Area of Circle

Area of a Circle:

Draw a circle. Divide it into equal (even) sectors and arrange them in line as shown below.

This shape looks like a rectangle of length 'pr' and breath 'r' units
Let us now learn about the area of a circle in detail.
Area of the circle = Area of the rectangle

= length x breadth

= ∏ r x r

= ∏ r2 Sq. units
Next in this Blog let us look at the formula of the Areas of a circle.
Area of a circle Formula is ∏ r2 Sq. units

Area of a semicircle = 21 ∏ r2 Sq. units

Area of a quadrant = 41 ∏ r2 Sq. units.
The last topic that we will learn about in this area of the Circles Problems,and we will also see the various how we can solve these problems.
Problems related to area of A Circle:If the area of the sector of a circle is 60 sq. cm and the length of the arc of the sector is 12 cm, what is the radius of the circle?

Solution:
Area of the sector = 1 lr = 60 cm2
2

l = 12cm.
A = 1 x 12 x r = 60
2

6r = 60

r = 10 cm.

The radius of the circle is 10 cm.

Hope you like the above example of Area of a Circle.Please leave your comments, if you have any doubts.

Symmetry

Definition:-

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning of Symmetry is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise.

In formal terms, we say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa).

Types of Symmetry:

In this Blog I will also share with you the information on the different types of Symmetry.In the following two different types of symmetry are given:

1. Symmetry in geometry
2. Symmetry in mathematics


Symmetry in geometry:

Symmetry definition in geometry it means a sub-group.Our next concern is the very important topic on Isometrics,the most common question here is-What does Isometrics consists of?? Isometrics consists of three or two dimensional space. In following operations:

1. Reflectional Symmetry(FLIP)
2. Rotational Symmetry (TURN)
3. Translational Symmetry (SLIDE)

Reflectional symmetry (FLIP):

Splits the image into one side of the half side of mirror image. It is also called line or mirror symmetry. A Reflectional symmetry is called FLIP.

Rotational symmetry (TURN):

To turn the center point of an object into degress. A Rotational symmetry is called TURN.

Translational symmetry (SLIDE):

In straight line is divided into sequence line. A Translational symmetry is called SLIDE.

Symmetry in Mathematics

In mathematical operation, to apply the object into operation. The set of operations to form a group. Two object form a group of operations. To apply the objects into symmetry. So it is called a symmetry definition in mathematics.

Hope you like the above example of Symmetry.Please leave your comments, if you have any doubts.

Fractions

In this Blog I will help you understand the concept of Fractions and also we we see how to solve the fractions.Fraction is an equal part of one whole object.The most common question is how do you denote a fraction???Fraction can be represented as " p/q " where 'p' denotes the value called numerator and 'q' denotes the value called denominator and q not equal to zero.

Introduction to fraction:

A fraction is a number that can represent part of a whole. The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. A much later development were the common or "vulgar" fractions which are still used today (½, ⅝, ¾, etc.) and which consist of a numerator and a denominator.

In this Blog we are going to see how to change 385 to fraction and 385 into decimal,what is mean by improper fraction and some other examples based on fraction.

385 to Fraction:

385 to Fraction:

In this Blog we are going to see how to convert 385 to fraction and what is mean by improper fraction and how to convert 385 into decimal.Let us now solve few problems related to Fractions to understand the concept even better.

Problem 1:

Convert 385 to fraction.

Solution:

Given integer 385.

To convert the 385 to fraction we need to multiply and divide by the same number.We get only the improper fraction.

Improper fraction:

If [a/b] is improper fraction means, b < 10 =" [3850"> 38.5 × 10

(ii) 385 × (100/100)

=> 3.85 × 100

=> 3.85 × 102

(iii) 385 × (1000/1000)

=> 0.385 × 1000

=> 0.385 × 103

Problems on Fraction:

Problem1:

Add two fraction [3/4] and [2/5]

Solution:

Given , [3/4] and [2/5]

= [3/4] + [2/5]

To add fraction ,we need common denominator,

To make a common denominator , multiply 3/4 by 5 on both numerator and denominator and multiply 2/5 by 4 on both numerator and denominator.

= [3/4] × [5/5] + [2/5] × [4/4]

= [15/20 ] + [8/20]

= [(15+8) / 20]

= [23 /20]

Answer: [3/4] + [2/5] = [23 /20]

Problem 2:

Multiply the fractions 5/6 and 2/8

Solution:

Given, [5/6] and [2/8]

= [ 5/6] × [2/8]

= [ ((5)(2)) / ((6)(8))]

= [10/48]

= [5 / 24]

Answer: [5 / 24]

Problem 3:

[Divide the fraction 16/25 by 10/24]

Solution:

Given, [16/25] ÷ [10 /24]

We can divide by,

(i) Take the reciprocal for [10/24]

(ii) Multiply it with [16/25]

[16/25] ÷ [10 /24] = [16 /25] × [ 24 /10]

= [((16)(24)) / (( 25)(10))]

= [ 384 / 250 ]

= [192 / 125]

Answer: [192 / 125]

Hope you like the above example of Fractions.Please leave your comments, if you have any doubts.

