Category: Maths Notes PPT

A frustum can be constructed from a circular cone in a right angle. It has a cone tip formed by cutting height which is perpendicular, an upper base and lower base. These bases in a derivation of frustum are both circular and parallel in structure.

When considering a right cone in a circular shape, the problem can be comprehensive to other n-sided pyramids and cones. Let’s take an example to understand the structure of a frustum and its application. This will further help to gain an understanding of the function of the volume of a frustum. 

In the above figure, h represents height while r is the radius of an upper base, and R is the radius. One has to apply a frustum formula to derive the volume here.

When a combination is formed by taking solids, one has to add volumes of two adjacent shapes. This will give the required volume of a structure or volume of frustum of a cone. In the case of a frustum, a cone will be cut into smaller cone ends. To find the value, one needs to subtract this separated part.

This segment explains this concept of finding volume and surface area of frustum with examples and theory.

Let’s check what the volume of frustum formula is and how it works.

How to Apply Volume of Frustum of Cone Formula? 

The sliced part of a cone can be termed as a frustum. Therefore, the calculation of volume requires finding two circular cone’s volume differences. 

From the figure above, the total height becomes H’ = H+h 

Here the total slant height becomes L =l1 +l2

We know that the radius of the cone is C and the radius of the cut cone is r. 

This makes the volume of the total cone to be 1/3 π C2 H’ which is equal to 1/3 π C2 (H+h)

The volume of the tip of a cone here will be 1/3 πr2h. Now to find the volume of a frustum, one has to calculate the dissimilarity between two circular cones in a right angle.

This gives 1/3 π C2 H’ -1/3 πr2h

Following the rules, it gives us 1/3π C2 (H+h) -1/3 πr2h

We find that 1/3 π [ C2 (H+h)-r2 h ].

After seeing the cone, a student can assess the sliced part, which gives us a result that right angle of the whole cone Δ BAD is similar to that of sliced cone Δ BPQ. 

This gives us, C/ r = H+h / h.

That is  H+h = Ch/r. Replacing the value of H+h in the frustum of a cone formula we get

1/3 π [ C2 (Ch/r)-r2 h ] =1/3 π  [C3h/r-r2 h]

 1/3 π h (C3/r-r2 )  =1/3 π h (C3-r3 / r)  

Similar Property of Triangles to Find Derivation of Volume of Frustum 

If we use a similar diagram and properties, we can evaluate the value of h, C/ r to be equal to (H+h)/ h.

We have seen that here h is [r/(C-r)] H 

Replacing the value of h in this equation gives us the solution 1/3 πH [r/(C-r)][(C3-r3)/ r)]

 Now we get 1/3 πH [(C3-r3)/(R-r)]

 Which gives us 1/πH [(C-r)(C2 +Cr+r2)/ (C-r) ]

 Finally, the value as 1/πH (C2 +Cr+r2)

 Consequently, the V or the conical frustum volume will be 1/3 πH (C2 +Cr+r2 ).

How to Find Total Surface Area and Curved Surface Area in a Volume Truncated Cone  

In the figure above one can find the curved surface area of the frustum of a cone to be π(C+r)l1 

Here the total surface area of the frustum of a cone will be π l1 (C+r) +πC2 +πr2

We take the slant height to be l1 in both surface area of a cone. This gives us √ [H2 +(C-r)2

The resemblance of triangles equations characteristics has been calculated using two Δ BAD and Δ BPQ. 

Therefore, students need to procure information on all formulas of frustum to solve equations confidently. If they practice from quality study materials, they will be accustomed to the surface area, the volume of a frustum measurement, the volume of truncated cone derivation and more.

One can check to gather quality guidance on the topic volume of a frustum of a pyramid and related topics. They offer a wide array of notes, practice papers, live classes, and more. Student’s desiring to secure outstanding scores in the subject maths can download the app today!

Twin primes are a pair of prime numbers that have a difference of 2 between them. For example, 3 and 5, 41 and 43 are two common pairs of twin prime numbers.

Alternatively, you can also define twin prime numbers having a prime gap of 2. Now, what is meant by the prime gap?

Let’s study it in detail!

Twin prime numbers form the basis of mathematics, this concept is taught very early in schools and students should get an in-depth knowledge of twin prime numbers as these can help students critically understand the language of numbers and can help them get a good score in the examination.

