Circles

In this mini-lesson, we will explore the world of circles. We will learn various parts of a circle like a diameter, radius, circumference, circles chord, tangent circles, major arc, segment, sector, and discover other interesting aspects of it. You can try your hand at solving a few interesting interactive questions at the end of the page.

One of the earliest inventions of humankind - the wheel is proof of our ancestors’ knowledge about circles. 

However, before we begin, let's first understand the formation of a circle.

Try this simulation which shows how a circle is formed by the collection of points that are equidistant from the center. 

In this short lesson, we will learn all about circles and their properties.

Lesson Plan

What Is a Circle?

Definition

A circle is formed by the set of points that are at a constant or at a fixed distance (radius) from a fixed point (center) in the plane.

The fixed point is called the origin or center of the circle.

The fixed distance of the points from the origin is called the radius.

circle with center


Terms Related To Circles

Radius

Radius is the distance from the center of a circle to the boundary of the circle.

Let's refer to the circular park shown below.

Multiple radii shown in the park

A circle can have multiple radii as they all start from the center and touch the boundary of the circle at various points.

 

Diameter

Diameter is a straight line passing through the center that connects two points on the boundary of the circle.

A circle marked with diameter, centre, boundary

Let's go back to the figure of the park.

Multiple diameters shown in the park

We can see that \(AFB\) is a diameter.

We should note that there can be multiple diameters in that park, but they should:

  • pass through the fountain (center).
  • be straight lines.
  • touch the boundary of the park at two points.

Chord

Any line segment that touches the circle at two different points is known as the chord of a circle

Referring to the park, we can see that \(PQ\) is a chord that shows where the play area starts.

Circular park marked showing the chord


Tangent in Circles

If a line touches a circle at a unique point, then that line is called a tangent line to the circle. 

tangent to the circle touches at one point

In the above figure, line \(L\) is a tangent line to the circle. 

Secant

Do you know what is a secant line?

A line that intersects two points on an arc/circumference of a circle is called the secant line. 

Look at the figure shown below.

what is a secant line

The line\(l\) intersects the circle at two points, \(A\) and \(B\).

The line \(l\) is a secant line.

So, now you know what is a secant line!

Arc

An arc of a circle is a part of the circumference.

An arc of a circle is any part of the circumference.

Segment

The area enclosed by the chord and the corresponding arc in a circle is called a segment.

Look at the following figure.

Segment in a circle

The region APB is called the minor segment and the region AQB is called the major segment.

Sector

The area enclosed by two radii and the corresponding arc in a circle is called a sector. 

Look at the following figure.

Sector in a circle

The region OAPB is called the minor sector and the region OAQB is called the major sector.

Let's summarize the various parts of a circle with the help of the figure and table given below.

A circular park labelled with the circumference, radius, diameter, chord, play area, entrance gate, exit gate, and fountain.

In a Circle In our park Named by the letter
Centre Fountain \(F\)
Circumference Boundary  
Chord Play area entrance \(PQ\)
Radius Distance from the fountain to the Entrance gate \(FA\)
Diameter Straight Line Distance between Entrance Gate and Exit Gate through the fountain \(AFB\)
Minor segment The smaller area of the park, which is shown as the Play area  
Major segment The bigger area of the park, other than the Play area  
Interior part of the circle The green area of the whole park  
Exterior part of the circle The area outside the boundary of the park  
Arc Any curved part on the circumference.  

Formulas Of Circles

Let's see the list of important formulae pertaining to any circle.

Area of a Circle Formula

The area of a circle is defined as the amount of space covered by the circle.

The area of a circle depends on the radius.

Area of circle

The formula to calculate the area of a circle is given as:

Area = \( \pi \times r^2 \)

Circumference of a Circle Formula

The circumference is defined as the length of the boundary of the circle.

circumference of a circle with formula

The formula for the circumference of a circle is given as:

Circumference of circle = \( 2 \times \pi \times r \)

Arc Length Formula

As seen previously in the article, an arc is a section (part) of the circumference.

Understand Arc Length Formula

Here, \(\theta\) is in radians.

The arc length formula is given by:

Length of an arc = \(\theta \times r\)

Area of a Sector of a Circle Formula

The sector shown in the figure below makes an angle \(\theta\) (measured in radians) at the center.

Understand Area of a Sector of a Circle Formula

The area of a sector formula is given by:

Area of a sector of a circle= \(\dfrac{\theta \times r^2}{2}\)

Here, \(\theta\) is in radians.

Length of Chord Formula

The length of a chord can be calculated if the angle made by the chord at the center and the value of radius is known.

Understand Length of Chord Formula

Length of chord = \(2r \sin\left(\frac{\theta}{2}\right)\)

Here, \(\theta\) is in radians.

