Monday, March 25, 2013

Which Fraction is Larger


 Yes, which fraction is larger or smaller? The question has great importance. As in many of the competitive exams, we are asked to arrange a set of given fractions. Before learning, how to compare the fractions, we'll learn the basics of fractions. Then we'll learn two methods, that will help us to tell "which fraction is larger".

Fraction: A number of the form `a/b` , where a and b are integers and b`!=` 0,  is known as fraction. Here, a is called numerator and b is called denominator.

Example: `3/5` is a fraction, where Numerator = 3, and Denominator = 5.

7 is also a fraction, because it can be written as `7/1`

1st Method to find which fraction is larger


Method to compare two fractions: This method is known as Cross multiplication method. Steps are as follows:

Let `a/b` and  `c/d` be the two given fractions, that we have to compare and decide which fraction is larger?

Find the product of numerator of the first fraction and the denominator of the 2nd fraction. So, we get ad and bc.

i) If ad > bc, then `a/b`  >  `c/d` .

ii) If ad = bc, then   `a/b` = `c/d` .

iii) If ad < bc, then `a/b`  <  `c/d`

Ex: Compare, which fraction is larger `3/5` and `5/8`?

Products are 3X8= 24 and 5X5=25.

Here, 24 < 25,  So `3/5`` < ``5/8`

Now, we can say very easily, which fraction is larger.

Hence, `5/8` is larger fraction among the two.


2nd Method to know which fraction is larger:


This method is used to compare more than two fractions:

Step I: Find the LCM or LCD of the given fractions. Let it be m.

Step II: Convert all the given fractions into like fractions, each having m as denominator.

Step III: Now we compare the numerators. The fraction having greater numerator is greater and vice- versa.

Ex: Arrange the fractions  `2/5` , `3/10` and `9/14` in ascending order.

LCM of 5, 10 and 14 is 70.

Now, let us change each of the given fractions into an equivalent fraction having 70 as denominator.

Fraction problem


Practice problems of which fraction is larger


1. Which fraction is larger: `5/6` or `6/7` ?

2. Which fraction is larger: `21/25` or `3/5` ?

Monday, March 18, 2013

Study graph functions


Graphs of functions f is the collection of the all ordered pairs (x, f(x)). In particular, if x is a real number, a graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on the Cartesian plane is sometimes are called curve sketching. If the function input x is an ordered pair (x1, x2) of real numbers, the graph is the collection of all ordered triples (x1, x2, f(x1, x2)), and its graphical representation is a surface .

Example problems for study graph functions :


Study linear functions graph:
If a function f   : R → R is defined in the form f(x) = dx + e then the function is called a linear function. Here d and e are constants.

Problem 1: 

Draw the graph of the linear function f : R → R defined by f(x) = 7x + 1.
Solution:
Draw the table of some pairs (x, f(x)) which satisfy f(x) = 7x + 1.

x -2 -1 0 1 2
f(x) 15 8 1 -6 -13

Plot the points and draw the curve passing through these points. Note that, the curve is a straight line.

graph functions

Study functions of graph:

Graph of a functions: The graph of a function f is a graph of the equation y = f(x)

Problem 2:
Draw the graph of the function f(x) =2x2

Solution:
Draw a table of some pairs (x, y) which satisfy y = 2x2
x -3 -2 -1 0 -1 -2 -3
f(x) 18 8 2 0 2 8 18

Plot the points and draw the smooth curve passing through the plotted points.

graph functions

Note:

Note that if we draw a vertical line to the above graph, it meets the curve at only one point
i.e. for every x there is a unique y

Example problem for study logarithmic graph functions :


Draw the graphs of the logarithmic functions (1) f(x) = log 52x (2) f(x) = log 10ex (3) f(x) = log 53x
Sol:

The logarithmic function is the defined only for the positive real numbers. i.e. (0, ∞)
Domain: (0, ∞) Range: (− ∞, ∞)
log functions

Thursday, March 14, 2013

Study Online Exponentiation


Exponentiation should be the operation, which is written as the form of an. Where a and n is said to be base and exponent as well as n is any positive integer. In general, exponentiation means that repetitive multiplication. Otherwise, exponentiation an is the product of n factors of a. Online study should be the topic-oriented or else technical help for clarifying doubts that are convey through the computer software. Let us study properties and example problems for exponentiation.


