Geometry (polygon)

Polygons

A polygon is a closed figure where the sides are all line segments. Each side must intersect exactly two others sides but only at their endpoints. The sides must be noncollinear and have a common endpoint.

A polygon is usually named after how many sides it has, a polygon with n-sides is called a n-gon. E.g. the building which houses United States Department of Defense is called pentagon since it has 5 sides.

Kindle Unlimited Membership Plans

Types of Polygons

Regular or Irregular

A regular polygon has all angles equal and all sides equal, otherwise it is irregular

Regular		Irregular

Concave or Convex

A convex polygon has no angles pointing inwards. More precisely, no internal angle can be more than 180°.

If any internal angle is greater than 180° then the polygon is concave. (Think: concave has a "cave" in it)

Convex		Concave

Simple or Complex

A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself! Many rules about polygons don't work when it is complex.

Simple Polygon (this one's a Pentagon)		Complex Polygon (also a Pentagon)

Interior angles of polygon

An Interior Angle is an angle inside a shape

Sum of Interior Angles = (n-2) × 180°

If a polygon is regular (i.e. all of its sides are of equal length and all of its angles are of equal size), then the following rule is true:

Each Interior Angle of a Regular Polygon = (n-2) × 180° / n

Where 'n' refers to the number of sides.

Exterior angles of polygons

The exterior angle of a polygon is an angle between any side of a shape and a line extended from the next side.

When you add the exterior angle of a polygon and its corresponding and exterior angle on a straight line, it amounts to 180°.

As a rule;

The sum of the exterior angles of a simple polygon add up to 360°

This is referred to as a 'full revolution'.

The exterior angle of a polygon and its corresponding interior angle always add up to 180°. This is due to the fact that the two corresponding angles create a straight line:

Exterior angle of a regular polygon = 360° ÷ (number of sides)

This is due to the fact that the exterior angles of a regular polygon always add up to 360°.

Tables of polygon and their angles

Algebra in real world

Algebra in daily life

1. Key to solve daily algebraic problems as I discussed earlier.
2. It is the prerequisite for advanced training.
3. Widely used in scientific numericals.
4. Used in every day life.
5. While entering data for any observations.
6. Used in Banking field.
7. Used to calculate Interest rates, when you lend money to someone or borrow from someone.
8. In playing games, (to calculate score in volleyball, football etc.)
9. While doing shopping, to track how much money you have wasted.
10. In keeping the record of your balance and mobile data. 
11. Used in Advanced engineering.
12. In maintaining the boundaries of time in your daily schedule of busy life.

Algebra is everywhere in our everyday lives. It is impossible to step outside or inside our houses, let alone go to work every day, without encountering the practical effects of algebra.


Have you ever taken out a loan? If so, you had to pay interest on the loan. The formulas used to calculate that interest are built using the language of algebra. Businesses have to finance day-to-day operations as well as long term investments, like building a new factory or plant, and they do complex algebraic calculations to decide the lowest cost method of financing.

Are you saving money? If so, you are expecting a return on your money, and it will no doubt be compounded. This is calculated using the rules of algebra. Businesses do the same thing when they manage their investment portfolios.

Have you ever landscaped your backyard or done home improvements? If so, you had to figure out how much area to cover (with paint, with grass, or with weed inhibitor, for example) and how much it would cost for different alternatives. The calculations to cost out the alternatives and figure out how much you need to purchase are done using algebraic rules.

Do you need to make sales projections for next year and the year after? Almost certainly you’ll start out by drawing a line through your sales levels for the past few years and project that into the future. Once again, algebraic rules (in this case, for developing linear equations) will be the ticket.

Have you ever had to adjust a cooking recipe for a different number of people than the recipe was written for? You need to proportionally change the ingredients to match the needed servings. Restaurants have the same challenge. In both cases, algebra is used do the conversions.

Have you ever traveled abroad? If so, you probably had to convert some of your US dollars to the local currency of the country you were traveling in. Companies have to do something very similar if they have any foreign locations for sales or manufacturing or if they purchase any supplies from abroad or sell to any customers in other countries.

