Category: Maths Notes PPT

The term Arithmetic Addition in mathematics refers to adding two or more numbers together for forming the addition table and the mathematics symbol or operator, which denotes that the addition is the + plus sign, for example – 8 + 4 = 12.  

The term Sum also denotes the same operation of addition. We can use the summation symbol “∑” for denoting the addition of a series of numbers.

Understanding the Addition Table

 

Now, let us have a better understanding of using the addition table. Suppose you want to add 5 and 3, that is, 5 + 3. All you have to do is go down to the row, which has the number “5” written on it and then along the column, which has the number “3” written on it to get your answer, which is 8. As we know that the mathematical operation of arithmetic addition is commutative, you can also go down the row, which has the number “3” written on it, and then along the column, which has the number “5” written on it to get your answer, which is again 8 (5 + 3 = 3 + 5).

Explanation of the Addition Table Chart

 

 

 

 

 

 

 

 

 

 

Easy and Quick Tips to Keep in Mind for Performing Addition

1. If we add 0 to a number, then it makes no difference, for example – 7 + 0 = 7, 6 + 0 = 6, 89 + 0 = 0.

2. If we add a group of numbers together, the answer will be the same, irrespective of their order or sequence, for example – 8 + 7 + 3 + 2 = 20 and 7 + 2 + 3 + 8 = 20. < /span>

3. For performing the mathematical operation of addition on a long list of numbers, we arrange the numbers in a column and then obtain the result easily. 

4. If we add 1 to an even number, then the result is an added number, for example – 4 + 1 = 5, 8 + 1 = 9. 

5. If we add two even numbers, then the result is also an even number, for example – 8 + 2 = 10, 6 + 4 = 10.

6. If we add 1 to a number, then the result is the next number after the number in the question, for example – 7 + 1 = 8, 5 + 1 = 6.

Basic Mathematics is a part of one’s daily life, and because of that, the introduction of Mathematics also takes place with real-life examples. Everyone has come across the age-old question “if you have 6 candies and give your friend 2, how many candies are left with you?”. While with the advancement of classes, Maths becomes more about theorems and formulas, a part of it is always derived from practical experiences. The section, algebra deals with such real-world problems and algebra linear equations applications is one of the essential topics under it. 

Linear equations real-world problems let a student formulate their equations to solve them. Every sum will require a new approach, though it is very possible to categorise certain situations under one type of equation. To understand this type of equation, one first needs to know different aspects of it. 

Linear Equations

A linear equation, according to Mathematics, is an equation of first-order that is defined for lines in a coordinate system. 

The most common form of a linear equation is,

y= ax + b

Where a is the slope of the line and b is the y-intercept. 

This can also help one to understand the relation between two variables that is y and x. Different forms of this equation can be used to solve different types of linear algebra real-world problems. Let’s solve a real-world problem to understand how one can formulate a linear equation and solve it. 

Q1: Raj went to buy flowers pots in his local nursery. The shopkeeper said that there are two types of pots, terracotta and plastic. He also said that the price of a terracotta pot is three times the price of a plastic pot. If Raj can buy one of each at 100 rupees, what is the price of each pot?

Ans: First, understand the relation between the two costs. 

Now, let’s assume that the cost of a plastic pot is x, then the cost of a terracotta pot will be three times x that id 3x. 

It is stated that one of each pot can be bought using 100 rupees, that means

x + 3x = 100, this is the linear equation for this given problem.

Now solve it for x, that is

4x = 100

x = 100/4

x = 20

So, the price of a plastic pot is 25 rupees. Since a terracotta pot costs three times, its price will be 3 × 25 = 75 rupees. 

Under such circumstances where one only needs to find one variable, solving word problems with linear equations are usually like the example above. But if one needs to find the value of more than one variable, a system of linear equations is used. 

Applying Systems of Linear Equations

Linear system or system of linear equations is a set of one or more linear equations that involve a common set of variables. For example,

3x + 2y = 3

-2x + y = 4

These two equations are together a system of linear equations. Algebra linear equations applications are not much different from that of solving for a single variable, one has to formulate their own equations here as well. This can be better understood by practising real-life linear equations word problems. Let’s solve one such problem to see the course of equation formulation. 

Q2: The cost of passes for a theme park is 162 rupees for 12 children and 3 adults. For the same theme park, the cost of passes for 8 children and 3 adults was 122 rupees. What is the cost of one pass for each child and adult?