Properties - Study Online Exponentiation

We are having seven number of exponentiation properties that used for solving problems. In this properties, a, m and n are any integer values.

Product of like bases:

The product of powers with the same base means we can add the powers and keep the common base.

am an = am+n

Quotient of like bases:

To divide the powers with similar base, we can subtract the exponents and remain the common base.

`a^m/a^n` = am-n

Power to a power:

In case of raising the power to power, we need to keep same base and multiply the exponent values.

`(a^m)^(n)` = amn

Product to a power:

In case of raising the product to power, we need to raise each factor to the power value.

(ab)m = am bm

Quotient to a power:

In case of raising the quotient to power, we need to raise the numerator and denominator to the power value.

`(a/b)^n` = `a^n/b^n`

Zero exponent:

Any number that is raised with zero power should be equivalent to ‘1’.

a0 = 1

Negative exponent:

a-n = `1/a^n` or `1/a^(-n)` = an

These are the properties that are used for exponentiation problems in study math online.


Example Problems - Study Online Exponentiation


Example 1:

Solve 32 34.

Solution:

Given, 32 34.

This is in the structure of am an, so we need to use am an = am+n property.

Here, m = 2 and n = 4 and a = 3.

Thus, 32 34 = 32+4

= 36

= 3 × 3 × 3 × 3 × 3 × 3

= 9 × 9 × 9

= 729

Hence, the answer is 32 34 = 729.

Example 2:

Shorten the following `6^7/6^4` .

Solution:

Given, `6^7/6^4` .

This is in the structure of `a^m/a^n` , so we need to use `a^m/a^n` = am-n property.

Here, m = 7 and n = 4 and a = 6.

Thus, `6^7/6^4` = 67-3

= 64

= 6 × 6 × 6 × 6

= 36 × 36

= 1296

Hence, the answer is `6^7/6^4` = 1296.

That’s all about the study online exponentiation.

Monday, March 11, 2013

Simple Algebra Problems


Algebra is a branch of mathematics dealing with variables, equations , expressions.Here we are going to solve some of the simple algebra problems including some equations, expressions and algebraic identities.

Solve Simple algebra problems on numbers:

Ex 1: Subtract  – 4 from – 10.

Sol :
=– 10 – (– 4)

= – 10 + (additive inverse of – 4)

= – 10 + 4

= – 6 (see addition of two integers)

Ex 2: Find the product of  (a) (+ 6) × (– 5) (b) (– 12) × (+ 12) (c) (– 15) × (– 4)

Sol : 

(+ 6) × (– 5) = – 30 (positive × negative= negative)

(– 12) × (+ 12) = – 144 (negative × positive = negative) 

(– 15) × (– 4)= + 60 (negative × negative = positive) 

Ex 3: Find the sum of given algebraic expression 2x4 – 3x2 + 5x + 3 and 4x + 6x3 – 6x2 – 1.

Sol:

Using the associative and distributive property of the real numbers, we obtain (2x4 – 3x2 + 5x + 3) + (6x3 – 6x2 + 4x – 1)

= 2x4 + 6x3 – 3x2 – 6x2 + 5x + 4x + 3 – 1

= 2x4 + 6x3 – (3+6)x2 + (5+4)x + 2

= 2x4 + 6x3 – 9x2 + 9x + 2.

  The following scheme is helpful in adding two polynomials 2x4 + 0x3 – 3x2 + 5x + 3 0x4 + 6x3 – 6x2 + 4x – 1 2x4 + 6x3– 9x2 + 9x + 2 


Ex 4: Find the subtraction of given simple algebraic expression 2x3 – 3x2 – 1 from x3 + 5x2 – 4x – 6.