BODMAS rule

It gives us an order in which to perform the mathematical operations in a given problem or equation. It is actually an acronym and goes as follows

This is the order in which you perform the operations to arrive at the correct answer. You first solve the operations within the bracket. This is called “opening the bracket’. Then you move on to division, after which you perform multiplication if the equation requires it from left to right. And finally, you add the numbers and lastly perform the subtraction.

The best way to understand the BODMAS rule is through examples.

Given expression is,

24 + 13 – 5 × ½ of 10 – {45 ÷ (17 – 2)} =?

⇒ ? = 24 + 13 – 5 × ½ of 10 – {45 ÷ (15)}

⇒ ? = 24 + 13 – 5 × ½ of 10 – {3}

⇒ ? = 24 + 13 – 5 × (½ × 10) – 3      (of means ‘×’)

⇒ ? = 24 + 13 – 5 × 5 – 3

⇒ ? = 24 + 13 – 25 – 3

⇒ ? = 9

Hence, the required answer is 9.

240 ÷ 8 × 512 ÷ 4 + ½ of {1800 ÷ (11 × 24 ÷ 8 × 3 – 69)2} = ?

According to the BODMAS, first we need to solve the Brackets.

Here in this equation first we will solve curly bracket and inside the bracket, we will again follow BODMAS Rule. In the curly bracket once again another bracket appears. So we solve that bracket first. Within these smaller brackets, there is no ‘of’ or ‘order’, so we proceed with the following steps in BODMAS.

We see multiplication, division and subtraction signs. Since multiplication and division are of the same rank, we go left to right to solve this. First we multiply, then we divide. Then we perform the second multiplication. After that, we proceed to the subtraction.

Here are the steps to illuminate this process.

⇒ 240 ÷ 8 × 512 ÷ 4 + ½ of {1800 ÷ (264 ÷ 8 × 3 – 69)2} = ?

⇒ 240 ÷ 8 × 512 ÷ 4 + ½ of {1800 ÷ (33 × 3 – 69)2} = ?

⇒ 240 ÷ 8 × 512 ÷ 4 + ½ of {1800 ÷ (99 – 69)2} = ?

⇒ 240 ÷ 8 × 512 ÷ 4 + ½ of {1800 ÷ 302} = ?

Now we have reached the end of the small brackets. We encounter our first exponent. So we need to solve this ahead of division.

Now solving curly bracket,

⇒ 240 ÷ 8 × 512 ÷ 4 + ½ of {1800 ÷ 900} = ?

⇒ 240 ÷ 8 × 512 ÷ 4 + ½ of 2 = ?

So far we have cleared all the brackets. Now we move to the next step. We come across our first ‘of’. Here, we simply treat ‘of’ as product i.e. multiplication. But note that, ‘of’ will be solved before a regular multiplication.

⇒ 240 ÷ 8 × 512 ÷ 4 + ½ × 2 = ?

⇒ 240 ÷ 8 × 512 ÷ 4 + 1 = ?

Now, we have cleared our expression of all brackets, exponents and ‘of’s. We now move to the 3rd step. Here we perform all the divisions and multiplications. Note that these two operations are the same rank. So we perform either in the order we come across them in, from left to right.

⇒ 30 × 512 ÷ 4 + 1 = ?

⇒ 15360 ÷ 4 + 1 = ?

⇒ 3840 + 1 = ?

Now we have cleared the expression of all multiplication and division operations as well. All we are left with is addition and subtraction.

⇒ ? = 3841

Hence, the required answer is 3841.

Operating rational numbers

ADDITION

Addition of rational numbers with same denominators: Consider two rational numbers 2/9 and 3/9, here both have different numerators but same denominators. To add these, we add both the numerators keeping common denominator.

Addition of rational numbers with different denominators:

To add rational numbers with different denominators convert them to rational numbers with same denominators and then add.

In order to convert rational numbers to numbers with same denominators take the LCM of the denominators. Then multiply both rational numbers by this LCM and we will get equivalent rational numbers with this LCM as the denominator.

