A circle is a closed shape formed by tracing a point that moves in a plane such that its distance from a given point is constant. The word circle is derived from the Greek word ** kirkos**, meaning hoop or ring. In this article, we cover the various circle formulas, properties of a circle & important terms related to circles.

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## Definition of a Circle

When a * set of all points* that are

*are joined then the geometrical figure obtained is called circle.*

__at a fixed distance from a fixed point__Let us now learn a bit about the terminology used in circles.

## Terms related to Circles & Circle formulas

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### Center

The fixed point in the circle is called the center.

- So, the set of points are at a fixed distance from the center of the circle.

### Radius

Radius is the fixed distance between the center and the set of points. It is denoted by __“R”____.__

### Diameter

The diameter is a line segment, having boundary points of circles as the endpoints and passing through the center.

- So, logically a diameter can be broken into two parts:
- One part from one boundary point of the circle to the center
- And, the other part from the center to another boundary point.
- Hence, Diameter = Twice the length of the radius or
__“D = 2R”__

- Hence, Diameter = Twice the length of the radius or

### Circumference

It is the measure of the outside boundary of the circle.

So, the * length of the circle or the perimeter of the circle* is called Circumference.

### Arc of a circle

The arc of a circle is a portion of the circumference.

From any two points that lie on the boundary of the circle, two arcs can be created: A Minor and a Major Arc.

**Minor arc:**The shorter arc created by two points.**Major Arc:**The longer arc created by two points.

### Sector of a circle:

A Sector is formed by joining the endpoints of an arc with the center.

- On joining the endpoints with the center, two sectors will be obtained: Minor and Major.
- By default, we only consider the Minor sector unless it is mentioned otherwise.

### Semi-circle

A semi-circle is half part of the circle *or,*

- A semi-circle is obtained when a circle is divided into two equal parts.

Now that we know all the terminologies related to the circles, let us learn about the properties of a circle.

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## Important Properties of Circles – Related to Lines

### Properties related to Lines in a Circle | Properties of circle

#### Chord

A chord is a line segment whose endpoints lie on the boundary of the circle.

##### Properties of Chord

- Perpendicular dropped from the center divides a chord into two equal parts.

#### Tangent

Tangent is a line that **touches** the circle at any point.

##### Properties of Tangent

- Radius is always perpendicular to the tangent at the point where it touches the circle.

## Important Properties of Circles – Related to Angles

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### Properties related to Angles in a circle | Properties of Circle

#### Inscribed Angle

An inscribed angle is the angle formed between two chords when they meet on the boundary of the circle.

##### Properties of Inscribed Angles

1. Angles formed by the same arc on the circumference of the circle is always equal.

2. The angle in a __semi-circle__ is always 90°.Central Angle

A central angle is the angle formed when two-line segments meet such that one of the endpoints of both the line segment is at the center and another is at the boundary of the circle.

#### Property of Central Angles

- An angle formed by an arc at the center is twice the
__inscribed angle__formed by the same arc.

## Important Circle Formulas: Area and Perimeter

The following are some mathematical formulae that will help you calculate the area and perimeter/circumference of a circle.

### Perimeter:

- Perimeter or the Circumference of the circle = 2 × π × R.
- Length of an Arc = (Central angle made by the arc/360°) × 2 × π × R.

### Area:

- Area of the circle = π × R²
- Area of the sector =(Central angle made by the sector/360°) × π × R².

## Summary of all the Properties of a Circle

Here is a summarized list of all the properties we have learned in the article up to this point.

Property | Element | Description |

Lines in a circle | Chord | Perpendicular dropped from the center divides the chord into two equal parts. |

Tangent | The radius is always perpendicular to the tangent at the point where it touches the circle. | |

Angles in a circle | Inscribed Angle | 1. Angles formed by the same arc on the circumference of the circle is always equal. 2. The angle in a semi-circle is always 90. |

Central Angle | The angle formed by an arc at the center is twice the inscribed angle formed by the same arc. | |

Important Formulae | Circumference of a circle | 2 × π × R. |

Length of an arc | (Central angle made by the arc/360°) × 2 × π × R | |

Area of a circle | π × R² | |

Area of a sector | (Central angle made by the arc/360°) × π × R² |

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## Application of the circle properties & circle formulas in the questions

### Question 1

The lengths of two sides in a right-angle triangle other than hypotenuse are 6 cm and 8 cm. If this right-angle triangle is inscribed in a circle, then what is the area of the circle?

- 5 π
- 10 π
- 15 π
- 20 π
- 25 π

#### Solution

**Step 1: Given**

- The lengths of two sides other than hypotenuse of a right triangle are 6 cm and 8 cm.
- This triangle is inscribed in a circle.

**Step 2: To find**

- Area of the circle.

**Step 3: Approach and Working out**

Let us draw the diagrammatic representation.

By applying the property that the angle in a semi-circle is 90º, we can say that AB is the diameter of the circle.

- And, once we find the length of the diameter, we can find the radius, and then we can find the area of the circle as well.

Applying Pythagoras theorem in △ ABC,

- AB² = AC² + BC²
- AB² = 6² + 8²
^{ }= 36 +64 = 100 - AB = 10 cm

- AB² = 6² + 8²

Since AB is the diameter, AB = 2R = 10

- Hence, R = 5 cm.

Area of the circle = π × R²= π × 5² = 25 π.

Hence, the correct answer is option E.

### Question 2

In the diagram given above, O is the center of the circle. If OB = 5 cm and ∠ABC = 30^{0} then what the length of the arc AC?

- 5π/6
- 5π/3
- 5π/2
- 5π
- 10π

#### Solution

**Step 1: Given**

- OB = 5 cm
- ∠ABC = 30°

**Step 2: To find**

- Length of the arc

**Step 3: Approach and Working out**

- Length of the arc = (Central angle made by the arc/360°) × 2 × π × R.

To find the length of the arc, we need the value of two variable, the center angle made by the arc and the radius.

- We are already given radius as OB = 5cm
- We need to find the ∠AOC

On visualizing the diagram, the inscribed angle by the arc AC is ∠ABC, and the center angle by arc AC is ∠AOC.

- Hence, we can apply the property that the angle made at the center by an arc is twice the inscribed angle formed by the same arc.
- Thus, ∠AOC = 2 × ∠ABC = 2 × 30°
^{ }= 60°

Now, we know the central angle formed by the arc as well.

- Hence, length of the arc AC =(Central angle made by the arc/360°) × 2 × π × R.
- =(60°/360°) × 2 × π × 5.
- =(1/6) × 2 × π × 5.
- =(5π/3) cm

Thus, the correct answer is option B.

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