Sol: Using associative and distributive properties, we have (x3 + 5x2 – 4x – 6) – (2x3 – 3x2 – 1)

= x3 + 5x2 – 4x – 6 – 2x3 + 3x2 + 1

= x3 – 2x3 + 5x2 + 3x2 – 4x – 6 + 1

= (x3 – 2x3) + (5x2 + 3x2) + (–4x) + (–6+1)

= –x3 + 8x2 – 4x – 5.

The subtraction can also be performed in the following way:

Line (1): x3 + 5x2 – 4x – 6.

Line (2): 2x3 – 3x2 – 1.

Changing the sign of the polynomial in Line (2), we get Line (3): –2x3 + 3x2 + 1.

Adding the polynomial in Line (1) and Line (3), we get –x3 + 8x2 – 4x – 5 

Ex 5: Find the product of   given simple algebraic expression x3 – 2x2 – 4 and 2x2 + 3x – 1.

Sol:

(x3 – 2x2 – 4) (2x2 + 3x – 1)

= x3 (2x2 + 3x – 1) + (–2x2) (2x2 + 3x – 1) + (–4) (2x2 + 3x – 1)

= (2x5 + 3x4 – x3) + (–4x4 – 6x3 + 2x2) + (–8x2 – 12x + 4)

= 2x5 + 3x4 – x3 – 4x4 – 6x3 + 2x2 – 8x2 – 12x + 4

= 2x5 + (3x4 – 4x4) + (–x3 – 6x3) + (2x2 – 8x2) + (–12x) +4

= 2x5 – x4 – 7x3 – 6x2 – 12x + 4.


Algebra Identities to solve some simple problems :


 Algebraic identity for (x + a)(x + b)

 By using the distributive properties of numbers,

(x + a )(x + b ) = x(x + b) + a(x + b) = x2 + xb + ax + ab

= x2 + ax + bx + ab= x2 + (a + b)x + ab.

 (x + a)(x + b) ≡ x2 + (a + b)x + ab

(x – a)(x + b) ≡ x2 + (b – a)x – ab

(x + a)(x – b) ≡ x2 + (a – b)x – ab

(x – a)(x – b) ≡ x2 – (a + b)x + ab

(a + b)2 ≡ a2 + 2ab + b2

(a – b)2 ≡ a2 – 2ab + b2

(a + b)(a – b) ≡ a2 – b2

(a + b)3 ≡ a3 + 3a2b + 3ab2 + b3

(a − b)3 ≡a3−3a2b+ 3ab2−b3

a3 + b3 ≡ (a + b)3−3ab(a + b)

a3 − b3 ≡ (a −b)3+ 3ab(a−b)

Solve the following simple algebra problems using above identities:


Pro 1: Find the product:  (x + 3) (x + 5)

Sol: (x + 3) (x + 5) = x2 + (3 + 5) x + 3 × 5 = x2 + 8x + 15.


Pro 2:  Find the product:  (p + 9) (p – 2)

Sol: (p + 9) (p – 2) = p2 + (9 – 2) p – 9 × 2

= p2 + 7p – 18.


Solve simple algebra problems on the cubic identity


Algebra problems on cubic identity:

(i) (3x+2y)3 = (3x)3 + 3(3x)2(2y) + 3(3x)(2y)2 + (2y)3

= 27x3 + 3(9x2)(2y) + 3(3x)(4y2) + 8y3

= 27x3 + 54x2y + 36xy2 + 8y3.

 (ii) (2x2– 3y)3 = (2x2)3 – 3(2x2)2 (3y) + 3(2x2) (3y)2 – (3y)3

= 8x6 – 3(4x4)(3y) + 3(2x2)(9y2) – 27y3

= 8x6– 36x4y + 54x2y2 – 27y3


Pro 1: If the values of a+b and ab are 4 and 1 respectively, find the value of a3 + b3.

Solution:a3+ b3 = (a+b)3 – 3ab(a+b) = (4)3 – 3(1)(4)

= 64 – 12

= 52.


Pro 2: If a – b = 4 and ab = 2, find a3 – b3.

Solution:a3 – b3 = (a–b)3+3ab(a–b)

= (4)3+3(2)(4) =64+24

=88.

