class 10 maths

Class 8 Maths: Chapter Wise Step-By-Step Solutions

We are having solution on Class 8 Maths. Make sure you read each and every chapter in detail so that you won’t miss anything.

In the upcoming chapters, we are presenting step-by-step solution for each chapter.Till you don’t cover your basics in your primary class you can’t succeed further  in your higher class,it same goes with the 8 maths. If we talk about maths, basics are very much important for this subject and these basics will build you in your higher classes i.e your 9th,10th boards.
So, here is the best solution for class 8th maths!
Let’s dive inside!

Class 8 Maths: Rational Numbers

So in this chapter you will learn all type of numbers i.e
1) Natural numbers
2) Whole numbers
3) Integers
4) Rational and Irrational numbers
5) Real numbers

1) Natural numbers – Simple counting numbers on the number line starting from one to infinity are called natural numbers.
Thus N = {1,2,3,4,5…….} is the set of natural numbers.

2) Whole numbers – Natural numbers starting with zero to infinity are called Whole numbers.
Thus W={0,1,2,3,4,5……..} is the set of whole numbers.

3) Integers Negative natural numbers to the positive natural numbers forms integers.

 Where set I or Z is used to denote integers.
Thus Z = {……..-3,-2,-1,0,1,2,3………}
Where Z+= {1,2,3,4,5….} = N, is the set of positive integers.
            Z= {-1,-2,-3,-4,-5,…..} is the set of negative integers.

i) Addition of integers: 

➡️ The sum of two positive numbers is a positive number obtained by adding the numerical.
➡  The sum of two negative integers is a negative integer obtained by adding the numerical value.
➡️ To find the difference between the numerical value we add a positive and negative integer and give the sign of the integer with greater numerical value to it.

ii) Multiplication of integers:
➡️  The multiplication of two positive integers or two negative integers is a positive integer.
➡️ The multiplication of one positive integer and one negative integer is a positive integer.

iii) Division of integers:

➡️ If a and b are integers,
Then a÷b is not necessarily an integer.

➡️ 0÷a=0,Where a≠0
➡️(a÷b)÷c≠a÷(b÷c),unless c=1
➡️a÷1=a

4) Rational numbers – The numbers that can be expressed in the fraction (p/q), Where p and q are integers and q≠0 are called rational numbers.

Properties of rational numbers
1)  Every integer is a rational number.
2)  Every terminating decimal is a rational number.
3)  Every fraction is a rational number.
4)  Every recurring decimal is a rational number.
5)  The square root of every perfect square is a rational number.
6) Rational number between two rational numbers x and y is (x+y)/2.
7) Reciprocal of any non-zero integers a is 1/a.

If x,y,z are rational numbers then

PropertyAdditionMultiplication
1. Commutativex + y = y + xx x y = y x x
2. Associative(x + y) + z = x + (y + z)x x (y x z) = (x x y) x z
3. Identityx + 0 = 0 + x = xx x 1 = 1 x x = x
4. Inversex + (-x) = 0x x (1/x)= 1  (x≠0)

Irrational numbers : Irrational means not rational.Number which can neither be expressed as a repeating decimal.

Properties of irrational number:

1) π is an irrational number.
2)  It is not compulsory that the sum of two irrational will be irrational.
3) The product,quotient and the difference of two irrational is not necessarily irrational.
4) If the product of two irrational numbers is rational then each one is called the Rationalising factor of the other.
5)Addition, subtraction, multiplication and division of two irrational numbers can be either a rational or irrational number.

5) Real numbers: All rationals and irrational together form real numbers.
There is an existence of infinitely many real numbers between any two real numbers.

Properties of order relation on Real numbers:
1. If p and q are two real numbers then only one of the relations holds good.

i.e. p = q or p < q or p > q

2. If p< q and q < r then p < r 

3. If p < q then p + r < q + r

4. If p < q and r > 0 then pr < qr and If r < 0 then pr > qr.

Surds (Irrational root)

If n is an integer greater than 1 and if a is a positive real number and nth root of a is x then it is written as xn = a or na = x

If a is a positive rational number and nth root of a is x and if x is an irrational number then x is called a surd.