Ans: From the question, it is clear that these two situations will translate into two equations. Now, put your understanding into these two situations and form equations just the way it was done above. System of equations real-world problems can have multiple variables; in this case, there are two children and adults.

Let the cost of a child’s pass be x, and the cost of an adult’s pass be y. So, according to the first situation,

12x + 3y = 162

And according to the second situation 

8x + 3y = 122

Now, subtract the first equation from the second one, like

12x + 3y = 162

 – 8x + 3y =122

    4x         = 40

x = 40/4

x = 10

Put the value of x in any of the equations to y’s value, like

8(10) + 3y = 122

80 + 3y = 122

3y = 42

y = 14

So, the cost of a child’s pass is 10 rupees and 14 rupees for an adult’s pass. 

Algebra linear equations applications simplify relations between two or some times more variables. Yet, there are several aspects that a student should understand before solving its questions. provides live coaching classes on a variety of mathematical topics including linear equations and its different applications. Check their sample sheets to practice more real-world problems and master the skill of equation formulation. 

Analytics Geometry of Two and Three Dimensions

Analytic Geometry is an invention of Descartes and Fermat. It is also referred to as coordinate geometry. It is a mathematical subject where algebraic symbolism and methods are used to represent and solve geometry problems.

Analytic geometry basically establishes a connection between geometric curves and algebraic equations. It is often referred to as a branch of algebra that is used to model geometric objects. It helps in the reformulation of algebraic problems in geometry and vice versa, where the methods and principles of both fields can be used to solve the problems in the other.

Analytical geometry of three dimensions forms the foundation of various modern fields of geometry that include algebraic, differential, discrete, and computational geometry. The application of Analytic geometry spreads across a wide range of fields including physics and engineering, and also in aviation, rocketry, space science, and spaceflight.

Trigonometry and analytic geometry encompass some very important subjects such as midpoints and distance, parallel and perpendicular lines on the coordinate plane, dividing line segments, distance between the line and a point, etc. This field is often deemed as the next level of mathematics, that helps in the field of learning logical thinking and improving problem-solving skills of an individual.
In this article, we will discuss the most important aspects of analytic geometry including formulas, cartesian plane, analytic geometry in three dimensions, its applications, and some solved sums for a better and more in depth understanding.

What is Analytic Geometry?

In analytic geometry, points are defined as ordered pairs of numbers, for instance, (x, y), and the straight lines are consequently defined as the sets of points that satisfy linear equations. 

Analytic geometry defines geometric axioms as derivable theorems. 

Here is an example, for any two distinct points, say (x1, y1) and (x2, y2), there is a single line ax + by + c = 0 that passes through these points. The coefficients (a, b, c) can be found (up to a constant factor) from a linear system of two equations:

ax1 + by1 + c = 0

ax2 + by2 + c = 0,

or directly from the determinant equation. It is a contradiction to symmetric geometry, where no candidates or formulas are used.

Both two dimensional and three-dimensional geometry are used in coordinate geometry. The most common terms used in analytic geometry are:

Planes

For a clear understanding of analytical geometry of three dimensions or two, and the importance and applications of analytic geometry, we need to be clear about the definition of a plane. A plane is a flat surface that continues infinitely in both directions. Analytic Geometry helps locate any point on this plane with the coordinates of the point in the X and Y plane.

Coordinates

Two ordered pairs which define the location of any given point in a plane can be defined as coordinates. The box below will help us get a better understanding.

In this grid, the location of letter x is B2 i.e. Column B and row 2. So, B and 2 are the coordinates of this box, x.

Analytical Geometry of three dimensions has different kinds of coordinates. The most important ones are:

Cartesian Coordinates

The Cartesian coordinate is the most well-known coordinate system in which every point has an x-coordinate and y-coordinate which represents its horizontal position, and vertical position respectively.

Polar Coordinates

A coordinate system where each point in a plane is denoted by the distance ‘r’ from the origin and the angle θ from the polar axis is known as the polar coordinates.

Cylindrical Coordinates

In cylindrical coordinates, all the points are expressed by their height, radius from z-axis, and the angle projected on the xy-plane with respect to the horizontal axis. The height, radius, and angle are denoted by h, r, and θ, respectively.

Spherical Coordinates

In this type of coordinates the point in space is denoted by its distance from the origin ( ρ), the angle projected on the xy-plane with respect to the horizontal axis (θ) and also another angle, with respect to the z-axis (φ).

Did you Know?

1. The origin is the point of intersection of the axis(x-axis and y-axis). Both x and y-axis are zero at this point.

2. x-axis – The values at the right-hand side of the x-axis are positive and ones on the left-hand side are negative.