Twin prime numbers can be defined as a set of two numbers that have only one composite number between them. Another definition of twin prime numbers is – the pair of numbers with a difference of two are also called twin prime numbers. The term twin prime was coined by Stackle in 1916. If we put it in a simple manner, twin prime numbers are numbers where two numbers have a difference of two.

The first few twin primes are n+/-1 for n=4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, etc. Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19) etc.

Properties of Twin Primes

• We have already studied that twin prime numbers are the pairs of prime numbers that have a difference of two. Here are some of the properties of twin primes-

• 5 is the only prime digit which has positive and negative prime differences of 2. Hence, 5 has two prime pairs – (3,5) and (5,7).

• Both 2 and 3 are not a pair of twin prime numbers as there is no composite number between them.

• The basic form in which prime numbers are represented is {6n-1, 6n+1}, 3 and 5 being an exception.

• Prime pairs when added, their sum is always divisible by 12, 3 and 5 being an exception.

Twin Prime Conjecture

Twin prime conjecture is also called Polignac’s conjecture in terms of number theory. The definition of twin prime conjecture states that there are infinite twin prime pairs with a difference of two. The conjecture states that a positive even number m, has infinite pairs of two consecutive prime numbers with difference n. The twin prime conjecture basically says that infinite twin primes exist.

This means as the numbers get larger, primes start to get less frequent and therefore twin primes get rarer. Polignac’s conjecture is named after Alphonse de Polignac in 1849 as he introduced it. Alphonse discovered that any even number can be expressed in infinite ways as the difference between two consecutive primes. It is also known as Euclid’s twin prime conjecture.

Concepts related to twin primes that will help to get a better understanding of the concept are as follows-

Properties of Twin Primes

Prime Gap

The prime gap is nothing but the difference between two consecutive prime numbers. In mathematical form, it can be expressed as:

Prime gap = prime number + 1 – prime number

What are Twin Primes Properties?

The only prime digit having both positive and negative prime differences of two is 5. Therefore, it occurs in two prime pairs – (3, 5) and (5, 7).

As no composite number exists between 2 and 3, these two successive digits are not a pair of twin prime numbers.

All other pairs of twin prime numbers stay in the form of {6n – 1, 6n + 1}, except the pair (3, 5).

If you add two numbers of a prime pair, the result will be divisible by 12. Again, for this case, the pair (3, 5) is an exception.

What is Twin Prime Number Conjecture?

Alphonse de Polignac, a French mathematician, introduced the first statement of twin prime conjecture in 1846. He stated that any even numeral could be illustrated in an infinite number of ways. An example of the same is the difference between two prime numbers (13 – 11 = 5 – 3 = 2).

This theory is sometimes referred to as Euclid’s twin prime conjecture, but it was proved that an infinite number of primes might exist. However, this is not the case for twin prime numbers.

First Hardy-Littlewood Conjecture

This conjecture is named after two English mathematicians, namely G. H Hardy and John Littlewood. Involved in prime constellation distribution, which includes twin primes, this conjecture generalises the twin conjecture.

Consider π2(x) to be the number of prime digits provided p is lesser than equals to x, such that p + 2 also gives a prime number. Therefore, the constant of twin prime C2 can be represented as the following –

C2 = [pi left ( 1-left ( frac{1}{left ( p-1 right )^{2}} right ) right )]

Here, p is a prime number and greater than equals to 3.

The approximate answer is 0.660161815846869573927812110014….

Furthermore, the unique of the initial Hardy-Littlewood conjecture can be illustrated as:

π₂(x)～2C₂ x(lnx)2

～2C₂∫x2

dt(lnt)2

Note: The integer ‘2’ is the one and only even prime number.

As Polignac’s conjecture stated that there are many twin prime pairs with a difference of 2, but Yitang Zhang proved that there is an infinite number of prime pairs, which holds a gap of not less than 70 million.

Various other Prime Types

Cousin Prime

When the difference between two prime numbers is 4, they are termed as cousin primes.

Prime Triplet

The set of prime triplets contains three numbers such that the largest and smallest number has a difference of 6. Two exceptions are (2, 3, 5) and (3, 5, 7).

Solved Numerical on Twin Primes

Problem 1: What are Twin Primes Between 1 and 100?

Solution. The twin prime pairs between 1 and 100 are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61) and (71, 73).

Do it Yourself

Problem 2: Find out whether the following numbers are the addition of twin prime numbers. (a) 36 (b) 120 (c) 84 (d) 144

Solution. You can use the formula p + (p + 2) to find out whether the above-mentioned numbers are twin primes. Take a look!

