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, area and circumference of a circle in detail.

## Circle Definition

The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle. Here, the fixed point is 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” or “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” or “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 on the circumference of a circle is called the 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 of a circle subtend equal angles at the centre
- 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 can also be 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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