Circle: Definition, Properties, Formulas, Theorems & Example

Circle: The collection of all the points in a plane, which are at a fixed distance from a fixed point is called a circle. The fixed point is called the center of the circle whereas, the fixed distance is called the radius. In everyday life, we come across various objects that are in the shape of a circle, be it a clock, a coin, or a bangle. A circle is a two-dimensional geometrical shape. The circle is divided into three parts namely, interior (inside of the circle), the circle, and exterior (outside of the circle).

Students are always advised to have deep knowledge of the various concepts of circle and its theorems. Understanding the concepts will help them to solve the problems in a fraction of seconds with a positive outcome. We have provided detailed information on Circle and its structure, properties, formulas, and examples in this article. Read on to find out.

Circle: Definition

A circle is a two-dimensional figure where all the points are at a fixed distance from a fixed point in a plane. The circle is divided into three parts – inside the circle, the circle, and outside the circle. All students must know how to draw a circle by using the following steps:

1. Take a compass with a pencil attached to it.
2. Take a sheet of paper, and place the pointed tip of the compass in the middle of the paper.
3. Open the other leg of the compass to a particular distance.
4. Keep the pointed tip on the same point and rotate the other leg slowly taking one revolution.
5. Keeping one fixed point, you drew all the points that were fixed at a distance in a plane.

Circle: Properties

The basic properties of circles are as follows:

1. A circle is the collection of all points which are equidistant from a fixed point.
2. Equal chords of the circle subtend equal angles at the circle.
3. The diameter of the circle is the largest chord.
4. Circles of different sizes comprise different radii.
5. The diameter of the circle divides into two equal parts.

Circle: Structure

The different parts of the circle are described below:

1. Center: The fixed point of the circle is called the center.
2. Radius: The fixed distance of any point on the circle from the center is called the radius.
3. Chord: If there are two points P and Q on a circle, then the line segment joining these points are called a chord.
4. Arc: A segment of a circle between two points on it is called an arc. The longer segment is called the major arc and the shorter one is called the minor arc.
5. Circumference: The length of the complete circle (its boundary) is called the circumference.

Circle: Formulas

The circle is a two-dimensional figure.

Area of the Circle: It is the amount of space occupied by the circle. The formula used to measure the area of a circle is as follows:

Where, π = 3.1415 and r is radius.

Circumference of the Circle: The length of the complete circle is called the circumference.

Where, π = 3.1415, d is diameter and r is radius.

Circle: Theorems

Check the following theorems of circles:

1. Equal chords of a circle subtend equal angles at the center.
2. If the angles subtended by the chords of a circle at the center are equal, then the chords are equal.
3. The perpendicular from the center of a circle to a chord bisects the chord.
4. The line drawn through the center of a circle to bisect a chord is perpendicular to the chord.
5. There is one and only one circle passing through three given non-collinear points.
6. Equal chords of a circle (or of congruent circles) are equidistant from the center (or centers).
7. Chords equidistant from the center of a circle are equal in length.
8. The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
9. Angles in the same segment of a circle are equal.
10. If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).
11. The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.
12. If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.

Circle: Example

Check a few of the solved examples of circles:

Example 1: Find the area of a circle whose radius is 12 cm.

Given: Radius = 12 cm

Area= πr²

= 3.14 x 12 ² = 452.16

FAQs Regarding Circle

The frequently asked questions on circle are given below:

We hope this article on Circle helps you. If you have any questions, feel to ask in the comment section below. We will get back to you at the earliest.