Circles

In our daily routine, we come across many shapes that are round, such as a wall clock, a wheel, the sun, the moon, the earth, a coin, bangles, rings, etc. All these shapes are in the form of a circle. A circle is a 2-dimensional shape defined as a collection of all the points in a plane that are equidistant from a given fixed point.

An interesting way to imagine a circle is thinking of it as a single line segment that is bent in a circular shape.

In this article, we will study what a circle is, how to draw a circle, related terms, properties of a circle, and examples of circles.

Circle Definition

A circle is a collection of points in a plane that are equidistant from a given fixed point.

The fixed point is called the centre of the circle. And the fixed distance from the centre is called the radius of a circle. From the figure O is the centre of the circle, r is the radius of a circle and the blue lines indicate the circumference of the circle that is the perimeter of the circle.

The circle formula is given as:

(x - h)² + (y - k)² = r²

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

How to Draw a Circle?

The following steps are required to draw a perfect circle:

• Take plain paper.

• Plot a point on it.

• Take a compass to measure some length on a ruler.

• Now, place the sharp end of the compass on that point.

• Keeping the pointed leg on the point, rotate another leg one complete round drawing a line.

• You will see that a circle is drawn. And thus, we get a collection of points on the line that is equidistant from a given fixed point, and that is what defines a circle.

Terms Related to a Circle

Some important terms related to circles are explained below. It will prove helpful to you in an example of a circle.

Centre: The fixed point of a circle is called the centre of the circle,

Radius: The fixed distance from the centre to any point on the circle is called the radius of a circle.

Chord: a line segment joining any two points on the circle is called a chord. In the figure below the green line indicates the chord. 

Diameter: A chord that passes through the centre of the circle is called the diameter of a circle. The orange line indicates the diameter. Also, diameter is the longest chord and can be expressed as 2r where r is the radius of a circle. There are infinite numbers of diameter in a circle.

Secant: A chord that intersects the circle in two points is called a secant. In the figure below, the blue line indicates a secant.

Tangent: A line that touches a circle at one point is called tangent to the circle, and that point is called the point of contact. In the figure below, the red line indicates the tangent.

Arc: A piece of the circle between two points is called an arc. The red curve line is called the arc of a circle.

Circumference: The length of the complete circle is called the circumference of a circle. It is also said to be the perimeter of the circle. The circumference of a circle is equal to 2πr, where r is the radius of the circle.

Segment: The region between the chord and the arc is called the segment of the circle. In the below figure, the purple region indicates the segment. The segment containing the minor arc is called the minor segment and the segment containing the major arc is called the major segment.

Sector: The region between the arc at the two radii is called the sector. The blue region indicates the sector.

Interior and Exterior of a Circle: The circle divides the plane into three parts:

• The ‘inside’ of the circle is also called the interior of the circle, as B lies in the interior of the circle.

• The ‘outside’ of the circle is also called the exterior of the circle, as point A lies outside the circle.

• ‘On’ the circle is also called the circumference of the circle, as point C lies on the circle.

Properties of a Circle

Some of the properties of a circle are as follows:

• Circles with the same radii are said to be congruent.

• The longest chord of a circle is the diameter of a circle.

• The diameter of a circle is double the radius.

• The diameter divides the circle into two equal semicircles.

• The radius that is drawn perpendicular to the chord bisects the chord.

• A circle can be inscribed in a square, triangle or kite.

• The chords that are equidistant from the centre are equal in length.

• The distance from the centre of the circle to the longest chord (diameter) is zero, i.e., the longest chord is the diameter.

• The perpendicular distance from the centre of the circle decreases when the length of the chord increases.

• An isosceles triangle is formed when the radii join the ends of a chord to the centre of a circle.

Formulas Related to a Circle

Some of the important circle formulas that are widely used in solving examples of circles are as follows:

Diameter d: The diameter of a circle is twice the radius of a circle. The formula for the diameter is given as follows:

Circumference: The outer boundary of a circle is the circumference of the circle. The formula for circumference is given by-as follows :

Area: Area of a circle is defined as the space occupied by the circle. The formula for area is given as follows:

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Solved Problems Related to Circles

Example 1: Find the area and the circumference of a circle whose radius is 7 cm. (Consider the value of π to be 22/7).

Solution:

Given radius r  = 7 cm.

Area A = π r² 

= 22/7  × 7 x 7

Area  = 154cm²

Circumference C = 2πr

     = 2  x 22/7 x 7 

Circumference =  44 cm

Example 2 : If the diameter of a circle is 12 cm, then find its area.

Solution:

It is given that the diameter d is equal to 12 cm.

So, radius r = d /2 

= 12/2 

= 6cm

Hence, area A = πr²

          A = π x (6)²

  = 3. 14 x 36

          A = 113.04 cm²

Did you Know?

Circles are special shapes that have the largest areas for any given length of their perimeters. Circles are shapes with high symmetry. Each line through a circle’s centre forms a line of reflection symmetry. It also has rotational symmetry around its centre for each and every angle.

Quiz Time

It is time for you to test your knowledge of what you have learnt in this article. Solve the following questions to check if you are ready to take up challenges in problems related to circles.

1. A circular garden has a radius of 21 m. The owner wants to put a plastic edge around the garden, and so the owner wants to know what the circumference of the garden is. Find the garden’s circumference. (Consider the value of π to be 22/7).

2. A circle with a radius of 21 cm is given. Determine its area and its circumference.

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Conclusion

Circles are an important component in Geometry and in Mathematics as a whole. Several theorems related to circles are constantly referenced in order to solve problems in exams or even in real-life situations like construction and architecture. Experts always rely on the terms and concepts, which have already been discussed in this comprehensive article on the topic, to put them into practical use. Circles are sure to constitute a significant portion of your Maths syllabus, and skipping this topic will cause you to lose precious marks that can otherwise be easily scored. Hence, it is advised to read through and internalise as many formulas and properties related to circles as possible, to be able to score good marks in your exams.