Circles in Maths

Circles in Maths: A circle is a two-dimensional shape where all points on the circumference are the same distance from the centre. In other words, it is a collection of all points in a plane that are the same distance away from a fixed point, called the centre. Its area is equal to pi times the square of its radius.

In this article, you will understand more about circles in math, including their formulas, examples, parts of circles, and practice problems on circles.

What is a Circle in Maths?

A circle is a perfectly round, flat shape with no corners or ends. Every point on the circumference of the circle is the same distance away from the center, which is called the radius. A line going all the way through the center and touching both sides of the circle is the diameter, and it’s twice as long as the radius. Circles are fundamental shapes in geometry, used to calculate areas, circumferences, and more. They’re also all around us, from wheels to pizzas!

Circle Definition

Circle is a two-dimensional geometric shape that consists of all points in a plane that are equidistant (at the same distance) from a fixed point called the centre. Distance from the centre to any point on the circle is called the radius of the circle.

Circle Examples

Various objects that we observe in real life are circular in shape. Some examples of circular-shaped objects are, Chapattis, Coins, Wheels, Rings, Buttons, CDs / DVDs, Bangles, Plates, etc.

How to Draw a Circle?

Drawing a circle is the most basic construction problem that is taught to students. Construction a circle requires some basic tools such as,

• Pencil
• Compass
• Scale or Ruler

To draw a circle follow the steps below,

Step 1: Attach the pencil to the compass and make sure the tip of the compass and pencil are aligned properly.

Step 2: Mark a point on the paper and call it the centre of the circle.

Step 3: Measure the radius of the circle using the ruler and adjust the compass accordingly.

Step 4: Draw the circle using the compass from the centre point marked in Step 2.

Interior and Exterior of Circle

If we draw a circle it divides the 2-D plane into three parts which are:

Position	Description
Inside the Circle	A point whose distance from the circle’s center is less than the radius; known as the interior point.
On the Circle	Points whose distance from the circle’s center is equal to the radius; these lie on the circumference.
Outside the Circle	Points whose distance from the circle’s center is greater than the radius; known as exterior points.

Parts of Circle

Circle has various parts and some of the important parts of the circle are:

• Center
• Radius
• Diameter
• Chords
• Tangent
• Secant
• Arc
• Segment
• Sector

Center of Circle

Center of a circle is defined as the point from which any point on the circumference of the circle is at a fixed distance. The center of a circle is located inside the circle

Radius of Circle

Radius of Circle is the distance between any point on the circumference of the circle and the fixed point called the centre. The constant distance between any point on the circle and its centre is called the radius.

Radius = Diameter/2 

r = D/2

Diameter of Circle

A chord passing through the centre of the circle is called the diameter of the circle. It is the largest chord of the circle and every diameter has an equivalent length. It is denoted by the letter ‘D’.

Diameter = 2 × Radius

D = 2r

Chord of Circle

Any line which touches the circle at two points on its circumference is the chord of the circle. The diameter is also a chord of the circle. The longest chord of the circle is the diameter of the circle.

Tangent of Circle

A line that touches the circle only at one point on its circumference is called the tangent of the circle. It is always perpendicular to the radius of the circle at the point of tangency.

Secant of Circle

A line intersecting the circumference of the circle is called the secant on the circle. We also call it the extended chord.

Arc of a Circle

Any portion of the circumference of the circle is called the arc of the circle.

Segment in Circle

The chord divides the circle into two parts and each part is called the segment of the circle. There are two segments formed by a chord that is,

• Major Segment
• Minor Segment

Sector of a Circle

The area between two radii and the corresponding arc in a circle is called the sector of the circle. There are two types of sectors

• Major Sector
• Minor Sector

Image added below shows the parts of a circle.

Properties of Circle

Some of the properties of the circle are :

• Circle with the same radii are called the congruent circle.
• Equal chords are equidistant from the centre of the circle.
• Equidistant chords from the centre of the circle are always equal.
• The perpendicular drawn from the centre of the circle to the chord always bisects the chord.
• We can draw two tangents from an external point to a circle.
• Tangents drawn from the endpoints of the diameter are always parallel to each other.

Circle Formulas

There are various formulas related to the circle. Let the radius of the circle is ‘r’ then some of the important formulas related to the circle are:

Formulas of Circle	Expressions
Area of Circle	πぱいr2
Circumference of Circle	2πぱいr
Length of Arc of Circle	θしーた × r
Area of Sector of Circle	(θしーた × r2) / 2
Length of Chord	2 r sin(θしーた/2)
Area of Segment	r2(θしーた – sinθしーた)/2

Note: Value of πぱい is taken to be 3.14 or 22/7

Area of Circle Proof

We can easily prove the formula for the area of the circle using the area of the triangle formula. For this first, we have to draw various concentric circles inside the given circle. Then open all the concentric circles to form a right-angled triangle.

If the radius of the given circle is r, then the outer circle would form the base of the right triangle having length 2πぱいr.

Height of the triangle is ‘r’

Area of the right-angled triangle so formed is equal to the area of a circle.