Friday, March 8, 2013

Median Test Learning


Median Definition :If the generally number of values within the case is even, next the median is the mean of the two center numbers. The median is a supportive number within cases wherever the distribution contain extremely large huge values which would or else twist the data. Represent the center value of a given set of values is called median.

To locate of values set within lower value toward higher value (rising order). If the collections of data contain an odd number of ways; the median is the middle value of the place behind arrangement the listing into rising order.


Median test learning


Method for median computation

`M=(n+1)/2`

Wherever,

M- Median,

n – Entirety number of set

But here is an odd number of values, then the median is the center value of the set.

If an even number to find median first

For example the given set{2,6,8,10}

Total number of value (n) =4

Median`M=(n+1)/2`

`M=(4+1)/2`

`M=(5)/2`

M=2.5

Consequently median must appear among second and third value.

Second value=6

Third value =8

Mean of second and third value is the Median of this set

Median`M=(14)/2`

`M=(7)/2`

M=3.5


Examples for median test learning


Example 1 for median test learning

Compute the median of this set {20, 10, 12, 40, 60,50,30}.

Solution:

First values can be arranged in order

Therefore the given set={10, 12, 20, 30, 40, 50, 60}

Here, total number of value (n) =7.

Median`M=(n+1)/2`

`M=(7+1)/2`

`M=(8)/2`

M=4

Therefore the fourth value is median that is 30.

Example 2 for median test learning

Compute the median of this set {10, 12, 13, 17, 18, and 20}

Solution:

Total number of value (n) =6

Median`M=(n+1)/2`

`M=(6+1)/2`

`M=(7)/2`

M=3.5

Consequently median must appear among Third and Fourth value.

Third value=13

Fourth value =17

Mean of third and fourth value is the Median of this set

Median`M=(13+10)/2`

`M=(30)/2`

M=15

Example 3 for median test learning
Compute the median of this set {26, 32, 36, and 55}

Solution:

Total number of value (n) =4

Median`M=(n+1)/2`

`M=(4+1)/2`

`M=(5)/2`

M=2.5

Consequently median must appear among second and third value.

Second value=32

Third value =36

Mean of second and third value is the Median of this set

Median`M=(32+36)/2`

`M=(68)/2`

M=34

Learning Weighted Median


In calculation of weighted median, the importance of all the items was considered to be equal. However, there may be situations in which all the items under considerations are not equal importance. For example, we want to find average number of marks per subject who appeared in different subjects like Mathematics, Statistics, Physics and Biology. These subjects do not have equal importance. Learning the formula to find weighted median by giving Median.


Learning Median Definition:


The arithmetic median computed by specific importance of every object is known weighted arithmetic median. To consider the every importance object, we can assume number known as weight to every object is directly proportional to its specific importance.

Weighted Arithmetic Median is computed by using following formula:

`sum` Yw = `(sum ty) / (sum t)`

Where:
Yw Stands for weighted arithmetic median.
y   Stands for values of the items and
t   Stands for weight of the item

Learning the important three methods of weighted median:

1.Arithmetic Median

2. Weighted Average

3. Average speed.

The formula for weighted average is:

Weighted Average = Sum of weighted objects / total number of objects


Learning to solve examples of Weighted median:


Ex 1:A student obtained 30, 40, 50, 60, and 35 marks in the subjects of Math, Statistics, Physics, Chemistry and Biology respectively. Assuming weights 1, 3, 5, 4, and 2 respectively for the above mentioned subjects. Find Weighted Arithmetic Mean per subject.

Solution:



Subjects
Marks Obtained
      y
Weight
   t
`sum` ty
Math
30
130
Statistics403120
Physics
50
5
250
Chemistry
60
4
240
Biology
35
270
Total
`sum` t = 15`sum` y= 710


Now we will find weighted arithmetic mean as:
`sum` Yw = `( sum ty)/(sum t) = 710/ 15`   =  47.33marks/subject

Ex 2: A class of 30 students took a math test. 15 students had an average (arithmetic median) score of 90. The other students had an average score of 70. What is the average score of the whole class?