(a) 36

p + (p + 2) = 36

2p + 2 = 36

2p = 36 – 2

P = [mathrm{frac{34}{2}}]

P = 17

So, substituting the value in (p + 2) we get 17 + 2 = 19.

(b) 120

p + (p + 2)

2p + 2 = 120

P = [mathrm{frac{120-2}{2}}]

P = 59

Again, substituting the value in (p + 2) we get 59 + 2 = 61.

(c) 84

p + (p + 2) = 84

2p + 2 = 84

p = [mathrm{frac{84-2}{2}}]

p = 41

Puttin
g the value of p in p + 2 we get 41 + 2 = 43

(d) 144

p + (p + 2) = 144

2p + 2 = 144

p = [mathrm{frac{144-2}{2}}]

p = 71

Putting the value of p in p + 2, we get 71 + 2 = 73

Conclusion:

By going through the information provided above, you must have been able to understand what twin primes are. You can also refer to solved questions for practice.

In Mathematics, sets are defined as the collection of objects whose elements are fixed and can not be changed. In other words, a set is well defined as the collection of data that does not carry from person to person. The elements can not be repeated in the set but can be written in any order. The set is represented by capital letters.

The empty set, finite set, equivalent set, subset, universal set, superset, and infinite set are some types of set. Each type of set has its own importance during calculations. Basically, in our day-to-day life, sets are used to represent bulk data and collection of data. So, here in this article, we are going to learn and discuss the universal set.

What is Set, What are Types of Sets, and Their Symbols?

A set is well defined as the collection of data that does not carry from person to person.

1. Empty Sets 

The set, which has no elements, is also called a null set or void set. It is denoted by {}.

Below are the two examples of the empty set.

Example of empty set: Let set A = {a: a is the number of students studying in Class 6th and Class 7th}. As we all know, a student cannot learn in two classes, therefore set A is an empty set.

Another example of an empty set is  set B = {a: 1 < a < 2, a is a natural number}, we know a natural number cannot be a decimal, therefore set B is a null set or empty set.

2. Singleton Sets

The set which has just one element is named a singleton set.

For example,Set A = { 8 } is a singleton set.

3. Finite and Infinite Sets

A set that has a finite number of elements is known as a finite set, whereas the set whose elements can’t be estimated, but has some figure or number, which is large to precise in a set, is known as infinite set.

For example, set A = {3,4,5,6,7} is a finite set, as it has a finite number of elements.

Set C = {number of cows in India} is an infinite set, there is an approximate number of cows in India, but the actual number of cows cannot be expressed, as the numbers could be very large and counting all cows is not possible.

4. Equal Sets

If every element of set A is also the elements of set B and if every element of set B is also the elements of set A, then sets A and B are called equal sets. It means set A and set B have equivalent elements and that we can denote it as:

A = B

For example, let A = {3,4,5,6} and B = {6,5,4,3}, then A = B

And if A = {set of even numbers} and B = { set of natural numbers} then A ≠ B, because natural numbers consist of all the positive integers starting from 1, 2, 3, 4, 5 to infinity, but even numbers start with 2, 4, 6, 8, and so on.

5. Subsets

A set S is said to be a subset of set T if the elements of set S belong to set T, or you can say each element of set S is present in set T. Subset of a set is denoted by the symbol (⊂) and written as S ⊂ T.

We can also write the subset notation as:

S ⊂ T if p ∊ S ⇒ p ∊ T

According to the equation given above, “S is a subset of T only if ‘p’ is an element of S as well as an element of T.” Each set is a subset of its own set, and a void set or empty set is a subset of all sets.

6. Power Sets

The set of all subsets is known as power sets. We know the empty set is a subset of all sets, and each set is a subset of itself. Taking an example of set X = {2,3}. From the above-given statements, we can write,

{} is a subset of {2,3}

{2} is a subset of {2,3}

{3} is a subset of {2,3}

{2,3} is also a subset of {2,3}

Therefore, power set of X = {2,3},

P(X) = {{},{2},{3},{2,3}}

7. Universal Sets

A set that contains all the elements of other sets is called a universal set. Generally, it is represented as ‘U.’

For example, set A = {1,2,3}, set B = {3,4,5,6}, and C = {5,6,7,8,9}.