Area of a Circle = Area of Triangle = (1/2) × base × height = (1/2) × 2πぱい r  × r

Therefore, 

Area of Circle = πぱいr²

Types of Circles

Apart from our normal Circle, there are 5 other types of circles based on their shapes:

Semicircle

A semicircle is a two-dimensional geometric shape that is half of a complete circle. It is a shape formed by cutting a circle exactly in half. It has the following properties:

• It is a one-dimensional locus of points, meaning all its points lie on a single curved line.
• It has a curved boundary called an arc, which measures 180 degrees (πぱい radians).
• It has a straight line segment called the diameter, which passes through the centre of the circle and connects the two endpoints of the arc.
• It has only one line of symmetry, which is the perpendicular bisector of the diameter.

The area and perimeter of a semicircle can be calculated using the following formulas:

• Area: A = (πぱいr²)/2, where r is the radius of the circle.
• Perimeter: P = πぱいr + 2r, where r is the radius of the circle.

Quarter Circle

A quarter circle, also known as a quadrant, is a shape formed by dividing a circle into four equal parts. It shares many similar properties with a semicircle, but with some key differences:

• Curved boundary: Like a semicircle, it has a curved boundary called an arc, but this arc only measures 90 degrees (πぱい/2 radians), which is one-fourth of the full circle’s circumference.
• Straight sides: It has two straight sides instead of one diameter. These sides are radii of the original circle, extending from the centre to the endpoints of the arc.
• Symmetry: Just like a semicircle, it has only one line of symmetry, which is the perpendicular bisector of one of its radii.

The area and perimeter of a quarter circle can be calculated using the following formulas:

• Area: A = (πぱいr²)/4, where r is the radius of the circle. This represents one-fourth of the area of the whole circle.
• Perimeter: P = πぱいr + 2r, where r is the radius of the circle. This is the same formula as for a semicircle, as it includes the length of the curved arc and the two straight radii.

Tangent Circles

Tangent circles are two circles in the same plane that meet at exactly one point, without overlapping or intersecting further. This single point of contact is called the point of tangency. There are two main types of tangent circles:

• Internally Tangent Circles: These circles share the same interior space and touch each other inside that shared area.
• Externally Tangent Circles: These circles have separate interiors and touch each other outside those regions.

Properties of Tangent Circles:

• Tangent Lines: Each circle has a tangent line that passes through the point of tangency and is perpendicular to the radius drawn from the centre to that point.
• Radical Circle: All circles tangent to a given pair of circles lie on a common circle called the radical circle. This circle’s centre is the midpoint of the line segment connecting the centres of the original circles, and its radius is half the difference of their radii.
• Apollonius Problem: Finding the circles tangent to three given circles is a famous geometric problem known as Apollonius’ problem. It has several elegant solutions with various applications in fields like astronomy and engineering.

Concentric Circles

Concentric circles are two or more circles that have the same centre point but different radii. Imagine dropping pebbles of different sizes into a still pond, creating ripples that share the same starting point but expand outwards at different rates.

Following are some key features of concentric circles:

• Shared Centre: Their central point, where all radii meet, serves as the common foundation for all circles.
• Differing Radii: Each circle has a unique radius, determining its size and distance from the centre. The larger the radius, the bigger the circle.
• Non-Intersecting: Concentric circles never overlap or intersect, as they maintain their distinct boundaries defined by their individual radii.
• Symmetrical Layers: They create a visually pleasing arrangement of nested circles, radiating outwards from the centre in a harmonious and symmetrical fashion.

Other Types of Circles

There are many other types of circles, some of which are:

• Circumcircle
• Incircle
• Excircle
• Fractle Circle

Circumcircle

A circumcircle refers to the unique circle that passes through all the vertices of a given polygon, such as a triangle, quadrilateral, or any other polygonal shape.

This circle’s center lies at the intersection of the perpendicular bisectors of the sides of the polygon, and its radius is the distance from the center to any of the vertices.

Inscribed circle or Incircle

An inscribed circle, also known as an incircle, is a circle that is tangent to all sides of a given polygon.

In the case of a triangle, the inscribed circle is the largest circle that fits snugly within the triangle, touching all three sides. The center of the incircle, called the incenter, is the point of concurrency for the angle bisectors of the triangle, and its radius is known as the inradius.

Excircle or Escribed Circle

An excircle, or escribed circle, is a circle that lies outside a given polygon and is tangent to one of its sides and the extensions of the other two sides.

For example, in the context of a triangle, there are three excircles, each tangent to one side of the triangle and the extensions of the other two sides. The center of an excircle lies at the intersection of the external angle bisectors of the polygon, and its radius is known as the exradius.

Fractal Circles

Fractal circles are geometric figures that exhibit self-similarity and intricate patterns at multiple scales, characteristic of fractals.

They are created using recursive processes or algorithms that generate complex structures by repeatedly applying a set of rules.

Articles related to Circles in Maths:

• Sector of Circle
• Tangent of Circle
• Important points about Circle
• Circle Formulas For Diameter, Area and Circumference

Solved Examples on Circles

Few examples on circles are:

Example 1: If the diameter of a circle is 142.8 mm, then what is the radius of the circle?

