒ (2 – a)² + (1 – a)² = a²

⇒ 4 – 4a + a² + 1 – 2a + a² = a² 

⇒ a² – 6a + 5 = 0

⇒ (a – 5) (a – 1) = 0

⇒ a = 5 or a =1

radius a = 1 and a = 5

The equation of circle is- 

(x – 1)² + (y – 1)² = 1² and (x – 5)² + (y – 5)² = 5² 

x² + y² – 2x – 2y + 1 = 0 and x² + y² – 10x – 10y + 25 = 0

Hence the correct answer is (B) x² + y² – 2x – 2y + 1 = 0

Question 4: Find the coordinates of the center and radius of the circle x² + y² + 12x – 16y – 44 = 0.

Solution: We know that the general equation of circles is

x² + y² + 2gx + 2fy + c = 0      …..(i)

Center = (–g, –f)

Radius = √g² + f² – c

We have given equation of circle as x² + y² + 12x – 16y – 44 = 0.

On comparing with eq(i), we get

Hence, g = 6, f = –8, c = –44

So, Centre = (–6, 8)

Radius = √g² + f² –c

= √36 + 64 + 44

= √144

Radius = 12

Question 5: Find the equation of the circle passes through the points (1, 2), (3, –4), and (5, –6).

Solution: Given that, the circle passes through points P (1, 2), Q (3, –4), and R (5, –6).

We know that the general equation of circle is-

x² + y² + 2gx + 2fy + c = 0      ….. (i) 

We have, P (1, 2), Q (3, −4) and R (5, −6)

Since P, Q and R lies on (i)

1 + 4 + 2g + 4f + c = 0    …(ii)

9 + 16 + 6g – 8f + c = 0   …(iii)

25 + 36 + 10g – 12f + c = 0    …(iv)

Solving (ii), (iii) and (iv), we get,

g = –11, f = –2 and c =25

from (i)

The equation of circle is-

x² + y² − 22x − 4y + 25 = 0

10.0Sample Questions on Circle

1. What is the formula for the area of a circle?

Ans: The formula of the area of a circle is: Area = π × radius².

2. How do you find the equation of a circle? 

Ans: The equation of a circle in a coordinate plane is: (x − h)² + (y − k) ² = r², where (h, k) represents the center of the circle is specified by the coordinates (h, k), and r is the radius.

3. How do you calculate the area of a sector of a circle?

Ans: The formula for area of a sector of a circle is:

Area of Sector = $\frac{\text{ CentralAngle }}{360^{\circ }}\times \pi {\mathrm{r}}^2$

where the central angle is the angle formed by the arc at the circle's center, measured in degrees, and r is called the radius of the circle.