How to calculate Area of Ellipse?

An ellipse is nothing but the planar curve which is resulted by the intersection by a plane of a cone in a way such that it will produce a closed curve. Circles can be said as a special case of ellipse which can be obtained by cutting in orthogonal plane of the cone’s axis. Ellipse can also be said as the locus of all the points in the plane in which the sum of the distances of two fixed points will be a constant. Before going into the formula for area of an ellipse, it is essential to know the elements of ellipse which forms the structure of it. An ellipse is said to be a smooth and closed curve symmetric about its vertical and horizontal axis. The longest diameter of an ellipse is called as major axis and the shortest diameter of an ellipse is called as the minor axis. These both axes are the lines that pass through the centre of an ellipse.

With the measurement of both these axis only, the area of a ellipse will vary. Each of the above said axes bisects perpendicularly with the other. Also, the sum of the distance from the two focus of the ellipse to any point P on the ellipse will be equal to the major axis. It can also be said that each axis cut the other equally into two parts and they cross at right angles to each other. Also, if these both axes are equal in length, then it is called as circle.

Area of an Ellipse Formula
The standard equation for representing the ellipse equation is given as,
X^2 / a^2 + Y^2 /b^2 = 1.

This equation is a standard equation in which when the ellipse is centred with origin.
But algebraically the ellipse area formula is given in other terms with the help of semi major axis (half of major axis) and semi minor axis (half of minor axis). Thus the formula for area of ellipse is given as Pi*a*b, where ‘a’ is the semi major axis and ‘b’ is the semi minor axis. This formula is actually arrived from the formula of circle Pi multiplied by radius square. Here the radius is split into semi major and semi minor axis.
Also at some special cases, when an ellipse is given by the implicit equation which is given as,
AX2 + Bxy + CY2 = 1, then the area of the ellipse would be 2*Pi whole divided by square root of 4AC subtracted with B square.

Understanding Fractions Better

Numbers have always fascinated humankind. Indians are credited with the invention of zero. Indians have always been good with numbers and the world looks at us with awe. We have great strides and come a long way. From natural numbers, whole numbers, integers, real numbers and complex numbers. We have also studied fractions. Let us what are fractions to understand the concepts of denominator numerator. A number can written in the form of A/B where A is an integer and B is an integer. A can be called the numerator and B can be called the denominator. So we see numerator v/s denominator to extend the concept further. The numerator is written on top of the horizontal line and the denominator is written below the horizontal line. So we understand what is a numerator and denominator if we didn’t know it earlier.

We must understand numerators and denominators to understand fractions better. Operations of addition, subtraction, multiplication and division between the fractions can be performed. Here both numerator and also the denominator come into picture. If the denominators are dissimilar they must be made similar to proceed further with the operations. For this we need to take the LCM of the denominators. Taking the LCM is a simple operation. Unless the denominators are equal we cannot proceed further. Once the denominators are equal, we can just add the numerators if the operation to be performed is addition. If subtraction is to be performed we just need to subtract the numerators after making the denominators equal. So we must know the definition of numerator and denominator to perform these operations. Only when we know it, we can do something.

After making the denominators equal if they are not, we proceed to perform addition, subtraction or any operation with fractions. Sometimes the fractions can be in the form of mixed fractions. We need to be careful in such cases. We must first convert them into improper fractions without proceeding. They cannot be directly used. For the process of finding LCM we need improper fractions in our hand. Once we get the improper fractions we first check whether their denominators are same or not. If they are same, our problem becomes very simple and easy. The numerators can be just added or subtracted as the case may be. If they are not equal only then the process of taking LCM is to be performed and addition or subtraction is to be done.

Distributive Property of Addition

The distributive property is expressed in the mathematics expressions as following equation: a (b + c) = ab + ac. You can understand this as the sum of a (b + c) is corresponding to the sum of a times b and a times c. Distributive property of addition would be inaccurate to multiply ab and just add c, or to multiply ac and add b.

Order of Operations:

The distributive property of addition that remind that all contained by the parenthesis needs to be multiplied by the outside number. Distributive property, when they are knowledge the order of operations.

Concept of that the problems anywhere present are different mathematical operations, such as multiple, addition, subtraction, parenthesis, you have to work in a certain order to get the right answer. This arrange is the parenthesis, exponent, multiplication,division , addition and subtraction, that may be abbreviate to the  PEMDAS.

Example Problems (distributive Property of Addition):

Example problems:

When you include a mathematics problem that use parenthesis you need to solve what’s in the parenthesis first, by you can move about on to solve further problems. If the mathematics problem largely has known numbers, it is somewhat easy to solve. 2(10+5) becomes 2(15) or is also equal under the distributive property of addition is 2(10) + 2(5). What obtain additional difficult is when you are functioning with variables (such as a, b, x, y, and so on) in algebra, and when these variables cannot be joint together.