3. y-axis – The values above the origin are positive and the ones below the origin are negative.

4. You need two numbers to locate a plane in the order of writing the location of the X-axis first and Y-axis after that. This denotes a unique position on the plane.

Analytic Geometry Formulas

To find measurements of geometric figures, formulas for analytic geometry are extremely useful. Graphs and coordinates are the vast fields of application of analytic geometry. Application of analytic geometry in the engineering field and sciences involves the study of the rate of change in varying quantities and helps to show the relationship between the quantities involved. Check out the analytic geometry formulas here.

Distance Formula

Let the two points be A and B, having coordinates to be (x1,y1) and (x2,y2) respectively.

Thus, the distance between two points is given as-

d = √[(x2-x1)2+(y2-y1)2]

Midpoint Theorem Formula

Let A and B are some points in a plane, which is joined to form a line, having coordinates (x1,y1) and (x2,y2), respectively. Suppose, M(x,y) is the midpoint of the line connecting the point A and B then its formula is given by;

M(x,y) = [(x1+x2/2),(y1+y2/2)]

Angle Formula

Let two lines have slope = m1 and m2, and θ be the angle formed between the two lines, A and B. This is represented as;

tan θ = m1-m2/1+m1m2

Solved Examples

How to find the distance between two points A and B such that the coordinates of A and B are (5, -3) and (2, 1)?

Given that, the coordinates are:

• A = (5, -3)  = (x1, y1)

• B = (2, 1) = (x2,y2)

The formula to find the distance between two points is given as:

• Distance,d = √[(x2-x1)2+(y2-y1)2]

• d = √[(2-5)2+(1- (-3))2]

• d =√[(-3)2+(4)2]

• d =√[9+16]

• d =√(25)

• d = 5

Thus, the distance between two points A and B is 5.

In simple mathematics, a polygon can be defined as any 2-dimensional shape that is formed with straight lines. In the case of quadrilaterals or triangles and pentagons, they are all perfect examples of polygons. The interesting aspect is that the name of any kind of polygon highlights the number of sides the polygon possesses.

 

A pentagon has 5 straight sides and the shape must also be closed (all the lines should connect to each other):

 

Types of Pentagon

1. Regular Polygon

2. Irregular Polygon

3. Concave polygon

4. Convex Polygon

 

The General Rule

Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total sum.

 

What is a Regular Polygon?

If all the sides of a polygon are equal and all the angles of a polygon are equal then the polygon is known as a regular polygon.

 

Different Shapes, Number of Sides, Sum of Interior Angles and the Measure of each Angle-

 

 

Properties of a Regular Pentagon

A regular pentagon has the following properties:

• Interior angles that measure 108°

• Exterior angles that measure 72°

• A regular pentagon has an area of approximately 1.7204774 × s2 (where s is equal to the side length)

Any pentagon has the following properties:

 

Sum of Angles in a Pentagon

 

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To find the sum of the angles in a pentagon, divide the pentagon into different triangles. There are three angles in a triangle. Since the sum of the angles of the triangles is equal to 180 degrees.

 

3 x 180 = 540 degrees

Therefore, the sum of angles in a pentagon is 540 degrees.

 

Now to find the measure of the interior angles of the pentagon, we know that the sum of all the angles in a pentagon is equal to 540 degrees (from the above figure)and there are five angles. (540/5 = 108 degrees)

 

So, the measure of the interior angle of a regular pentagon is equal to 108 degrees.

 

How to Measure the Central Angles of a Regular Pentagon?

 

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To find the measure of the central angle of a regular pentagon,we need to make a circle in the middle of the pentagon.We know that a circle is 360 degrees around. Now ,divide that by five angles. Now the measure of each central angle is equal to 360/5 = 72 degrees.

 

So, the measure of the central angle of a regular pentagon is equal to 72 degrees.

 

Pentagon with Right Angle

• A pentagon has five sides.

• A regular pentagon has no right angles ( It has interior angles each equal to 108 degrees).

• An irregular pentagon has at most three right angles because a fourth would leave 180 degrees to be used for the final angle that is (540 degrees – 360 degrees), which is a straight line.

• For a pentagon to have any right angles, however, the pentagon has to be uneven, since the sum of all interior angles of a pentagon always sums up to 540 degrees.