In a surd, na the symbol is called radical sign, n is the Order of the surd and a is called radicand.

(1) Let a = 9, n = 5, then 59 is a surd because 59 is an irrational number.

(2) Let a = 125, n = 3 then 3125 is not a surd because 3125 = 5   is not an irrational number.

Comparison of surds

Let p and q are two positive real numbers and

If p < q then p × p < q × p

If p2< pq…(1) Similarly pq < q2 …(2)

∴ p2< q2 …[from (1) and (2)]

But if p[ > q then p2> q2 and if p = q then p2 = q2

hence if p < q then p2< q2

Here p and q both are real numbers so they may be rational numbers or surds.

Rationalize the denominator:

Rationalizing the denominator is the term where we have to multiply the denominator with the term itself in the numerator as well as in the denominator. 

Example:

Rationalize the denominator of 1/6

1/6= (1/6) x (6/6) = 6/6 …(multiply numerator and denominator by 6)

We can make use of rationalizing factors to rationalize the denominator.

It is easy to use the numbers with rational denominators, that is why we rationalize it.

Absolute value:

If p is a real number, then absolute value of p is its distance from zero on the number line

which is written as IpI, and IpI is read as Absolute Value of p.

If p > 0, then IpI = p If p is positive then the absolute value of p is p.

If p = 0, then IpI = 0 If p is zero then absolute value of p is 0.

If p < 0, then IpI = -p If p is negative then its absolute value is opposite of p.

Class 8 Maths: Linear Equations

Definition:

Equation of degree one containing only one variable is called a simple linear equation.
Mathematically  we can Express that equation as
a x + b = 0, Where a≠0.

Some examples of simple Linear Equations are as follows:
i. 5x-3=8-7x
ii. 3(a-7)=8.9
iii. 2q+5=(q/8)+9
iv. (y-1)/9+ (2y+9)/3=7/11

Rules for solving equations:

1. Same number can be added or subtracted to the both sides of an equation.

2. The same non zero number can be multiplied or divided to the both sides of an equation.

3. Any term of an equation can be taken to the other side with the sign change; this process is  called transposition.

It means that we can shift the variable from one side of the equation to the other side by changing its sign. If the equation has a positive sign then after shifting the variable from one side to the other the sign will be negative.

Let’s take an example:
In an equation 3x-5=2x+8 we will transpose 2x from right hand side to left hand side and -5 from left hand side to right hand side.Then +2x will became -2x and -5 will became +5 i.e

⇒ 3x-2x=8+5
⇒ x=13

Learn more about class 8 maths

Class 8 Maths: Types of Quadrilaterals

A Closed figure bounded by 4 line segments is called quadrilateral.

Let’s take a quadrilateral PQRS.
  A quadrilateral PQRS has:

I. Four vertices namely P,Q,R,S
II. Four sides namely PQ,QR,RS,SR
III. Four angles, namely ∠P,∠Q,∠R,∠S.
IV. Two diagonals, namely PR,QS.

Class 8 Maths Chapter 3
Adjacent sides(PQ, QR),(QR,RS),(RS,SP),(SP,PQ)
Opposite sides(PQ,SR),(PS,QR)
Adjacent angles(∠P,∠Q),(∠Q,∠R),(∠R,∠S),(ㄥS,ㄥP)
Opposite angles(∠P,∠R),(∠Q,∠S)

Properties of Quadrilateral:

1.
The sum of the angles of quadrilateral is 360.
Let’s take a quadrilateral 🗆ABCD

Quadrilateral ABCD

So according to the property,
ㄥA+ㄥB+ㄥC+ㄥD=360

2. Parallelogram:  If both the pairs of opposite sides of a quadrilateral are parallel then it is termed as parallelogram.

class 8 maths book

Properties of parallelogram:

Opposite sides are parallelAB II DC, AD II BC
Opposite sides are equalAB = DC , AD = BC
Opposite angles are equal∠A = ∠C, ∠B = ∠D
Adjacent angles are supplementaryㄥA+ㄥB=180,ㄥB+ㄥC=180,ㄥC+ㄥD=180,ㄥD+ㄥA=180
Diagonals bisect each otherAE = EC , DE = EB
Each diagonal bisects the parallelogram i.e  divides it into two congruent triangles
ABC≌ CDA and ABD≌CDB

3.Rectangle: Each angle measuring 90of a  parallelogram is called a rectangle.