Solution:Step 1: To get the sum of weighted terms, multiply each average by the number of students that had that average and then sum them up.

90 × 15 + 70 × 15 = 1350 + 1050 = 2400

Step 2: Total number of objects = Total number of students = 30

Step 3: Using the formula

Weighted Average = Sum of Weighted objects / Total Numbers of objects.

=  `2400 / 30`

= 80.

Answer: The average score of the whole class is 80.

Thursday, March 7, 2013

Learn Linear Functions in Real Life


In real life learning, linear function has single degree polynomial in its equation. The degree of the linear function always one. It cannot be increased. Linear function is  very useful in real life. In learning, linear function does not dependent any variable degrees. The linear function with the degree of one, always gives the straight line. In real life, linear function has the different variable equations. 'In learning of linear function in real life', we learn about linear equations and its degrees.

Forms of Linear function in real life


The function, which has straight line graph and that line function is known as linear function. In learning, linear functions are in three main forms.

1. Slope-Intercept Form is given by y = mx +b.

Example: y = 3x + 2

Here, slope m = 3 and y intercept b = 2

2. Point Slope Form is given by m = (y - y1) / ( x – x1)

Example: (y – 3) = 2(x – 1)

3. General Form is given by ax + by + c = 0

Example: 2x + 3y -1 = 0

Examples of set of real life linear function:

Example 1:

Solve for x and y. where, y = 3x + 4 and y = 7x.

Solution:

Plug, y = 3x + 4 in y = 7x

3x + 4 = 7x

x = 1

Now plug, x = 1 in y = 3x + 4

y = 3 + 4

y = 2

Answer:

The solutions are x = 1 and y = 2.

Example 2:

5x - y + 11 = 0 and x - y - 5 = 0, solve for x and y.

Solution:

5x - y + 11 = 0     (1)

x - y - 5 = 0         (2)

Subtract Equation (2) form Equation (1), we get,

4x + 16 = 0

4x = - 16

x = - 4

plug x = - 4 in equation (1)

- 4 – y + 13 = 0

- y + 9 = 0

y = - 9

The solutions are x = - 4 and y = - 9.


Learn linear function in real life - Practice problems:


Problem 1:

3x - y = 9 and 4x + y = 5. Solve for x, y.

A) (2, 3) B) (- 2, 3) C) (2, -3) D) (3, -2)

Answer: C

Problem 2:

2x + y = 12 and y = 3x - 3. Solve for x, y.

A) (3, 6) B) (3, - 6) C) (4, 5) D) (6, - 3)

Answer: A

Problem 3:

Solve the linear function 2x - y = 12 and y = 4x - 2

A) (- 5, 22) B) (5, - 22) C) (- 5, - 22) D) (5, 22)

Answer: C

Wednesday, March 6, 2013

solving online area of a circle


CIRCLE:

A line forming a closed loop, every point on which is a fixed distance from a center point.

A circle is a plane continuous figure connecting points which are equidistant from a given point, known as the center.

The word circle originates from the Latin word 'circus'. Chariot races, very popular in the Roman era were either circular or oblong and eventually the word was used to describe the shape as well.


Area of a circle - Important terms

In order to find the area of a circle, it is essential that we first understand a few important terms related to circles.

The outermost portion of the circle which separates the interior of the circle from the exterior is known as the circumference.

The center, as already mentioned, is the point inside the circle which is equidistant from all points lying on the circumference.

The distance between any point on the circumference from the center is known as the radius.

A line segment connecting any two points of lying on the circumference on the circle is called a chord.

The longest possible chord of a circle is known as a diameter. The diameter equals twice the radius.


Solving the area of a circle

Consider a circle with center P. Let us denote the radius of the circle by 'r' and the diameter by 'd'. Pi (denoted as π) is a numerical value, which is a crucial number used to calculate various attributes of circular figures. Its value up to 2 decimal points is 3.14. In terms of the radius.

we define the area (A) of the circle as:                  A = π r²

In terms of the diameter, which is twice the radius, this equation can be re-written as:

A = 0.25 π d²

The area in terms of the circumference (C) of the circle is given as:

A = C²/ 4π

Examples and exercises of area of circle.