Consider the equation 5(12a + 2), but we don’t know what the variable a stand for, we can’t add 12a + 2, but using the distributive property still allow us to just this expression because we identify this equation is equal to 5(12a) + 5(2). In order to simply the expression we can take each part separately and multiply it to 5, and we get 60a + 10.

Addition, Subtraction, Multiplication and Division

As the learning session for numbers is completed, the next session starts with addition, subtraction, multiplication and division. Understanding these calculations is the basics of learning mathematics. It is quite challenging to make kids’ understand the concepts of addition, subtraction, multiplication and division followed by order of operations, simplification and so on. In that case providing fun examples related to kids can be helpful. Let’s give it a try.

Addition: Addition denoted by the plus sign, “+”, is a mathematical operation that combines two or more collection forming larger collections.

For example: It is Tina’s birthday and she has received many beautiful dresses for girls – one from Annie aunty, another from grandpa and another one from her mother. Now adding all three we get, 1+1+1 = 3 dresses for girls. This mathematical operation is addition.

Subtraction: Subtraction denoted by the minus sign, “-”, is a mathematical operation that is inverse of addition. In this operation, a part is taken away from a collection and thus forming a smaller collection.

For example: Father brought 50 return gifts for kids on Tina’s birthday. Only 45 of her friends came for the birthday party. Now after subtracting the two, we can find, 50-45 = 5 return gifts for kids remaining. This mathematical operation is subtraction.

Multiplication: Multiplication denoted by the sign “x” is a mathematical operation that scales one number by another.

For example: Mother is planning to organize a fancy dress costumes competition on next Sunday. There are 5 groups of two members and therefore each group requires two costumes. Now by multiplying, we get 5 X 2 = 10 fancy dress costumes. This mathematical operation is multiplication.

Division: Division denoted by the sign “/” is a mathematical operation that includes a dividend, quotient, divisor and remainder.

For example: Tina has 5 best friends and she has 20 chocolates. She wanted to distribute the chocolates among them. Now diving the numbers we get, 20/5 = 4. Each of her friends will get 4 chocolates. This mathematical operation is division.

These are the basic operations of mathematics that form the first steps in learning mathematics for kids.

Problems related to consecutive integers

Many a times in arithmetic we face problems related to finding consecutive integers the sum of which is given to us. The said integers may be odd or even or neither. In this article we’ll try to understand how to solve such type of problems.

Formula for sum of consecutive integers:
As such there is no specific formula for finding sum of consecutive integers. Such problems are solved using algebra and arithmetic approach. The methods may however, vary from problem to problem. Let us look at the following example to understand this better.

Example 1: Three consecutive integers are there whose sum is 66. Find consecutive integers.
Solution: Suppose we assume that the middle or the second integer is n, then the first integer would be n-1 and the third integer would be n+1, because the integers are said to be consecutive, and we know that consecutive integers differ by 1. The three integers are therefore now: n-1, n and n+1. The sum of these integers is given to be 66. Therefore,
n-1 + n + n+1 = 66
3n = 66
n = 66/3
n = 22.
Since n was our middle integer, the other two would be n-1 = 21 and n+1 = 23. So the three consecutive integers are 21, 22 and 23.
Odd consecutive integers:
Odd consecutive integers are such that each of the integer in the list is an odd number. It may be negative or positive, but has to be an odd number. For example, …. -11, -9, -7, -5, -3, -1,  1, 3,  5,  7,  9…These are odd consecutive integers. They go up to infinity in both directions.

Example 2: The sum of three consecutive odd integers is -39. Find the integers.
Solution: Here again, assume that the middle integer is n. Then the odd integer before n would be n-2 and the odd integer after n would be n+2. So the sum
= n-2 + n + n+2
= 3n = -39
n = -39/3
n = -13.
Therefore the other two odd integers would be -15 and -11. So solution: -15, -13, -11 are the required three odd integers.
Even consecutive integers:
The method of solving problems of even consecutive integers is exactly same as that for odd integers.

Example 3: Find the three even integers whose sum is 0.
Solution: If the middle integer is n, then the other two would be n-2 and n+2. Therefore the sum
= n-2 + n + n+2
= 3n = 0
n = 0/3
n = 0.
So the integers would be -2, 0, and 2

Integers: Positive and Negative

Integers are the numbers consisting of positive numbers, zero and also the negative numbers.  The positive numbers and negative numbers are called as positive negative integers. Positive integers are the integers with positive sign for example, 2, 3, 1, 5 etc. and negative integers are the integers with negative sign. Examples of Negative Integers are -21, -3, -1, -5 etc. On a number line the integers are shown as given below

The following are the Positive and Negative Integers Rules:
Addition rules of Integers: 
When two positive integers are added the sum would be a positive integer; (3+5=8)
When two negative integers are added the sum would be a negative integer; [(-3) + (-5)= -(3+5)= -8]
When a positive integer and a negative integer are added, the numbers are subtracted and given the sign of the larger number;[ (3)+(-5)= -2]