Rectangle abcd

Properties of rectangle:

Opposite sides are equalAB = DC , AD = BC
Each angle measures 90.ㄥA=ㄥB=ㄥC=ㄥD=90
Diagonals are equalAC = BD
Diagonals bisect each otherAO=OC,OB=OD

4.Square: Rectangle having all sides equal is called a square.

Properties of square:
Let’s assume a square ◻️ PQRS

1. The first property of square is all sides are equal i.e  if we assume square ◻️PQRS then,
PQ=QR=RS=SP.
2. Second property is each angle measures 90.i.e ∠P=∠Q=∠R=∠S=90.
3. The third property is that the diagonals are equal i.e PO = OR and QO = OS.
4. And the last property is diagonals intersect at right angles, i.e. PR QS.

5.Rhombus:  A parallelogram having all sides equal is called a rhombus.

Properties of Rhombus:

Let’s assume a rhombus ◻️ABCD

1. Opposite sides are parallel i.e. AB II DC , AD II BC
2. All sides are equal,i.e AB=BC=CD=DA
3. Diagonals bisect each other at right angles i.e AO=OC,BO=OD and AC BD.
4. Diagonals bisect the angles of the Rhombus i.e AC bisects ∠A and ∠C; BD bisects ∠B and ∠D.

6.Trapezium: Quadrilateral in which one pair of opposite sides is parallel is termed as trapezium

Properties of trapezium:
Let’s assume a Trapezium ABCD

1.One pair of opposite sides is parallel i.e. AB II DC

2.Two pairs of adjacent angles are supplementary i.e ∠A+∠D=180, ∠B+∠C=180

Isosceles trapezium:  If two non parallel sides of a Trapezium are equal it is termed as isosceles trapezium

Properties of isosceles trapezium:

1.One pair of opposite side is parallel

2.Two pairs of adjacent angles are supplementary

3. Angles on the same base are equal

4. Diagonals are equal in length

7. Kite

Kite: In kite, two pairs of adjacent sides are equal.

Properties of kite:
Let’s assume a kite ◻️ABCD

1.Two pairs of adjacent sides are equal i.e AB = AD, BC = DC

2.One pair of opposite angles is equal i.e ∠B=∠D

3.Diagonals intersect each other at right angles i.e, AC 丄 BD

4.One of the diagonals bisected by the other i.e OB = OD

5.One of the diagonals bisect two opposite angles of the kite i.e AC bisects ∠A and ∠C.

6.One of the diagonals bisect the kite in such a way that it divides it into two congruent triangles   i.e Diagonal AC bisects the kite and so ΔABC≅ ΔADC.

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Class 8 Maths: Data handling

Which chapter is all about tabulation of raw data. Frequency Tally. Frequency distribution and column graph based on Frequency distribution.
Data obtained in original form is raw data.
In this chapter we introduce a new topic that is median and mode.

Data:  Information in the form of a numerical figure is termed as data.

Example:  the marks obtained by 10 students of class in a monthly test are:21,26, 30, 17, 41, 12, 26, 45, 30, 31.
From the above example we get a clear figure of some number of marks related to 10  students.

Statistics: Statistics is the science which deals with collection, presentation, analysis and interpretation of some numerical data.

Raw data:Data obtained in original form is raw data.

Array: The arrangement of numerical figures of a data in ascending or descending order is termed array.

Observation: Each numerical figure in a data is termed as observation.

Frequency: A particular observation which occurs a number of times in a given data  is termed as frequency.

Class interval: Each group into which raw data is condensed  is called class-interval.

Class mark: (Upper limit +  lower limit)/2

Mean:
When the frequency distribution is given in the form of classes, we take x as a class mark and the mean using the formula

Mean = (Sum of observations)/(number of observations)= Σ (fi.xi)/Σ fi

Range: The difference between the maximum value and the minimum value of the observation is called range.