Solution:

Diameter = 142.8 mm

By Formula, 

Diameter = 2 radius

Radius = (142.8 ÷ 2) = 71.4 mm

Thus, radius of circle is 71.4 mm

Example 2: Distance around a park is 21.98 yd. What is the radius of the park?

Solution: 

Circumference of the Park = 21.98 yd

We know that,

Circumference = 2πぱい × Radius

Radius = Circumference / 2πぱい

Radius = 21.98 / 2×3.14  = 3.5

Thus, radius of circle is 3.5 yd

Example 3: Inner circumference of a circular track is 440 m, and the track is 14 m wide. Calculate the cost of levelling the track at 25 rupees/m².

Solution:

Let radius of inner circle be r m. 

Now, 

Inner Circumference = 440 m 

2πぱいr = 440 

2 × 22/7 × r = 440

r = 440 × 744 

Inner Radius, r = 70 

Width of track = 14 m

Outer radius (R) = Inner Radius(r) + Width = (70 + 14) = 84 m

Area of Track = πぱい(R²− r²)

= πぱい (84² – 70²)

= 22/7 × (7056 – 4900) 

= 6776 m²

Cost of levelling at 25 rupees per square meter =  6676 × 25 = 169400 Rupees

Thus, cost of levelling track is 169400 Rupees

Example 4: Find the length of the chord of a circle where the radius is 7 cm and the perpendicular distance from the chord to the centre is 4 cm.

Solution: 

Given,

• Radius, r = 8 cm

Distance of Chord to Centre, d = 3 cm 

Chord Length = 2√(r² – d²) 

= 2√(8² – 3²) 

= 2√(64 – 9) 

= 2√55 

 = 2 × 7.416

Chord length = 14.83 cm

Thus, length of chord is 14.83 cm

Example 5: If radius of a circle is 5 cm and measure of the angle of the arc is 110˚, what is the length of the arc? 

Solution: 

Arc Length = (2 × πぱい × r) × angle / 360°

= 2 × 3.14 × 5 × 110/360° 

= 9.6 cm

Thus, length of arc is 9.6 cm.

Example 6: If the area of a sector with a radius of 6 cm is 35.4 cm². Calculate the angle of the sector. 

Solution: 

Area of Sector = Angle/360°  × (πぱい × r × r)

Angle/360° × (πぱい × 6 × 6) = 35.4 

Angle = (35.4/36πぱい) × 360° = 112.67°

Thus, angle of sector is, 112.67°

Practice Problems on Circles in Maths

Some practice problems on circles are,

Question 1. Calculate the circumference of a circle with a radius of 5 centimetres.

Question 2. Given a circle with a radius of 8 meters, find the diameter, circumference, and area.

Question 3. Find the length of an arc on a circle with a central angle of 60 degrees and a radius of 10 centimetres.

Question 4. Calculate the area of a sector of a circle with a central angle of 45 degrees and a radius of 6 inches.

Question 5. Determine the radius of a circle if its circumference is 31.4 meters (use πぱい ≈ 3.14).

Question 6. Given a circle with a diameter of 18 centimetres, find the length of a chord that is 8 centimetres away from the centre of the circle.

MCQs on Circles in Maths

Answer the following MCQs:

Q1. What is Name of Line segment that passes through Centre of a circle and has endpoints on Circle?

1. Radius
2. Diameter
3. Chord
4. Tangent

Q2. Distance from Centre of a circle to any point on its circumference is known as:

1. Circumference
2. Radius
3. Diameter
4. Arc Length

Q3. What is Relationship between Radius and Diameter of a circle?

1. Diameter = Radius
2. Diameter = 2 × Radius
3. Diameter = 0.5 × Radius
4. Diameter = Radius²

Q4. If a circle has a radius of 5 units, what is its diameter?

1. 2 units
2. 5 units
3. 10 units
4. 25 units

FAQs on Circles in Maths

What are Circles in Geometry?

We define a circle in geometry as the locus of a point which is always at a fixed distance from a fixed point, called centre of circle.

What are Circle Formulas?

A circle has various formulas but, important formulas for circle are,

• Area of Circle(A) = πぱいr²
• Circumference of Circle(C) = 2πぱいr

What is Chord in Circle?

A line segment joining two points of circumference of circle is called chord of circle. A chord passing through centre of circle is called radius of circle.

What is Difference Between Chord and Secant in Circle?

A chord is a line segment which joints any two points on circumference of circle, whereas secant is a line joining any two points on circumference of circle, it is also called as extended chord.

What is Circumference of Circle?

Length of boundary of circle is called Circumference of Circle. It is calculated using the formula, C = 2πぱいr

What is Area of Circle?

Space occupied inside circumference of circle is called Area of Circle. It is calculated using the formula, A = πぱいr²

What are Properties of Circles?

The properties of the circle are,

• It has the infinite number of radii.
• The radius of the circle is always perpendicular to the tangent of the circle.
• Diameter is the longest chord of the circle

What are Types of Circles in Maths?

We can categorize circles into various categories but important types of circles are,

• Concentric Circles: Circles having different radii but the same centre are called concentric circles.
• Congruent Circles: Circles having the same radius but different centres are called congruent circles.