CHAPTER 12:AREAS RELATED TO CIRCLES

Get the complete notes on an area related to circles for class 10 is provided here. The concepts related to circles such as area, circumference, segment, sector, angle and length of a circle, area for the sector of a circle is provided here. Also, the visualization of some plane and solid figures areas are discussed here.

Introduction

Area of a Circle

Area of a circle is $\pi r^2$, where $\pi =$$\mathrm{22/7\; o}r\approx 3.14$ (can be used interchangeably for problem-solving purposes)and r is the radius of the circle.
$\pi$ is the ratio of the circumference of a circle to its diameter.

Circumference of a circle

The perimeter of a circle is the distance covered by going around its boundary once. The perimeter of a circle has a special name: CIrcumference, which is π times the diameter which is given by the formula $2\pi r$

The segment of a circle

A circular segment  is a region of a circle which is “cut off” from the rest of the circle by a secant or a chord

A sector of a circle

A circle sector/ sector of a circle is defined as the region of a circle enclosed by an arc and two radii. The smaller area is called the minor sector and the larger area is called the major sector.

The angle of a Sector

The angle of a sector is that angle which is enclosed between the two radii of the sector.

Length of an arc of a sector

The length of the arc of a sector can be found by using the expression for the circumference of a circle and the angle of the sector, using the following formula:

$L=\; ($θ/360°)$\times 2\pi r$

where $\theta$ is the angle of sector and $r$ is the radius of the circle.

Area of a Sector of a Circle

Area of a sector is given by

$($θ/360°)$\times \pi r^2$

where $\angle \theta$ is the angle of this sector(minor sector in the following case) and r is its radius

Area of a sector

Area of a Triangle

Area of a triangle is,
$Area=(1/2)\times base\times height$
If the triangle is an equilateral then
$Are\mathrm{a=\surd }$$3/4\times a^2$  where $a$ is the side of the triangle.

Area of a Segment of a Circle

Area of the segment

Area of segment APB (highlighted in yellow)
= (Area of sector OAPB) – (Area of triangle AOB)

$=[(2\varnothing /360°)\times \pi r2$] – [(1/2)$\times AB\times OM]$

[To find the area of triangle AOB, use trigonometric ratios to find OM (height) and AB (base)]

Also, Area of segment APB can be calculated directly if the angle of the sector is known using the following formula.

 

where $\theta$ is the angle of the sector and $r$ is the radius of the circle

Visualisations

Areas of different plane figures

–    Area of a square (side l) $=l^2$
–    Area of a rectangle $=l\times b,$ where $l$ and $b$ are the length and breadth of the rectangle
–    Area of a parallelogram $=b\times h,$ where $b$ is the base and $h$ is perpendicular height.

parallelogram

Area of a trapezium $=[$$(a+b)\times \mathrm{h]/2},$

where

$a$ & $b$ are the parallel sides length

h is the trapezium height

Area of a rhombus $=$$pq/2,$ where $p$ & $q$ are the diagonals

Areas of Combination of Plane figures

For example: Find the area of the shaded part in the following figure: Given the ABCD is a square of side 28cm and has four equal circles enclosed within.

Area of the shaded region

Looking at the figure we can visualise that the required shaded area = $A\left(squareABCD\right)-4\times A\left(Circle\right)$.

Also, the diameter of each circle is 14 cm.
$=\left(l^2\right)-4\times \left(\pi r^2\right)$
$=\left({28}^2\right)-[4\times (\pi \times 49\left)\right]$
$=784-[4\times \frac{22}{7}\times 49]$
$=784-616$
$=168c{m}^2$