Volume of Cone

Volume of Cone:
In general terms when we speak about a cone the first thing that comes to our mind is an ice-cream cone as depicted in the picture on the right hand side.

But to be more specific the meaning of a cone is given below "A cone is a three-dimensional geometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex.n geometry, there are two types of shapes." They are two dimensional and three dimensional.

Two dimensional shapes are rectangle, square, parallelogram, rhombus etc.

Three dimensional shapes are cone, sphere, pyramid, cube, prism.


Properties of cone:

There are two properties of cone . They are volume of cone, surface area of cone. By using the properties of cone, we can calculate volume and surface area.

A cone has radius (r) and height (h) , slant height (s).

Volume of cone V = 1/3 * π * radius2 * height where π = 3.14

Volume of cone V = 1/3 * Volume of cylinder

Surface area of cone SA = π*radius* [sqrt(radius^2+height^2)]

Total surface area of cone = Surface area + π*radius2.

In this blog we will also see some example problems related to Volume of a cone.By solving few problems we can understand how to calculate the surface of a cone.
Problem 1:

Find the volume of cone for the radius = 1 meter and height = 1 meter.

Solution:

Given radius = 1 meter

Height = 1 meter

Volume of cone = [1/3] * π * radius2 * height

= [1/3 ] * 3.14 * (1)2 * (1)

= [1/3] * 3.14 * 1 * 1

= [3.14/3]

= 1.05 m3.

Problem 2:

Find the Total Surface area of cone for the radius is 3 meter and height is 4 meter.

Solution:

Given: Radius = 3 m

Height = 4 m

Surface area of cone = π*radius* [sqrt(radius^2+height^2)]

= 3.14 * 3 * [sqrt(3^2+4^2)]

= 3.14 * 3 * [sqrt(9+16)]

= 3.14 * 3 * [sqrt(25)]

= 3.14 * 3 * 5

= 47.1 m2.

Total surface area of cone = Surface area + π*radius2.

= 47.1 + 3.14*32

= 47.1 + 3.14 * 9

= 47.1 + 28.26

= 75.36 m2.

Problem 3:

Find the height of a cone for the volume of cone is 100 cubic inches and radius of a cone is 5 inches.

Solution:

Given: Volume of cone V = 100 cubic inches

Radius of cone = 5 inches

Volume of cone = [1/3 ] * π * radius2 * height

100 = [ 1/3 ] * π * 52 * height

100 = [1/3] * 3.14 * 25 * height

100 = [1/3] * 3.14 * 25 * height

100 = [(3.14*25)/3] *height

100 = [78.5/3] *height

100 = 26.17 * height

Height = [100/26.17]

Height = 3.82 inches

Hope you like the above example of Surface area of cone.Please leave your comments, if you have any doubts.

Coordinate Planes

Coordinate Plane:

In this Blog before we get into the details of coordinate planes,let us first understand the definition of a coordinate plane.The basic meaning of a coordinate plane is Coordinate plane is a plane formed by the intersection of a horizontal number line.

Definition of Coordinate Plane:

Coordinate plane is a plane formed by the intersection of a horizontal number line with a vertical number line. They intersect at their zero points. This point of intersection is called the origin and written as (0, 0).

On a coordinate plane, the horizontal number line is called the x-axis and the vertical number line is called the y-axis.

This is a coordinate plane. It has two axes and four quadrants. The two number lines form the axes. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. The center of the coordinate plane is called the origin. It has the coordinates of (0,0). Locations of points on the plane can be plotted when one coordinate from each of the axes are used. This set of x and y values are called ordered pairs.




















With the help of another example we can get a clear understanding about the coordinate Planes.


In the coordinate plane there are four quadrants.

Quadrant 1.

In quadrant 1 both X axis and Y axis values are positive. [( +x, +y )]

Quadrant 2

In quadrant 2 X axis values are negative and Y axis values are positive. [( - x, +y )]

Quadrant 3

In quadrant 3 X axis values are positive and Y axis values are negative. [( + x, - y )]

Quadrant 4

In quadrant 4 both X axis and Y axis values are negative. [( - x, - y )]
Hope you like the above example of Coordinate Planes.Please leave your comments, if you have any doubts.