# NCERT Solutions for Class 9 Maths Chapter 10 – Circles PDF

## Free PDF of NCERT Solutions for Class 9 Maths Chapter 10 – Circles

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## Circles

A circle is a curved plane figure. Every point on the circle is equidistant from a fixed point known as the center of the circle. It is a 2D shape and is measured in terms of radius. The word ‘Circle’ is derived from the Latin word 'circulus' meaning small ring.

## What is Circle?

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 (center) on the plane. The fixed point is called the origin or center of the circle and the fixed distance of the points from the origin is called the radius.

## 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 of Circle: 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.
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 center and divides it into two equal parts.

Tangent: A tangent is a line that touches the circle at a unique point and lies outside the circle.

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, that is a part or portion of its circumference.

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.

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.

## Properties of Circle

Let us move ahead and learn about some interesting properties of circles that make them different from other geometric shapes. Here is a list of 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 center of the circle.
• The perpendicular bisector of a chord passes through the center of the circle.
• When two circles intersect, the line connecting the intersecting points will be perpendicular to the line connecting their center points.
• Tangents drawn at the endpoints of the diameter are parallel to each other.

## Circle Formulas

### Let's see the list of important formulae pertaining to any circle.

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 → Area = πr² square units.

Circumference of a Circle Formula: The circumference is the total length of the boundary of a circle → Circumference = 2πr units.

Arc Length Formula: An arc is a section (part) of the circumference. Length of an arc = θ × r. Here, θ is in radians.

Area of a Sector Formula: If a sector makes an angle θ (measured in radians) at the center, then the area of the sector of a circle = (θ × r2) ÷ 2. Here, θ 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 radius is known. Length of chord = 2 r sin(θ/2). Here, θ 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 = r2(θ − sinθ) ÷ 2. Here, θ is in radians.