Mode: The observation which occurs maximum number of times is termed as mode.

Median: After arranging the data in ascending and descending order of magnitudes, the value of the middle term is called the median.

Square and Square Root

In this chapter we will learn about square and square root where square is a term where number is multiplied by the original number.

Square: The square of a number is that number raised to the power 2.

Examples:
1. Square of 8= 8²=8 x 8=64.

2. Square of (9/7)=(9/7)²=(9/7) x (9/7)=8149

3. Square of 0.2=0.2²=0.2 x 0.2=0.04 

Perfect square: Natural number is called perfect square.
Examples
we have 1²= 1, 2²=4 , 3²=9, 4²=16
Therefore each of the numbers 1, 4, 9, 16 etc is a perfect square.

Some properties of squares of a numbers:-

1. Square of an even number is always an even number.

Example 

2 is even and two square is 4 which is even.

4 is even and four square is 16 which is even.

2. The square of an odd number is always an odd number.

Example

3 is odd and 3 square is 9 which is odd.  5 is 9 and 5 square is 25 which is odd.

3. The square of an proper fraction is a proper fraction less than the given fraction.

Example
1/2 Is a proper fraction  1/2  Square is  1/4 Which is less than  1/2

4. The square of a decimal fraction less than one is smaller than the given decimal.

5. A number ending in 2,3,7 or 8 is never a perfect square.

Example
The number 72, 243, 576 and 1098 end in 2,3,7 and 8 respectively.
So none of them is a perfect square.

6. A number ending in an odd number of zeros is never a perfect square.

Example
The numbers 690  87000 and 4900000 end in one zero,three zeros and five zeros respectively. So, none of them is a perfect square.

 Square root: The square root of a number x is that number which when multiplied by itself gives x as the product we denote the square root of a number x by x.

Example:  7 x 7 equal to 49,  so  49=7 i.e  the square root of 49 is 7.

Methods of finding the square root of numbers

1. How to find the square root of a given perfect square number using prime factorization method?

Step 1. solve the given number and find out its prime factors.

Step 2. If you see similar factors then make a pair of those similar factors.

Step 3. When we choose one out of every pair the product of prime factors give the square root of the given number.

Note: A given number is a perfect square, if it can be expressed as the product of pairs of equal factors.

2. How to find the square root of a given number by Division Method?
Step 1. Mark of the digits in pairs starting with the units digit.each pair and remaining one digit is called period.

 Step 2. think of the largest number whose square is equal to or less than the first period.Take this number as the divisor as well as quotient.

Step 3. Subtract the product of divisor and quotient from the first period and bring down the next period to the right of the reminder. This becomes the new dividend.

Step 4. Now, a new divisor is obtained by taking twice the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.

Step 5. Repeat steps 2, 3 and 4 till all the periods have been taken up. Now, the question to obtain is the required square root of the given number.

Square root of a number in a decimal form:
Make the number of decimal places even, by affixing a zero, if necessary. Now, mark periods (starting from the right most digit) And find out the square root by the long division method. Put the decimal point in the square root as soon as the integral part is exhausted.

Here are some square and square roots of a number starting from 1 to 30.

class 8 maths squareroot

Cube and Cube Roots

Cube:The cube of a number is raised to the power of 3.
Examples:
1.  Cube of 9=9³=9 x 9 x 9=729.
2. Cube of (7/5)= (7/5)³= (7/5) x (7/5) x (7/5)=343125.
3. Cube of 0.8=(0.8)³=0.8 x 0.8 x 0.8=0.512

Perfect cube: A natural number is said to be a perfect cube, if it is the cube of some natural number.
Example:
The cube of 1 is equal to 1.
Cube of 2 is equal to 8.
Cube of 3 is equal to 27.
Cube of 4 is equal to 64.

Cube root: The cube root of a number x is that number which when multiplied by itself three times gives  x as the product.
Example:
5 x 5 x 5= 125, so 3125=5, i.e The cube root of 125 is 5.

Method of finding the cube root of numbers:
How to find out cube root of a given number by prime factorization method:

Step 1:  Find out the prime factors of the given number.
Step 2:  Make a group of three of a similar factor.
Step 3: The product of prime factors chosen one out of every triplet, gives the cube root of the given number.

