# Circles

## Circle

A circle is one of the most perfect figures in geometry. It is also one of the simplest shapes in geometry, and is formed by the set of all points which are at constant distance (**radius**) from a fixed point (**center**) in the plane. The centre of a circle is a point within the circle from which every point on the circle is at the same distance. The radius of a circle is the distance from the centre of the circle to any point on the circle.

In the figure above, O is the center and *r* is the radius of the circle. Any point on the circle will at a distance of *r* from the center O.

Any circle will divide the plane into: the **interior** of the circle, and the **exterior** of the circle. Suppose that the center of a circle is O, and its radius is *r*. Consider any point P in the plane.

- If P lies in the interior of the circle, then \(OP{\rm{ }} < r\)
- If P lies
*exactly*on the circle, then \(OP{\rm{ }} = r\) - If P lies in the exterior region of the circle, then \(OP{\rm{ }} > r\)

### Arc

An **arc** is a portion of the circumference of the circle. The figure shows both the minor arc as well as the major arc of the circle.

### Chord

A **chord** of a circle is a line segment whose end-points lie on the circumference.

### Diameter

The diameter of a circle is a chord that passes through the centre of the circle. \(AB\) is the **diameter** of the circle.

The **diameter** of a circle is a length equal to twice its radius. The word diameter can be used interchangeably – it could mean a segment, or it could mean a length. Consider the following figure:

### Circumference

The **circumference** of a circle is its boundary. The length of the *complete arc* of a circle is called its **circumference**.

### Segments of the circle

A chord of a circle divides the circle into two parts, which are called **segments** of the circle.

A **sector** of a circle is the region enclosed by two radii and an arc as shown.

## Tips and tricks

- The value of \(\pi \) can be approximated as \(\begin{align}22\over 7 \end{align} \) when the radius of the circle is divisible by 7. Otherwise, it can be approximated to 3.14. This can be used to calculate the area or the circumference.
- While calculating the area of a circle it is helpful to write the radius twice instead of squaring it. Squaring would make the calculations tedious. This holds true for radius with larger values. For example, if the radius of a circle is 21 cm it would be easier to calculate \(\begin{align}{22\over 7} \times 21 \times 21 \end{align} \) than calculating \(\begin{align}{22\over 7} \times 441. \end{align} \)

### Common mistakes or misconceptions

**Misconception 1:** Any given chord is a diameter

A diameter is a chord, as it is a line segment whose endpoints lie on the circumference. However, not all chords are diameters, as a chord may or may not pass through the center of the circle.

## Test your knowledge

1.

The shaded portion is a ________________.

2.

The shaded portion is a ________________.

3.

\(BC\) is ____________________________.

## Important Topics

- What is a Circle?
- Ratio of Circumference to Diameter
- Area of a Circle
- Concentric Circles
- Chords and Diameters
- Perpendicular Bisector of a Chord
- Symmetry of Any Circle
- Constructing Circles
- Equal and Unequal Chords
- Arcs and Subtended Angles
- Concyclic Points
- Cyclic Quadrilaterals
- What is a Tangent?
- Tangent as a Special Case of Secant
- Uniqueness of the Tangent at a Point
- Tangents From An External Point
- Circles Touching Each Other
- Alternate Segment Theorem
- Common Tangents