Then, we will write the universal set as, U = {1,2,3,4,5,6,7,8,9,}.

Note: According to the definition of the universal set, we can say that all the sets are subsets of the universal set. 

Therefore,

A ⊂ U

B ⊂ U

And C ⊂ U

8. Disjoint Sets

If two sets X and Y do not have any common elements, and their intersection results in zero(0), then set X and Y are called disjoint sets. It can be represented as;, X ∩ Y = 0.

Union, Intersection, Difference, and Complement of Sets

1. Union of Sets

The union of two sets consists of all their elements. It is denoted by (⋃).

For example, set A = {2,3,7} and set B = { 4,5,8}.

Then the union of set A and set B will be:

 A ⋃ B = {2,3,7,4,5,8}

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2. Intersection of Sets

The set of all elements, which are common to all the given sets, gives an intersection of sets. It is denoted by ⋂.

For example, set A = {2,3,7} and set B = {2,4,9}.

So, A ⋂ B = {2}

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3. Difference of Sets

The difference between set S and set T is such that it has only those elements which are in the set S and not in the set T. S – T = {p : p ∊ S and p ∉ T}

Similarly, T – S = {p: p ∊ T and p ∉ S}.

4. Complement of a Set

Let U be the universal set and let A ⊂ U. Then, the complement of A, denoted by A’ or (U – A), is defined as 

A’ = {x   U : x A}

X A’ x A

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Every set has a complement of sets. Also, for a universal set, the empty set is known as the complement of the universal set. The empty set contains no elements of the subset and is also known as null set, which is denoted by {Ø} or {}.

Questions to be Solved

Question 1. If set A = {a, b, c, d} and B = {b, c, e, f} then, find A-B.

Answer: Let’s find the difference of the two sets,

A – B = {a, d} and B – A = {e, f}

Question 2. Let X = {David, Jhon, Misha} be the set of students of Class XI, who are in the school hockey team. Let Y = {Zoya, Rahul, Riya} be the set of students from Class XI who are in the school football team. Find X U Y and interpret the set.

Answer:  (U union – combination of two sets)

Given X = {David, Jhon, Zoya}

           Y = {Zoya, Rahul, Riya}

Common elements (Zoya) should be taken once

                      X U Y = {David, Jhon, Zoya, Rahul, Riya}.

This union set is equal to the set of students from Class XI who are present in the hockey team or the football team or both of the teams.

A sector of a circle is a closed figure bounded by an arc of a circle and two of its radii. Each sector has a unique central (sector) angle that it subtends at the centre of the circle. Minor sectors subtend angles less than 180° while major sectors subtend angles more than 180°. The semicircular sector subtends an angle of 180°. The following diagram shows a minor sector of a circle of radius r units whose central angle is θ.

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Firstly, it is important to realise that a circle is a perfectly symmetric planar figure. It has infinite lines of symmetry passing through its centre. This means that all sectors of the same circle or of congruent circles, which have the same central angles are congruent sectors. We will use this fact to derive the formulas for the perimeter and the area of a sector. Consider the following diagram which shows two adjacent congruent sectors of a circle.

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Sectors OAB and OCB have the same central angle (θ), hence their arc lengths AB and BC are equal (l), as well as their areas. Together they form a bigger sector OAC. Clearly the central angle of sector OAC is 2θ, while its arc length AC is 2l . Therefore, we realise that the arc length of a sector is directly proportional to its central angle. The same applies for the area of the sector as well.

So let’s consider a sector of a circle of radius r and central angle θ. We know that for a central angle of 360°, the sector is actually the complete circle. So we can say that for a central angle of 360°, the sector’s arc length is equal to the perimeter of the circle and its area is equal to the area of the circle. Thus, using the concept of direct proportions, we arrive at the following results.

[frac{l}{theta } = frac{2pi r}{360^{circ}}]

⇒ l = [frac{theta }{360^{circ}}times 2pi r]

[frac{area}{theta } = frac{pi r^{2}}{360^{circ}}]

⇒ area = [frac{theta }{360^{circ}}times pi r^{2}]

The term [frac{theta }{360^{circ}}] can be thought of as the fraction of the total central angle of the circle (360°) covered by the sector. We can even relate the area of the sector to its arc length by using the above two formulas to obtain a simple formula for the area, as shown below.

area = [frac{theta }{360^{circ}}times pi r^{2} = frac{1}{2}times left ( frac{theta }{360^{circ}}times 2pi r right )times r = frac{1}{2}times lr]

To find the perimeter of the sector, we add the lengths of the two radii to the arc length.

perimeter = l + 2r 

Let’s look at an example to see how to use these formulas.

