Lines and circles are the important elementary figures in geometry. We know that a line is a locus of a point moving in a constant direction whereas the circle is a locus of a point moving at a constant distance from some fixed point. The theoretical importance of the circle is reflected in the number of amazing applications. Here we will discuss the properties of a circle and area and circumference of a circle in detail.

## Circle Definition

A circle is a collection of points where all the points are equidistant from the given point called the centre “O”. Some of the important terminologies used in the circle are as follows:

Terms |
Description |

Circumference |
The boundary of the circle is known as the circumference |

Radius |
The line from the centre “O” of the circle to the circumference of the circle is called the radius and it is denoted by “r” |

Diameter |
The line that passes through the centre of the circle and touches the two points on the circumference is called the diameter and it is denoted by the symbol “D” |

Arc |
Arc is the part of the circumference where the largest arc is called the major arc and the smaller one is called the minor arc |

Sector |
Sector is slice of a circle bounded by two radii and the included arc of a circle |

Chord |
The straight line that joins any two points in a circle is called a chord |

Tangent |
A line that touches the circumference of a circle at a point is called the tangent |

Secant |
A line that cuts the circle at the two distinct points is known as the secant |

## Circle Properties

Some of the important properties of the circle are as follows:

- The circles are said to be congruent if they have equal radii
- The diameter of a circle is the longest chord of a circle
- Equal chords and equal circles have the equal circumference
- The radius drawn perpendicular to the chord bisects the chord
- Circles having different radius are similar
- A circle can circumscribe a rectangle, trapezium, triangle, square, kite
- A circle can be inscribed inside a square, triangle and 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
- The perpendicular distance from the centre of the circle decreases when the length of the chord increases
- If the tangents are drawn at the end of the diameter, they are parallel to each other
- An isosceles triangle is formed when the radii joining the ends of a chord to the centre of a circle

## Circle Formulas

Area of a circle, A = πr^{2} square units

The circumference of a circle = 2πr units

The circumference of a circle formula is also written as πd

Where,

Diameter = 2 x Radius

D = 2r

Here “r” represents the radius of a circle.

## Circle Problem

The sample example to find the area and circumference of a circle is given below.

### Question:

Find the area and circumference of a circle having the diameter value of 10 cm?

### Solution:

Given:

Diameter, D = 10 cm

We know that diameter = 2 x Radius

Therefore, radius, r = d/2

r = 10/2 = 5

So, the radius is 5 cm.

Area of a circle, A = πr^{2} square units

A = 3.14 x 5 x 5

Where ,

π = 3.14

A = 3.14 x 25

A = 78.5 cm^{2}

Therefore, the area of a circle is 78.5 square units

The circumference of a circle = 2πr units

C = 2 x 3.14 x 5

C = 10 x 3.14

C = 31. 4 cm

Therefore, the circumference of a circle is 31.4 units.

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