Example 1: Add 7/6 and 2/9

Solution: To evaluate7/6+2/9

• Convert this to rational numbers with same denominators.
• LCM of 6 and 9 is 18
• Multiply 18 by both rational numbers
• Add 21/18 and 4/18

Additive Inverse Opposite of a number is the number with same magnitude of opposite sign. -1 is the opposite of 1. The sum of a number and its opposite is always zero.  This is same for rational numbers also. Two rational numbers are additive inverse of each other when their sum is zero. Thus, all opposite numbers are additive inverse to each other.

Subtraction of rational number

Subtraction of rational numbers with same and/or different denominators can be done like addition. Subtraction of rational numbers will be easy by transforming subtraction problem into addition problem. This can be done in two steps;

• Change the subtraction sign (-) into addition sign (+)
• Take the additive inverse of the rational number come after the sign.

Simply, take the additive inverse of the number after the sign and add them.

Example 2: Subtract  5/2 and -4/7

Solution: We have to subtract 5/2 and -4/7

For this, first convert subtraction problem into addition problem

Take  the additive inverse of the number after the sign. Here, -4/7  is the number after the sign.

Additive inverse of -4/7 is 4/7. Now add 5/2+4/7.

Here, LCM of 2 and 7 are 14.

Add 35/14  and 8/14

Multiplication of rational number

Multiplication of rational numbers is as simple as multiplication of integers. In general, product of two rational numbers will equal to the product of the numerators divided by the product of the denominators.

To find the product of a rational number and an integer, keep the denominator unchanged and multiply the numerators. To find the product of rational numbers, multiply the numerators and multiply denominators and write in the form given above.

Example 3: Multiply 3/8 and -9/11

Solution: To find the product, multiply the numerators and multiply denominators

Example 4: Divide 11/12 and 7

Solution:To find the product, multiply the numerators and keep the denominator unchanged.

We know, 7=1/7

Division of rational number

Divisions of rational numbers are similar to the divisions of fractions. To find the quotient of rational numbers, we need to recollect the concept of reciprocal of a rational number. Reciprocal of a rational number is nothing but the swapping of the numerator and denominator of that rational number i.e. invert the given the rational number will give you the reciprocal of that rational number. For example, 7/6 is the reciprocal of 6/7 . The product of a number and its reciprocal will always be 1.

In order to divide rational number, first change the division sign  to . Then take the reciprocal of the number after the sign and multiply.

Example 5: Divide 11/12 and  7/4

Solution: To evaluate 11/12÷7/4

• Change the division sign ÷ to x.
• Take the reciprocal of 7/4 which is equal to 4/7.
• Multiply

Mensuration formulas of 2D and 3D shapes

1. Mensuration Formulas for RECTANGLE

• Area of Rectangle = Length × Breadth.
• Perimeter of a Rectangle = 2 × (Length + Breadth)
• Length of the Diagonal = √(Length2+ Breadth2)

2. Mensuration Formulas for SQUARE

• Area of a Square = Length × Length = (Length)2
• Perimeter of a square = 4 × Length
• Length of the Diagonal = √2 × Length

3. Mensuration Formulas for PARALLELOGRAM

• Area of a Parallelogram = Length × Height
• Perimeter of a Parallelogram = 2 × (Length + Breadth)

4. Mensuration Formulas for TRIANGLE

• Area of a triangle=(1/2)(Base × Height)=(1/2)(BC×AD)

For a triangle with sides measuring a, b and c, respectively:

• Perimeter = a + b + c
• s = semi perimeter = perimeter/2 =  (a+b+c)/2
• Area of Triangle, A=
  (This is also known as “Heron’s formula”)
  
• Area of isosceles triangle =
  (Where a = length of two equal side, b = length of base of isosceles triangle.)
  