Area of Segment Formula

As seen earlier, the segment is the region formed by the chord and the corresponding arc covered by the segment, as shown below.

Understand Area of Segment formula

Area of a segment = \(\dfrac{r^2(\theta - \sin{\theta})}{2}\)

Here, \(\theta\) is in radians.


Properties Of Circles

Here is a list of properties of a circle.

  1. Two circles can be called congruent if they have the same radius.
  2. Equal chords are always equidistant from the center of the circle.
  3. The perpendicular bisector of a chord must pass through the center of the circle.
  4. When two circles intersect, the line connecting the intersecting points will be perpendicular to the line connecting their center points.
  5. Tangents are drawn at the points where the diameter meets the circle is parallel to each other. 
  6. Two circles are said to be tangent circles if they touch each other at exactly at one point.

 
important notes to remember
Important Notes
  1. The name ‘Circle’ is derived from a Greek word meaning ‘ring’ or ‘hoop’.

  2. The diameter is the longest chord in a circle.

  3. If the radius of a circle is extended further and touches the boundary of the circle, it becomes the diameter of a circle. Therefore, Diameter \(= 2 \times \) Radius

  4. Circumference of a circle can also be expressed in terms of diameter as \(C=\pi d\)

Solved Examples

Example 1

 

 

Do you like cake?

Most people do.

David bought a circular chocolate cake from his favorite bakery.

A circular chocolate cake

He cut it into two parts, but the portions could not be cut equally.

Can you identify the major segment, the minor segment, the chord, and the arc in it?

Solution

The smaller part of the cake is the minor segment.

The bigger part of the cake is the major segment.

The curve is the Arc.

The line which separates the cake into two parts is the chord.

A chocolate cake cut into major and minor segments; arc and chord are labelled

\(\therefore\) The required parts of the circle are shown.
Example 2

 

 

John went swimming in a circular swimming pool.

After swimming, he runs one round along the boundary of the pool.

Swimming pool with a ball inside and a boy outside

If the radius of the pool is 35 feet, can you find the distance that John ran around the pool?

Solution

To find the distance that John ran, you need to know the circumference of the circle (pool).

For this, you need to know the value of \(\pi\) and \(r\), where ‘\(r\)’ is the radius of the pool.

Given:

  • \( r = 35 \) feet
  • \(\begin{align}\pi = {\frac{22}{7}} \end{align}\)

Using the formula, Circumference (C) \(= 2\pi r\)

\[\begin{align}C &= 2 \times {\frac{22}{7}} \times 35 \\\\ &= 220 \text{ feet}\end{align}\]

\(\therefore\) John ran \(220\) feet.
Example 3

 

 

You want to decorate your tabletop, which is in the shape of a circle, with a colorful sticker.

A circular wooden table

If the radius of the tabletop is 21 inches, find the amount of paper you need to cover its top surface.

Solution

Given,

  • Radius of the table top \(=\) 21 inches
  • \(\begin{align}\pi = {\frac{22}{7}} \end{align}\)

A circular sticker with designs

Using the formula, Area (A) \(= \pi r^2\)

\[\begin{align}A &= {\frac{22}{7}} \times 21 \times 21 \\\\ &= 1386 \text{ inches}^2 \end{align}\]

\(\therefore\) Area of table top \(= 1386 \text{ inches}^2 \)
Example 4

 

 

Identify the parts of a circle hidden in this wheel.

Find the radius, the diameter, and the circumference with the help of the given colors and clues.

A wheel with a coloured radius and diameter

Solution

Parts hidden in the wheel Parts of a Circle
\(OC\) (red) Radius
\(HD\) (green) Diameter
The Black boundary Circumference
 
Challenge your math skills
Challenging Questions
1. The radii of the circles are given as 1, 2, 3, 4, and 5 inches. Find the area of the shaded region. 

Area of the shaded region of a circle


Interactive Questions

Here are a few activities for you to practice.

Select/type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about circles with the examples and practice questions. Now, you will be able to easily solve problems on the circles.

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Frequently Asked Questions (FAQs)

1. Is a circle a polygon?

A circle is not a polygon because a circle doesn't have straight edges.

2. What are concentric circles?

Concentric circles are circles that have the same center.

The following figure shows two concentric circles with the same center \(O\).

Definition of Concentric Circles

3. What is the equation of a circle?

The equation of circle with center \((a,b)\) and with radius \(r\) is \((x-a)^{2}+(y-b)^{2}=r^2\).

4. What is a cyclic quadrilateral?

A quadrilateral is cyclic if its vertices are concyclic, that is, if all four of its vertices lie on a circle.

In the following figure, ABCD is a cyclic quadrilateral.

ABCD is a cyclic quadrilateral.

5. What is a major arc of a circle?

The major arc of a circle is the arc whose length is larger than that of a semicircle.

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