Solved Examples

Question: Find the perimeter and the area of a sector of a circle of radius 35 cm whose central angle is 72°.

Solution:

r = 35cm, θ = 72°

l= [frac{theta }{360^{circ}}times 2pi r = l= frac{72^{circ}}{360^{circ}}times 2 times frac{22}{7}times 35 = 44cm]

perimeter = l + 2r = 2 × 35 + 44 = 114cm

area = [frac{1}{2}times lr=frac{1}{2}times 44times 35=770cm^{2}]

Why don’t you try solving the following problem to see if you have mastered using these formulas?

Question: Find the central angle of a sector of a circle whose area and perimeter are respectively equal to 231cm2 and 61cm if it is given that its radius is a whole number.

Options:

(a) 105°

(b) 120°

(c) 135°

(d) none of these

Answer: (c)

Solution:

area = 231cm², perimeter = 61cm

perimeter = 2r + l =61

area= [frac{1}{2}lr=231Rightarrow lr =462]

Solving these two equations, we get the following results.

(61 – 2r) × r = 462

⇒ 61r – 2r² = 462

⇒ 2r² – 61r + 462 = 0

⇒ 2r² – 28r – 35r + 462 = 0

⇒ (2r – 33) (r – 14) = 0

⇒ r = 16.5m, 14cm

Since r is given to be a whole number, r = 14cm

Substituting in the area formula, we have the following equations.

[frac{theta }{360^{circ}}times frac{22}{7}times 196= 231cm^{2}]

[theta =231times 360^{circ}times frac{7}{22times 142}=135^{circ}]

Square root is one of the most important functions in Mathematics which has a wide range of applications in day to day life and also scientific calculations. Square root of any number in Mathematics is that number which when multiplied by itself gives the product equal to the number whose square root is to be determined. Square root of a number is represented as the number written within the symbol ‘√’. Square root of a number ‘x’ is written as √x. Square root of a number can be represented in exponential form as the number to the power ½. Square root of a number ‘x’ can be written in exponential form as (x)1/2.

What is a Perfect Square Number and Square Root Formula?

It is very important to understand what is a square root of a perfect square number before making yourself clear about what is a root in math. What is a perfect square number in Mathematics can be that number which is obtained as a product by multiplying any integer by itself. The square root formula when used for perfect square numbers will yield a number which is an integer as the answer. i.e. Square root of a perfect square number is always an integer.

What is a Root in Math? 

There are several methods to find the square root of a number among which a few familiar ones are:

1. Prime factorization method

2. Repeated Subtraction Method

3. Average Method

4. Guess and check method

5. Number line method

6. Long division method

Finding Square Root Formula by Prime Factorization Method

Prime factorization method is a method in which the numbers are expressed as a product of their prime factors. The identical prime factors are paired and the product of one element from each pair gives the square root of the number. This method can also be used to find whether a number is a perfect square or not. However, this method cannot be used to find the square root of decimal numbers which are not perfect squares.

Example: Evaluate the root of 576.

Solution: 

576 is factorized into its prime factors as follows.

So, 576 can be written as a product of prime numbers as:

576 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3

Square root of 576 = 2 x 2 x 2 x 3 = 24

Square Root Formula Using Repeated Subtraction Method

This is a method in which the number whose square root is to be determined is repeatedly subtracted by the consecutive odd number till the difference becomes zero. The number of subtractions gives the root of the number.This method can only be used to find the square root of perfect square numbers.

Example: Estimate the Square root of 16

Solution:

The number is subtracted from odd numbers starting from 1.

16 – 1 = 15

15 – 3 = 12

12 – 5 = 7

7 – 7 = 0

Number of subtractions here is 4. So, the square root of 16 is 4.

Average Method of Square Root Formula:

In this method, the concept of average is used to find the square root of a given decimal number. This method can be conveniently used to find the square root of whole numbers upto a few decimal places.

Example: Evaluate the square root of 3 using the average method. 

Solution:

The two square numbers in between which ;3’ lies are 1 and 4. So, the square root of 3 lies between 1 and 2. Find the average of these two numbers to get the square root of 3.