• Area of an equilateral triangle =
  (Where, a is the side of an equilateral triangle)

5. Mensuration Formulas for TRAPEZIUM

• Area of a trapezium = (1/2) × (sum of parallel sides) × (distance between parallel sides)
  = (1/2) × (AB+DC) × AE
• Perimeter of a Trapezium = Sum of All Sides

6. Mensuration Formulas for RHOMBUS

• Area of a rhombus=(1/2)×Product of diagonals

• Perimeter of a rhombus = 4 × l
  (where l = length of a side)

7. Mensuration Formulas for CIRCLE and SEMICIRCLE

In the following formulae, r = radius and d = diameter of the circle

• Area of a circle = πr2= (πd2)/4
• Circumference of a circle = 2πr = πd
• Circumference of a semicircle = πr
• Perimeter of the semicircle = (πr+d)
• Area of semicircle =(πr2)/2
  
• Length of an arc = (2πrθ)/360, where θ is the central angle in degrees.
• Area of a sector = (1/2) × (length of arc) × r = (πr2θ)/360

Mensuration Formulas for 3D Figures

8. Mensuration Formulas for CUBOID

In the following formulae, l = length, b = breadth and h = height

• Total surface area of cuboid = 2 (lb + bh + lh)
• Length of diagonal of cuboid= √(l2+b2+h2)
• Volume of cuboid = l × b × h

9. Mensuration Formulas for CUBE

In the following formulae, a = side of a cube

• Volume of cube = a3
• Total surface area of cube = 6a2
• Length of Leading Diagonal of Cube = a√3

10. Mensuration Formulas for CONE

In the following formulae, r = radius of base, l = slant height of cone and h = height of the cone (perpendicular to base)

• Slant height of a cone = l =√(h2+r2 )
• Curved surface area of a cone = C = π × r × l
• Total surface area of a cone = π × r × (r + l)
• Volume of right circular cone =1/3 πr2h

11. Mensuration Formulas for CYLINDER

In the following formulae, r = radius of base, h = height of cylinder

• Curved surface area of a cylinder = 2πrh
• Total surface area of a cylinder = 2πr(r + h)
• Volume of a cylinder = πr2h

12. Mensuration Formulas for SPHERE

In the following formulae, r = radius of sphere, d = diameter of sphere

• Surface area of a sphere = 4πr2 = πd2
• Volume of a sphere = (4/3) πr3 = (1/6)πd3

13. Mensuration Formulas for HEMISPHERE

In the following formulae, r = radius of sphere

• Volume of a hemisphere =(2/3)πr3
• Curved surface area of a hemisphere = 2πr2
• Total surface area of a hemisphere = 3πr2

14. Mensuration Formulas for HOLLOW CYLINDER

Hollow cylinder made by cutting a smaller cylinder of same height and orientation out of a bigger cylinder.

• Volume of hollow cylinder = πh(R2– r2)
  (Where, R = radius of cylinder, r = radius of cavity, h = height of cylinder)

15. Mensuration Formulas for FRUSTUM OF A RIGHT CIRCULAR CONE

Frustum is created when a plane cuts a cone parallel to its base.
In the following formulae, R = radius of the base of the frustum, r = radius of the top of the frustum,
h = height of the frustum, l = slant height of the frustum

• If a cone is cut by a plane parallel to the base of the cone, the lower part is called the frustum of the cone.
• Slant height of the frustum =l=√(h2+(R-r)2)
• Curved surface area of frustum = π(R + r)l
• Total surface area of frustum = π(R + r)l + π(R2 + r2)
• Volume of the frustum=(1/3)πh(R2+r2+Rr)

16. Mensuration Formulas for PRISM

• Prism consists of two polygonal bases which are parallel to each other.
• These bases are joined by lateral faces, which are perpendicular to the polygonal bases.
• The number of lateral faces is equal to the number of sides in the polygonal base. Thus, the base of a prism could be of various shapes, namely, triangular, quadrangular, pentagonal etc.
• Volume of prism = Base area × height
• Lateral surface area of prism = perimeter of base × height
• Total surface area of prism = Lateral surface area + (2 × base area)

17. Mensuration Formula for PYRAMID

• Pyramid consists of a polygonal base and triangles at its sides. These triangles are called faces.
• The base could be of any shape, whereas the faces are generally isosceles triangles.
• All these triangular faces meet in a single point called the apex.
• Total surface area of pyramid = base area + (number of sides × ½ × slant height × base length)
• Volume of pyramid = (1/3) × base area × height