Important points:

For any positive real numbers a and b we have:
I. √(ab)= √a x √b
II. 3√ab = 3√a x 3√b
III. √(a/b) = √a/√b
IV. 3√(a/b) = 3√a/3√b

So,here are cubes and cube roots from 1 to 30.

class 8 maths solutions

Class 8 Maths: Comparing Quantities

Ratio: The ratio of two quantities of the same kind and in the same units is the fraction that one quantity is of the other.

Ratio and their Properties:
1. The ratio of a to b is the fraction a/b,written as a : b.
2. In a:b,  we call a as the first term and b as the second term.
3. The ratio a:b  is said to be in simplest form,  if HCF  of a and b is one.
4. If a:b :: c:d, we say that  a,b,c,d are in proportion and we write , a : b : : c : d.
5. In a : b : : c : d, we call b and c as minterms and d and a as extreme terms.
6. Product of  means =  product of extremes.
7.  If a : b : : c : x, then x is called the fourth proportional to a,b,c.
8. If a : b : : b : c, then c is called the third proportional to a and b.
9. If a : b : : b : c, then b is called the mean proportional to a and c.
10. Mean proportional between a and c = ac

Unitary method
1. A method in which the value of a unit quantity is first obtained to find the value of any other required quantity.

2.If two quantities are so related to each other  that an increase (or decrease) in the first causes an increase (or decrease) in the second then the two quantities are said to vary directly.

3.Indirect variation The ratio of one kind of like terms is equal to the ratio of second kind of like terms.

4.If two quantities are so related  to each other that an increase (or decrease) in the first causes a decrease (or increase) in the other, then the two quantities are said to vary indirectly.

5.In Indirect variation in the ratio of one kind of like terms is equal to the inverse ratio of second kind of like terms.

While solving the problems based on direct variation:

When the value of a certain quantity of a variable is given then first we divide the given value by the given quantity to find the value of 1 unit. Now, to find the value of a certain quantity, we multiply the quantity with the value of 1 unit.

While solving problems based on indirect variation:

When the value of a certain quantity of a variable is given then first we multiply the given value by the given quantity to find the value of 1 unit. Now, to find the value of a certain quantity we divide the value of one quantity by this quantity.

Time and work:

We saw how to solve the problems based on ratio and proportion and unitary method now we will see how to solve problems based on time and work:

General rules while solving the problems on time and work:

1. A person’s 1 day’s work = 1/Total number of days he required to complete the work.
2. Total number of days required to finish the work = 1/1 day’s work.
3. Number of days required to complete the work = Remaining part of work/A’s one day’s work .
4. If a, b and c  work together, then the money paid to them is divided among them in the ratio of their 1 day’s work.
5. If the rate of working A and B is a:b,then the ratio of times taken by them to finish it is b:a.

Time and distance

Now in this topic we will say what is speed, distance and time

 Here are some formulas related to speed distance and time

Speed=Distance/Time
Distance=Speed x Time
Time=Distance/Speed
Average speed=Total distance traveled / Total time taken

Now you speed is expressed in terms of kilometre per hour (km/hr)

 Distance is expressed in terms of kilometre(kms)

 and time is expressed in terms of hours(hrs)

Rule 1. To convert a speed of kilometre per hour are into metre per second multiply by 5/18

Rule 2. To convert a speed of metre per second into kilometre per hour multiply by 18/5.

Now from rule 3 we will see how to solve the problems:
Rule 3: Time taken by a train X metre long in passing a stationary object= time taken by it to cover X metres.

Rule 4. Time taken by X metre long train to pass a platform of Y metre long tunnel=  time taken by it to cover (X+Y) meters.

Rule 5. Two moving objects are running in the same direction at X and Y  kilometre per hour then their relative speed is (X-Y) kilometre per hour.

Rule 6. Two objects that are x and y are moving in opposite directions then their relative speed is (x+y)  kilometre per hour.