Square root of 3 = (1 + 2)/ 2 = 3/ 2 = 1.5 which is not accurate. So, finding the average is further continued as

Square root of 3 = (1.5 + 2)/2 = 1.75 which is approximately equal to square root of 3. 

Fun Facts About Square Root Formula:

• Square and square root operations are inverse mathematical operations with respect to each other.

• Square root of a square of a number is the number itself.

• The square of square root of a number is the number itself.

Conclusion

The article has presented complete insight about the Square Root formula that will help students to practice and learn.

What is Polygon?

A polygon is a simple closed curve. A two-dimensional closed figure bounded with three or more than three straight lines is called a polygon. Triangles, square, rectangle, pentagon, hexagon, are some examples of polygons.

The segments are referred to as the sides of the polygon. The points at which the segments meet are called vertices. Segments that share a vertex are called adjacent sides. A segment whose endpoints are nonadjacent vertices is called a diagonal. See the picture below.

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The below figures show some of the examples of polygons or polygonal curves( a closed curve that is not a polygon).

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In figure you can see that all the shapes are polygons, as all the shapes are drawn joining the straight lines only. There are no curved lines..

But in figure the shape is not a polygon because it is not fully connected and also has curved lines and such shapes that have curves are called a closed curve that is not a polygon.

In this article let us study what is polygon and regular polygon formulas.

Types of Polygons

As we know what is the meaning of polygon let us understand different types of polygons. They are as follows,

• Regular Polygon

• Irregular Polygon

• Convex Polygon

• Concave polygon

Regular Polygon

Polygons whose sides and angles are of equal lengths are called regular polygons.

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Irregular Polygon

Polygons with different sizes and different interior angles are called irregular polygons.

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Convex Polygon

Polygons with interior angles less than 1800 are called convex polygons.

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Concave Polygons

Polygons with interior angles greater than 1800 are called concave polygons.

Some Popular Polygons

Here is the list of some of the regular polygons with the number of polygon sides, shapes, and measures of its interior angles.

Regular Polygon Formulas 

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Some of the regular polygon formulas related to polygons are:

• Interior Angle: An angle inside the polygon at one of its vertices is called the interior angle. Angle Q is an interior angle of quadrilateral.

• Exterior Angle: An angle outside the polygon formed by one of its sides and the extension of an adjacent side is called the exterior angle of the quadrilateral.in the above quadrilateral angle ADZ, YUA, YUX, and QUX are the exterior angles.

• The sum of the measures of all the exterior angles of a polygon is 360°.

• Interior angle + corresponding exterior angle = 1800

• The sum of the measures of the interior angles of a polygon with n sides is given by the general formula  (n–2)180.

Sum of Interior angles of Polygon(IA) = (n-2) x 180

Exterior angle of a regular polygon(EA) = 360/n

Interior angle of a regular polygon = [frac{(n-2)180}{n}]

Diagonal of Polygon = [frac{n(n-3)}{2}]

Perimeter of Polygon(P) = n x s

Area of Polygon(A) = s/ 2 tan (180/n)

Solved Examples

Example 1:  A polygon is an octagon and its side length is 6 cm. Calculate its perimeter and value of one interior angle.

Solution:

The polygon is an octagon, so we have, n = 8

Length of one side, s = 6 cm

The perimeter of the octagon

P = n × s

P = 8 × 6

   =48 cm.

Now, for the interior angle, we have

Interior angle of a regular polygon = [frac{(n-2)180}{n}]

= ( 8 – 2) x 180/ 8

= 6 x 180 /8

= 1350

Therefore the perimeter of the octagon is 48cm and the value of one of the interior angles is 1350.

Example 2: Calculate the measure of one interior angle of a regular hexadecagon (16 sided polygon)?

Solution:

The polygon is an hexadecagon, so we have, n = 16

Interior angle of a regular polygon (IA) = [frac{(n-2)180}{n}]

IA = ( 16 – 2) x 180 / 16

= 14 x 180 /16

= 157.50

Therefore measure of interior angle of a regular hexagon is 157.50.

Example 3: Calculate the measure of 1 exterior angle of a regular pentagon?

Solution: the polygon is a pentagon , so we have n = 5

An exterior angle of a regular polygon (EA) = 360/n

= 360/ 5

= 720

Therefore the measure of an exterior angle of a regular pentagon is 720.

Quiz Time

1. What is the measure of 1 exterior angle of a regular decagon (10 sided polygon)?

2. If each exterior angle measures 10°, how many sides does this polygon have?