This post is also available in: हिन्दी (Hindi)
There are many types of 2D figures you study in geometry and mensuration. A circle is one such type of a 2D plane curved figure, where every point on its boundary is equidistant from its centre.
Let’s understand what is the definition of a circle in geometry and what are its different parts and properties.
What is the Definition of a Circle in Geometry?
A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance (radius) from a fixed point (centre) on the plane. The fixed point is called the origin or centre of the circle and the fixed distance of the points from the origin is called the radius.
There are many objects we have seen in the real world that are circular in shape. Some of the examples of circles are clocks, coins, CD discs, wheels, rings, and buttons.
Properties of Circle
The following are the important properties of a circle.
- A circle is a closed 2D shape that is not a polygon. It has one curved face.
- Two circles can be called congruent if they have the same radius.
- Equal chords are always equidistant from the centre of the circle.
- The perpendicular bisector of a chord passes through the centre of the circle.
- When two circles intersect, the line connecting the intersecting points will be perpendicular to the line connecting their centre points.
- Tangents drawn at the endpoints of the diameter are parallel to each other.
Interior and Exterior of a Circle
A circle divides a plane into two regions
- Interior: The region which lies inside the boundary of a circle is called the interior of a circle. The points lying in the interior region of a circle are called interior points.
- Exterior: The region which lies outside the boundary of a circle is called the exterior of a circle. The points lying in the exterior region of a circle are called exterior points.
In the above figure, the external points are A, B, C, and D, and the internal points are E, F, G, and H.
Note:
- The distance of an external point from the centre is greater than the radius of a circle
- The distance of an internal point from the centre is less than the radius of a circle
- The distance of a point lying on the circle from the centre is equal to the radius of a circle
Parts of a Circle
There are many parts or components of a circle that we should know to understand its properties. A circle has mainly the following parts:
Circumference: It is also referred to as the perimeter of a circle and can be defined as the distance around the boundary of the circle.
Radius: Radius is the distance from the center of a circle to any point on its boundary. A circle has many radii as it is the distance from the center and touches the boundary of the circle at various points.
Diameter: A diameter is a straight line passing through the center that connects two points on the boundary of the circle. We should note that there can be multiple diameters in the circle, but they should:
- pass through the center.
- be straight lines.
- touch the boundary of the circle at two distinct points which lie opposite to each other.
Relation Between the Radius and Diameter of a Circle
The diameter (d) of a circle is double the radius (r) of a circle, or we can say that radius of a circle is half the diameter of a circle.
- $\text{Diameter } = 2 \times \text{ Radius}$ or $d = 2r$
- $\text{Radius } = \frac{1}{2} \times \text{ Diameter}$ or $r = \frac{1}{2}d$
Chord of a Circle: A chord is any line segment touching the circle at two different points on its boundary. The longest chord in a circle is its diameter which passes through the centre and divides it into two equal parts.
Note: The chord passing through the centre is called the diameter.
Tangent: A tangent is a line that touches the circle at a unique point and lies outside the circle.
Note: The tangent and the radius of a circle make an angle of $90^{\circ}$ at the point of contact.
Secant: A line that intersects two points on an arc/circumference of a circle is called the secant.
Arc of a Circle: An arc of a circle is referred to as a curve, which is a part or portion of its circumference.
Sector of a Cirlce: The sector of a circle is defined as the area enclosed by two radii and the corresponding arc in a circle. There are two types of sectors – minor sector, and major sector.
Segment in a Circle: The area enclosed by the chord and the corresponding arc in a circle is called a segment. There are two types of segments – minor segment, and major segment.
Circle Formulas
We frequently use different circle formulas in geometry and mensuration. The circle formulas are as follows.
- Area of a Circle Formula: The area of a circle refers to the amount of space covered by the circle. It totally depends on the length of its radius. $\text{Area | = \pi r^2$ square units.
- Circumference of a Circle Formula: The circumference is the total length of the boundary of a circle.$\text{Circumference } = 2 \pi r$ units.
- Arc Length Formula: An arc is a section (part) of the circumference. $\text{Length of an arc }= \theta \times r$, where, $\theta$ is the central angle and is in radians.
- Area of a Sector Formula: If a sector makes an angle $\theta$ (measured in radians) at the centre, then the area of the sector of a circle = $\frac{\theta r^2}{2}$. Here, $\theta$ is in radians.
- Length of Chord Formula: It can be calculated if the angle made by the chord at the center and the value of the radius is known. $\text{Length of chord } = 2 r \sin \frac{\theta}{2}. Here, $\theta$ is in radians.
- Area of Segment Formula: The segment of a circle is the region formed by the chord and the corresponding arc covered by the segment. The area of a segment = $\frac{r^2 \left( \theta − \sin \theta \right)}{2}$. Here, $\theta$ is in radians.
Practice Problems
- Define the following terms
- Circle
- Radius
- Diameter
- Chord
- Tangent
- Secant
- Arc
- Sector
- Segment
FAQs
What is a circle in Geometry?
A circle is a round 2-dimensional shape. It is a closed shape with a distance from center to circumference termed as radius $r$ and distance from one point on the circumference to another point passing through the centre termed as diameter $d$. One of the best examples of the circle in the real world is a clock dial.
What is a chord in a circle?
A chord is a line segment joining two points inside a circle on its arc. As the diameter also has two endpoints on a circle, so, it is the longest chord. All angles marked in the circle subtended by the same chord are equal.
What is the circumference of a circle?
The circumference of a circle is defined as the linear distance around its boundary or we can say, if a circle is opened to form a straight line, then the length of that line will be the circle’s circumference.
What are the major parts of a circle?
The list of the different parts of a circle includes tangent, chord, radius, diameter, minor arc, major arc, minor segment, major segment, minor sector, and major sector.
Conclusion
A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance (radius) from a fixed point (centre) on the plane. The major parts of a circle are tangent, chord, radius, diameter, minor arc, major arc, minor segment, major segment, minor sector, and major sector.
Recommended Reading
- How to Construct a Perpendicular Line (With Steps & Examples)
- How to Construct Parallel Lines(With Steps & Examples)
- How To Construct a Line Segment(With Steps & Examples)
- What are Collinear Points in Geometry – Definition, Properties & Examples
- What is a Transversal Line in Geometry – Definition, Properties & Examples
- What are Parallel Lines in Geometry – Definition, Properties & Examples
- What is Concurrent lines in Geometry – Definition, Conditions & Examples
- What is Half Line in Geometry – Definition, Properties & Examples
- What is a Perpendicular Line in Geometry – Definition, Properties & Examples
- Difference Between Axiom, Postulate and Theorem
- Lines in Geometry(Definition, Types & Examples)
- What Are 2D Shapes – Names, Definitions & Properties
- 3D Shapes – Definition, Properties & Types