Rule 7.  while solving the sums related to speed of a boat in a steel water where the speed of a Boat is X kilometre per hour and the rate of stream is Y kilometre per hour then
speed of boat downstream = (X+Y)km/hr speed of boat upstream = (X-Y)km/hr.

Percentage:
A percentage is a number or ratio that represents a fraction of 100.
Percent means out of hundred.

Thus 6%=6/100 and 19%

 Here are some important things you should remember while solving the problems of percentage.

  • x%=x/100
  • a/b = (a/b) x 100 %
  • (a:b) = (a/b) x 100 %
  • If a number a increased by x%, then increase number = {(1+x/100) x a}.
  • If a number a decreased by x%, then decrease number = {(1-x/100) x a}.

Profit and loss

  • The cost at which an article is bought is called its cost price (C.P)
  •  The cost at which an article is sold is called its selling price(S.P)
  • Gain= S.P-C.P
  • Loss=C.P-S.P
  • Gain % = (Gain / CP) 100
  • Loss % = (Loss / CP) 100
  • SP = [(100 + Gain%) / 100] x CP
  • SP = [(100 – Loss %) / 100] x CP
  • CP = [100 / (100 + Gain%)] x SP
  • CP = [100 / (100 – Loss%)] x SP
  • Discount = Marked Price – Selling Price
  • Discount % = [(Discount)/(Marked price)] × 100.

Simple interest and compound interest:

In simple and compound interest there are some important formulas related to principal,rate and time. 

  • Principal: The money borrowed out for a certain period of time.
Principal = (100 × Interest)/(Rate × Time)
  •  Interest: The additional money paid by the borrower instead of the money used by him.
  • Amount: The total money paid back by the borrower to the lender.
Amount =  Principal + Interest
  •  Rate:The interest on hundred rupees for a unit time.
Rate = (100 × Interest)/(Principal × Time)
  • Simple interest:Simple interest is a multiplication of principal by daily interest rate with the number of days.
SI = (P × R ×T) / 100
  • Time period:The time at which we have to give the money back.
Time = (100 × Interest)/(Principal × Rate)
  • Note: In calculating the time the day on which money is borrowed is not included and the day on which money is paid back is included, when a certain sum is lent at compound interest then the interest accrued during the first year is added to the principal and the amount so obtained becomes the principal for the second year. the amount at the end of the second year becomes the principal for the third year and so on.
Compound interest =  Final amount –  Original principal

Class 8 Maths: Algebraic expressions

  • Constant:  A symbol having a fixed numerical value.
  •  Variable:  A symbol which takes on various numerical values.
  •  Algebraic expressions: A combination of constants and variables connected by addition, subtraction, multiplication and division.
  •  Terms of expressions: Several parts of algebraic expressions are separated by addition or subtraction sign.
  •  Polynomial: An algebraic expression in which variables in the world have only non negative integral power.
  • Degree of polynomial:  A polynomial has two or more variables then the sum of powers of all the variables in each term is taken up and the highest sum obtained is degree of polynomial or the highest power of the variable is called the Degree of the polynomial.
  • Linear polynomial:  A polynomial of degree 1.
  • Quadratic polynomial: A polynomial of degree 2.
  • Cubic polynomial: Polynomial of degree 3.
  •  Constant polynomial:  Polynomial having one term consisting of a constant. The degree of constant polynomial is zero.
  •  Polynomials having one two three terms are called monomials, binomials and trinomials respectively.
  •  Like Terms: terms having the same literal coefficients.
  •  While solving the sums of brackets if there is Plus sign before the bracket then remove the bracket without changing the signs of internal terms.
  •  While solving the sums of brackets if there is a negative sign before the bracket then remove the bracket by changing the signs of the internal terms.
  • While solving these sums  we use the rule of BODMAS That is the bracket of division, multiplication addition and subtraction.
  • Examples of monomial, binomial and trinomial. 
8xy³+10x²y4Binomial
2a²b²c³ − 3abc + 2abTrinomial
11x³Monomial