 

Convex  Pentagon and Concave Pentagon

 

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If all the vertices of a pentagon are pointing outwards, then the pentagon can be known as a convex pentagon. If a pentagon has at least one vertex pointing inside, then the pentagon can be known as a concave pentagon.

 

Questions to be Solved-

Question 1) Is the Diagram Given Below a Pentagon?

 

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Answer: The figure given below cannot be known as a pentagon because from the properties of a pentagon we know that it should be a closed figure. But the figure given above is open therefore, it is not a pentagon.

The Pentagon and its Subtypes

A pentagon is a two-dimensional shape with 5 sides and 5 angles in geometry. An angle is produced in a pentagon when two of its sides share a common point. Because there are five vertices in a pentagon, there are five angles in a pentagon. In this article, we will go through the angles in a pentagon in-depth, including internal angles, exterior angles, the sum of angles in a pentagon, and so on, with numerous examples.

A pentagon is a two-dimensional closed polygon having five sides and five angles. A pentagon may be categorized into several varieties according to its qualities. They are as follows:

Regular Pentagon: A pentagon with equal sides and interior angles.

Irregular Pentagon: A pentagon’s sides are not all equal, and the inner angles are not all the same size.

Convex Pentagon: All of the interior angles are less than 180 degrees, and all of the vertices point outwards. A convex pentagon is a regular pentagon.

Concave Pentagon: A concave polygon is formed when one of the inner angles of a pentagon is larger than 180° and one of the vertices points inward.

The diagrams below show the definitions of a regular pentagon, an irregular pentagon, and a concave pentagon.

The sum of a Pentagon’s outside angles

We know that the formula for calculating the sum of a polygon’s inner angles is (n – 2) 180°.

As a result, each interior angle = {(n – 2) 180°}/n.

Each exterior angle is known to be supplementary to the inner angle.

Thus, each outside angle = {180°n -180°n + 360°}/n = 360°/n may be calculated using the preceding method.

As a result, the sum of a polygon’s exterior angles = n(360°/n).

Because a pentagon has five sides, n=5.

Interior View of a Standard Pentagon

The internal angles of a pentagon are the angles created by two consecutive pairs of sides.

The number of sides equals the number of vertices plus the number of internal angles, which equals five.

Neighboring angles or adjacent interior angles are two internal angles that share a common side.

All five sides of a regular pentagon are equal, as are all five angles. As a result, the formula below gives the measurement of each internal angle of a regular pentagon.

Each internal angle is measured as (n – 2) 180°/n = 540°/5 = 108°.

n = the number of sides in this case.

A Regular Pentagon’s Exterior Angle

When the sides of a pentagon are stretched, the angles produced outside the pentagon with its sides are called exterior angles. A pentagon’s outside angles are all equal to 72°.

Because the total of a regular pentagon’s outside angles equals 360°, the formula for calculating each exterior angle of a regular pentagon is as follows:

Each exterior angle of a pentagon is measured as 360°/n = 360°/5 = 72°.

Trigonometry is a branch of mathematics that studies the relationship between the ratios of a right-angled triangle’s sides and its angles. Trigonometric ratios, such as sine, cosine, tangent, cotangent, secant, and cosecant, are used to study these relationships.

Trigonometry is the most important concept in Mathematics. It plays an important role in almost all the fields, whether it’s Aviation, Physics, Criminology, Military, etc. Trigonometry is used for finding the angles or the sides of the triangle.

Basics of Trigonometry 

The measurement of angles and problems involving angles are covered in the basics of trigonometry. Trigonometry has three basic functions: sine, cosine, and tangent. Other essential trigonometric functions can be derived using these three basic ratios. These functions form the foundation for all of the fundamental topics in trigonometry. In conclusion, the six main functions of an angle that are commonly used in trigonometry are as follows:

• sine (sin), 

• cosine (cos), 

• tangent (tan), 

• cotangent (cot), 

• secant (sec), and 

• cosecant (csc).

Applications of Trigonometry

Trigonometry has been used in a variety of fields throughout history, including architecture, theoretical physics, and surveying. It can be used for a variety of things, including:

• Oceanography, seismology, meteorology, physical sciences, astronomy, acoustics, navigation, electronics, and many other subjects are among them.

• It may also be used to determine the length of lengthy rivers, measure the height of a mountain, and so on.

• Solar, lunar, and stellar locations have all been calculated using spherical trigonometry.