Subtracting Positive and Negative Integers:
A positive integer subtracted from a negative integer would give a negative integer; (-7)-(4)=-7+4= -3
A negative integer subtracted from a positive integer would give a positive integer, here the numbers are added and given the positive sign; (5)- (-7)=(5+7) = 12
When two negative numbers are subtracted it gives a negative and a positive integer. The numbers are subtracted and given the sign of the larger number; (-5) – (-7)=-5+7= 2; -7 – (-5)= -7+5= -2

Multiplication Rules of Integers:
A  Positive Integer multiplied with a positive integer, the product would be a positive integer
3x5=15
A positive integer multiplied with a negative integer, the product would be a negative integer
3x (-5)=-15
A negative integer multiplied with a positive integer, the product would be a negative integer
(-7)x3=-21
A negative integer multiplied with a negative integer, the product would be a positive integer
(-7)x(-5)= 35

Dividing Positive and Negative Integers:
A  Positive Integer divided by a positive integer gives a positive integer, 15/5=3
A positive integer divided by a negative integer gives a negative integer, 15/(-3) = -5
A negative integer divided by a positive integer gives a negative integer, -21/3 = -7
A negative integer divided by a negative integer gives a positive integer, (-21)/(-7) = 3

Positive and Negative Integers Word Problems
The temperatures recorded in the Sahara Desert and Thar Desert are 138 degrees Fahrenheit and -55 degrees Fahrenheit respectively. Calculate the difference in the temperatures.
Sahara Desert Temperature= 138 degrees Fahrenheit
Thar Desert Temperature= (-55) degrees Fahrenheit
Difference in temperatures = Temperature in Sahara Desert – Temperature in Thar Desert
      = 138 – (-55) = 138 +55 = 193 degrees Fahrenheit (Answer)

Standard Deviation

Standard deviation is denoted by a symbol of Greek letter (s) and is shows how much variation exists from the mean or average or expected value. If the standard deviation is low, that will indicate that the point tents to very close to the mean, whereas standard deviation is high, the data points are spread out over a large range of values. Standard deviation has very useful important properties such as unlike variance and is expressed in the same units as the data. Mean standard deviation is mainly used in the statistic conclusion to measure confidence. Also used to find how set of data spread out.
Standard deviation (s) is the square root of its variance (s2), which is the average of the squared difference from the average of the mean. Mean (µ) is simple average value of given set of data.
For example just take different height of dogs, find out mean, variance, and standard deviation.
Height of dogs = 700mm, 570mm, 180mm, 530mm, 400mm
To find the mean
Mean = 700+570+180+530+400/5
= 2380/5 = 476
The mean or average height of the dog is 476mm.
Then to calculate the variance first subtracts the value of mean from the every height of the dog and then square the resulting values. Add the sum of squared values and divide with total number of dogs, resulting value gives the variance.
Variance = (700-476)2 + (570-476)2+ (180-476)2 + (530-476)2 + (400-476)2 / 5
  = 50176+8836+87616+2916+5776 / 5
= 155320/5 = 31064
The square root of the variance gives the standard deviation for the height of the dogs.
Standard deviation (s) = vvariance
= v31064
= 176.24 mm


Finding the standard deviation for population and sample, following formulas are used,
Standard deviation for population (s) = v(1/n ?_(i=1)^n¦(xi- µ)2)
Standard deviation for sample (s) = v(1/(n-1) ?_(i=1)^n¦(xi- " " )2)
Where,
µ,   - mean
When we have n value of data, if we are calculating variance, we should divide by n for the population and divided by n-1 for a sample. Standard deviation is used to measure the investment volatility, in finance. It is also called as historical volatility.

Mean and Standard Deviation
Mean and standard deviation is mainly used to find the center of the data set. Mean is defined as; it is the simple average value of the given data set and is represented by a symbol of Greek letter (µ).
Mean for population (µ) = 1/n ?_(i=0)^(n-1)¦xi
Mean for sample ( ) = 1/n ?_(i=1)^n¦x

Where,
n - Size of the sample or number of item in the sample
x, xi - Observed value or set of value

Finding Standard Deviation
Finding standard deviation the following steps should be followed.
First calculate the mean of given set of data by sum of given data divided by total number of data.
Then subtract the mean from each observed value.
Square the each difference and then add all the squared values to get their total sum. The resulting value divided by one less then the number of data in the data set.
The resulting value gives the variance.
Finally standard deviation can get from square root of the variance.

Find Standard Deviation
Find standard deviation for the list of numbers, 1, 3, 4, 6, 9, 8
Mean (µ) = 1+3+4+6+9+8/6
= 5.16
Variance (s2) = (1-5.16)2 + (3-5.16)2 + (4-5.16)2 + (6-5.16)2 + (9-5.16)2 + (8-5.16)2  / ( 6 – 1)
= 17.31+4.66+1.35+0.71+8.07+14.75 / 5
= 46.85 / 5 = 9.37
Variance (s2) is 9.37
We know that standard deviation is the square root of the variance
So the standard deviation (s) = vs2
= v9.37
s = 7.81