Class 8 Maths: Exponents and Expansions

If a is any number and n is a positive integer, then and  a x a x a x a…..n times = anWhere an, We call a as the base and n as the exponent or index.
Here are some formulas related to exponents:

class 8 power

Here are some formulas related to expansion:

class 8 maths formula

Class 8 Maths: Circle

class 8 circles
  • A circle is a simple closed curve consisting of all points in a plane which are at a fixed distance from a fixed point inside it.
  • The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.
  • The set of all points of the plane which lie on the circle or inside the circle forms the circular region.
  • All the radii of a circle are equal.
  • A line segment whose endpoints lie on a circle is called a chord.
  • A chord of circle passing through its Centre is called diameter of circle.
  • Diameter is the longest chord of the circle.
  • Diameter =  2 X radius.
  • A line which intersects the circle at two distinct points is called a secant.
  • A line which touches a circle at one point only is called tangent to the circle at that point.
  • The tangent at any point of a circle is perpendicular to the radius through the point of contact.
  • One and only one tangent can be drawn to a circle at a point on the circle.
  • Two tangents can be drawn to a circle from a point outside the circle.
  • The lengths of tangents drawn from an external point to a circle are equal.
  • One-half of the whole arc of a circle is called a semi-circle.
  • The whole arc of a circle is called its  circumference.
  • Angle in a semicircle is a right angle.

Area and its perimeter

So basically what is the area
Area: The measure of the surface enclosed by a plane figure.

Refer this relation table:-

unit conversions

Perimeter and Area of Rectangle:

i. The perimeter of rectangle = 2(l + b).

ii. Area of rectangle = l × b; (l and b are the length and breadth of rectangle)

iii. Diagonal of rectangle = √(l² + b²)

Perimeter and Area of the Square:

i. Perimeter of square = 4 × S.

ii.Area of square = S × S.

 iii. Diagonal of square = S√2; (S is the side of square)

Perimeter and Area of the Triangle:

● Perimeter of triangle = (a + b + c); (a, b, c are 3 sides of a triangle)

● Area of triangle = √(s(s – a) (s – b) (s – c)); (s is the semi-perimeter of triangle)

● S = 1/2 (a + b + c)

● Area of triangle = 1/2 × b × h; (b base , h height)

● Area of an equilateral triangle = (a²√3)/4; (a is the side of triangle)

Perimeter and Area of the Parallelogram:

● Perimeter of parallelogram = 2 (sum of adjacent sides)

● Area of parallelogram = base × height

Perimeter and Area of the Rhombus:

● Area of rhombus = base × height

● Area of rhombus = 1/2 × length of one diagonal × length of other diagonal

● Perimeter of rhombus = 4 × side

Perimeter and Area of the Trapezium:

● Area of trapezium = 1/2 (sum of parallel sides) × (perpendicular distance between them)

                                = 1/2 (p₁ + p₂) × h (p₁, p₂ are 2 parallel sides)

Circumference and Area of Circle:

● Circumference of circle = 2πr

                                        = πd

                       Where, π = 3.14 or π = 22/7

                                     r is the radius of circle

                                     d is the diameter of circle

● Area of circle = πr²
● Area of ring = Area of outer circle – Area of inner circle.

Volume and surface area of solids

The body occupying space is called solids.The solid body occurs in various shapes such as cuboid, cube, cylinder, cone and sphere, etc.The space occupied by a solid body is called its volume,where volume is measured in terms of cubic centimetre or cubic metres.

Cuboid:

Volume of a Cuboid (V) = l× b × h

Total surface Area of a Cuboid (S) = 2(lb + bh +hl)

Diagonal of the cuboid = √(l²+b²+h²)

Are of the four walls of a room = sum of the four vertical (or lateral) faces

                                             = 2(l + b)h

Where l = Length, b = breadth and h = height.

Volume of cube:

 Cuboid whose length, breadth and height are equal is called a cube. So, the volume of the cube whose edge is a is expressed as

Volume of the cube = a x a x a = a³

 Diagonal of the cube = a3

Total surface area of the cube=6

Lateral surface area of a cube = 4a²

These were all chapter of class 8 maths which will help you get good knowledge and excel your merits. Make sure you practice and learn each concept of class 8 maths so that you won’t miss anything to get concept clear and get good marks in your exams.

If you have any queries, feel free to drop a comment below.

ALL THE BEST!

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