Example 1: The radius of a circle is 3 inches. What is the area?

A = π r²

A  =  (  3.14  ) ( 3 ) 2

A = (3.14) (9)

A =  28.26 in 2

Answer  the following :

1) The diameter of a circle is 8 centimeters. What is the area?

2) The radius of a circle is 5 feet. What is the area?

Tuesday, March 5, 2013

Half Angle Properties


Introduction:

Let we will discuss about the half angle properties. The half angle is the angle that is half of original angle. Tha is, the product of two half angles at an edge is equal to the original(full) angle at that same edge. We will develop the properties formulas for half angle of both sine and cosine.


Half angle properties formula for sine:


We should start with formula for cosine of double angle. That is,
cos 2θ = 1 - 2sin^2 θ

Half angle properties formula - sine:

Let us consider,


Then 2θ = α and the formula becomes:


Solve for,


That is, we acquire sin(α/2) lying on the left of equation and everything else on right,



Solving of equation gives the following sine of half-angle identity:


The sign of sin α/2 represents on quadrant in which α/2 lies.
If α/2 is in the first or second quadrants then formula uses the positive case:


If α/2 is in the third or fourth quadrants, the formula uses the negative case:


Example:

Find the value of sin 45o using the sine half-angle properties.

Solution:

The sine half angle formula is ,



Therefore,

sin 45o = `sqrt((1 - cos 90 )/(2))`

= `sqrt((1 - 0 )/(2))`

= `sqrt((1)/(2))`

= 0.707


Half angle properties formula for cosine:


Half angle properties formula - cosine:

Using same process, with the similar replacement of θ = α / 2. We want to substitute into the identity,


We get,


Reverse process of the equation should be,


Addition of 1 with both sides of an equation. We get,


Making division process on both sides of equation by 2


Solving for cos(α/2), we get,


As before, the sign we need depends on quadrant.
If α/2 is in first or fourth quadrants that formula uses in positive case:


If α/2 is in the second or third quadrants, the formula uses the negative case:


Example:

Find the value of cos 115o using the cosine half-angle formula.

Solution:

We need to find cos 115o

That is, α = 230°, and so α/2 = 115°.

Therefore, cos 115o = `sqrt((1 + cos 230 )/(2))`

= `sqrt((1-0.643)/(2))`

= `sqrt((0.357)/(2))`

= 0.422

We have seen about the half angle properties.

Monday, March 4, 2013

Geometric Area


The geometric area refers to the size of the interior of a planar (flat) figure. Area (A) is a two-dimensional measure. The square units that  an area is measured with square inches, square feet and square centimeters. Hectares and acres are also considered in some special cases. In the International System of Units (SI), the standard unit of area is the meter squared (m 2).

Geometric area of Square

Formula to find the area of the square:

Area of the square is calculated by multiplying the base times itself.
Area of square = side x side square unit.
Area = a2

1. The side length of the square is 10 feet; find the area of the square.
Solution:

Area = a2

Given: Side length= 10feet.
                               = 10 x 10
                   Area    =100 feet2

2. The side length of the square is 4.5 m; find the area of the square.
Solution:

Area = a2

Given: Side length= 4.5m.
                              = 4.5 x 4.5
                      Area =20.25 m2

Geometric area of Rectangle

Area of the rectangle is calculated by multiplying the base times the height.

 Area of rectangle     =  (length x width) square unit.

Area = l x w square unit

3. The length and breadth of the rectangle are 10 meters and 5 meters respectively .find the area of that rectangle.

Solution:

Area = l x w square unit
           
Given: Length= 10 meters
                        Breadth=5 meters
                                    =10x5
                        Area    = 50 m2      
     
4. The length and breadth of the rectangle are 11 feet and 6 feet respectively .find the area of   that rectangle

Solution:

Area = l x w square unit.
            Given: Length= 11feet
                        Breadth=6feet
                                    =11x 6
                        Area    = 66 ft2

Geometric area of Triangle


Area of the triangle is calculated by multiplying the base times the one-half the height.