Real-Life Applications of Trigonometry

Many real-life examples are used frequently in trigonometry. Let’s use an example to better understand trigonometry. A young boy stands beside a tree. “How tall is the tree?” he wonders as he looks up at it. The tree’s height can be determined without having to measure it. A right-angled triangle, or one with one of the angles equal to 90 degrees, is what we have here. If the distance between the tree and the boy, as well as the angle produced when the tree is viewed from the ground, trigonometric formulas can be used to compute the tree’s height.

Other Applications include:

• Trigonometry can be used to measure the height of a building or mountains

• Trigonometry is used video games

• Trigonometry is used in construction

• Trigonometry is used in flight engineering

• Trigonometry is used in physics

• Archeologists use trigonometry

• Trigonometry is used in criminology

• Trigonometry is used in marine engineering

• Trigonometry is used used in navigation

Conclusion

Trigonometry is the branch of mathematics that studies the relationship between angles and sides of a triangle. It is possible to calculate the heights of large mountains or structures using trigonometry; it is also used in astronomy to calculate the distance between stars or planets, and it is frequently used in physics, architecture, and GPS navigation systems. 

Introduction of Area and Perimeter 

In our everyday life, we come across a lot of objects of varying shapes and sizes, square, triangle, rectangle, sphere, circle, rhombus, and many others. Area and Perimeter give us valuable knowledge which helps us deal with these different forms of objects.

What are Area and Perimeter? 

So, what is the meaning of Perimeter and Area?

Let us see how we can define Perimeter and Area in the mathematical sense before we head to solving problems. Before getting to the trickier part of the topic, we must strengthen our conceptual understanding of Area and Perimeter –

Area – Area helps in determining the boundaries of a two-dimensional object or a plain. The term, “area” can be defined as the enclosed space within the frames of an object. This space is known as the Area of that particular shape. 

Perimeter- The sum total of the distance covered along the outer part of the shape gives us the Perimeter. Again, dimension plays a major role in this too as depending on the dimensions of the shapes, the Perimeter of different shapes can be equal to each other.  

How Can We Calculate Area and Perimeter? 

Now that we are clear on the area and perimeter meaning in math, let us see how we can calculate the Perimeter and Area of some of the most common shapes we encounter on a daily basis. 

Area and Perimeter of a Square –

A square’s all four sides are equal in length and are at an identical angle if 90 degrees. Therefore, to measure the Area covered by the square and it’s Perimeter, these following are the formulae to be used –

Area – a2

Perimeter – 4a

Let’s take a as the length of the 4 sides.

 

Area and Perimeter of A Rectangle – 

A rectangle is an elongated figure with its opposite sides matching in length and all the angles at an identical 90 degrees.

Rectangle’s Area – a×b

Rectangle’s Perimeter – 2(a+b)

Let a and b be the length and breadth respectively.

 

Area and Perimeter of the Triangle –

A triangle is made up of 3 sides. The Perimeter of a triangle is calculated by adding up the lengths of all 3 sides of the triangle.

Triangle’s Area- ½ b×h, let b be the base and h be the height.

Triangle’s Perimeter – a +b + c where all three represent the 3 sides of a triangle.

Area and Circumference of a Circle –

The circle ‘s Perimeter is known as the circumference.

Circumference = 2πr

Area of Circle = πr2 

Area And Perimeter Chart

The following chart contains all the formulas used to calculate the Area and Perimeter of different shapes in tabular data to make it easier for you to go through.

 

Here are some solved perimeter and area class 5 questions with answers for you to understand better –

Q. If the given radius of a circle is 21cm. Calculate its circumference and Area. 

Solution:

Given, radius = 11cm

Therefore, Area = π × r2

A = 22/7 × 11 × 11

A = 380.28 sq.cm.

Circumference, C = 2πr

C = 2 x 22/7 x 11 = 69.14 cm

Example 2:

Q. If the given square’s side’s length is 4 cm. Then measure its Area and also calculate the Perimeter.

Solution:

We know, length of the side, a = 4 cm

Area = a2 = 42 = 16 sq.cm

Total length of its boundary, Perimeter = 4a = 4 x 4 = 16 sq.cm.

Example 3:

Q. A given rectangular field measures 20m in length and 30m in breadth. Calculate the Area and Perimeter.

Solution:

Given, Length = 20m

Width = 30m

Since, Area = length x breadth = 20 x 30 = 600 sq.m.

Perimeter = 2 (length + breadth) = 2 x (20 +30) = 2 x 50 = 100 m. 

Did You Know – 

• Squares and rectangles always cover more Area, and the Perimeter always tends to be lesser than the Area covered but in triangles, the Perimeter is more than the Area covered.