Area of triangle= `(1)/(2)` (base x height)

Area = `(1)/(2)bh`

5. What is the area of triangle with base 5 and height 10 feet?

Solution:

Area = `(1)/(2)bh`

Given: Base= 5 feet.
         Heigh t= 10feet
                    = 1/2 (5x 10)
                    =1/2(50)

            Area =25 ft2

6. What is the area of triangle with base 7 and height 14 feet?
Solution:

Area = `(1)/(2)bh`

Given: Base= 7 feet.
         Height=14feet
                    =1/2 (7 x 14)
           Area =1/2 (98)
            Area =49 ft2

Geometric area of Circle

Area of the circle is calculated by multiplying pi ( π = 3.14) by the square of the radius
Area of the circle= π x r2

7. The radius(r) of a circle is 3 inches. Find the area of that circle?

Solution:

Area of the circle= π x r2
π=3.14

Given: r=3inches
          A=3.14 x (3)2
                   =3.14 x 9 inche2.
    Area= 28.26 in2.

8. The radius(r) of a circle is 7.5 meters. Find that area of the circle.

Solution:

Area of the circle is calculated by multiplying pi ( π = 3.14) by the square of the radius
Area of the circle= π x r2
π=3.14

Given: r= 7.5 meters
          A= 3.14 x (7.5)2
            =3.14 x 56.25
  Area =176.625 m2

Friday, March 1, 2013

circumference circle


In our real life, we could see many circular objects. For example, wheel of a vehicle, shape of a pizza, tennikoit ring,.. Take a rope or an inelastic thread and measure the circular path of the given object and that forms the circumference of the circular object. As the size of the circle increases, the circumference of circle also increases.

Introduction on circumference of a circle:

Circle is one of the most important shapes in geometry. Circumference is defined as the measure needed to make the closed curve. Circle has a fixed center. Distance between the center and a point on the circumference of the circle is called as radius. A line passing through center and any two points on the circumference of the circle is called as diameter.  If the radius or diameter is given then we can easily calculate the circumference of the circle. This article helps us to find the circumference of the circle with the given details.


Circumference of a circle:


The diagram below shows the circumference of circle.

Circumference of circle

The Circumference of a circle can be calculated using the following formula:

Circumference = 2  `pi`r  or `pi`

where r ----> radius of circle

d ----> diameter of circle

π---- > constant and value of π is  3.14 or 22/7

Diameter is twice the radius. So d = 2r and   r = `d/2`


Problems on finding circumference of the circle:

Ex 1: Find the Circumference of the circle with radius 20 cm.

Sol:

Step 1: Write the formula

Circumference of a circle  = 2  `pi`r units

Step 2: Plug the known values and calculate circumference

2π r               = 2 (22/7) 20

= 2 (3.14) 20

= 125.6 cm

Ex 2: Find the Circumference of the circle with radius 32 cm.

Sol:

Step 1: Formula for Circumference of a Circle

Circumference of a circle  = 2  `pi`r units

Step 2: Plug the known values and calculate circumference

2πr                  = 2 (22/7) 32

= 2 (3.14) 32

= 200.96 cm

Ex 3: Find the Circumference of the circle with diameter 52 in.

Sol:

Step 1: Write the formula

Circumference of a circle  = `pi` d

Step 2: Plug the known values and calculate circumference

πd             = (22/7) 52

= (3.14) 52

= 163.28 in

Ex 4: Find the Circumference of the circle with diameter 74 in.

Sol:

Step 1: Write the formula

Circumference of a circle  = `pi` d

Step 2: Plug the known values and calculate circumference

πd            = (22/7) 74

= (3.14) 74

= 232.36 in

I like to share this jee main 2013 sample paper with you all through my article.


Practice problems on circumference of the circle:


Find the Circumference of the circle with radius 18 cm. .
Answer: 113.04 cm

Find the Circumference of the circle with radius 28 cm.
Answer: 175.84 cm

Find the Circumference of the circle with diameter 63 in.
Answer: 197.82 in

Find the Circumference of the circle with diameter 81 in.
Answer: 254.34 in