Here you will find all the maths formulas for class 10. You can also download these formulas of class 10 in a PDF file and take the printout for fast practice. There are fifteen chapters in NCERT class 10 Maths book. We have provided all the formulas for these chapters, including Real Numbers, Polynomials, Linear Equations, Arithmetic Progressions, Quadratic Equations, Trigonometry, Area & Volume, and other important formulas.

Maths formulas for class 10 are the basic formulas that you will need to prepare for the mathematics exam and improve your mathematical skills for higher classes. These basic maths formulas are crucial for every student. We suggest you note down all the important maths formulas for class 10 or take the printout to save your time while studying.

Page Contents

## Real Numbers

Real number maths formulas include important theorems, algorithms, and Euclid’s division lemma to learn more about real numbers.

### Euclid’s Division Lemma

**Theorem**: For given positive integers a and b, there exist unique integers q and r satisfying,

*a = bq + r, 0 ≤ r < b.*

**Algorithm**: Series of well-defined steps which gives a procedure for solving a type of problem**Lemma:**Proven statement used for proving another statement

### HCF using Euclid’s Division Algorithm

Finding HCF of two number **c** and **d** where **c > d**

**Step 1**: Apply division lemma to get the whole numbers (q and r) such that, c=dq+r, 0≤r**Step 2**: If r=0, d is the HCF of c and d. If r ≠0, apply division lemma to d and r.**Step 3**: Continue the process till the remainder is zero. The divisor at the final stage will be HFC.

### Theorem of Arithmetic

Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

- The prime factorisation of a natural number is unique, except for the order of its factors.
- For any two positive number p and q,
**HCF (p,q) x LCM (p,q) = p x q**

### Rational Number Theorems

- If p is a prime number and a is a positive integer, then if p divides a
^{2}, the p also divides a. - √2 is irrational
- If x is a rational number whose decimal expansion terminates, then x can be expressed in the form of p/q, where p and q are coprime, and the prime factorisation of q is in the form 2
^{n}5^{m}, when n,m are non-negative integers.

LCM (p,q,r) = p.q.r HCF(p,q,r) / HCF(p,q) . HCF(q,r) . HCF(p,r)

HCF (p,q,r) = p.q.r LCM(p,q,r) / LCM(p,q) . LCM(q,r) . LCM(p,r)

## Polynomials

### Degree of polynomials

Highest power of x in a polynomial p(x) is the degree of polynomial.

**Linear polynomials**: Polynomials of degree 1, i.e., 2x-3**Quadric Polynomials:**Polynomials of degree 2, i.e., 3x^{2}-4x+7**Cubic Polynomials:**Polynomials of degree 3, i.e., 4x^{3}+3x^{2}+2x+9

If k is zero of polynomial p(x) = 2x-3, then p(k) = 2x-3 = 0.

Zero of the lineal polynomial ax+b is -b/a = -(constant term)/coefficient of x

- A quadratic polynomial in x with real coefficients is of the form ax
^{2}+ bx + c, where a, b, c are real numbers with a ≠ 0. - The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x) intersects the x-axis.
- A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.

If α and β are the zeroes of the quadratic polynomial ax^{2} + bx + c, then,

If α, β, γ are the zeroes of the cubic polynomial ax^{3}+ bx^{2}+ cx + d, then

**Division Algorithm:**

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that,

**p(x) = g(x) × q(x) + r(x),** *where r(x) = 0 or degree r(x) < degree g(x).*

## Pair of Linear Equations in two Variables

Here you will find maths formulas for class 10 for chapter 3. Download all the formulas of class 10 in PDF file or read online.

**Linear Equations**:

An equation between variables that produce a straight line when plotted on a graph is called linear equation.

No. of variables | Equation | Condition |

1 | ax+b=0 | a & b are real numbers, a≠0 |

2 | ax+by+c=0 | a,b,c are real numbers and a≠0, b≠0 |

3 | ax+by+cz+d=0 | a,b,c,d are real numbers and a≠0, b≠0, c≠0 |

### Pair of linear equations:

Two linear equations in the same two variables are called a pair of linear equations in two variables.

in two variables | a_{1}x+b_{1}y+c_{1}=0 a_{2}x+b_{2}y+c_{2}=0 | a_{1},b_{1},c_{1} & a_{2},b_{2},c_{1} are real numbers and a_{1}^{2} + b_{1}^{2} ≠ 0, a_{2}^{2} + b_{2}^{2} ≠ 0. |

### Graphical Method of solving pair of linear equations

It is represented by two lines on the graph.

**Consistent pair of equations:**If the lines intersect at a point, then that point gives the unique solution of the two equations.**Dependent (consistent) Pair**: If the lines coincide, then there are infinitely many solutions – each point on the line being a solution.**Inconsistent Pair:**If the lines are parallel, then the pair of equations has no solution.

### Algebraic Method

**Substitution Method:**- Find the value of one variable, say y in terms of the other variable, i.e., x from either equation, whichever is convenient.

- Substitute this value of y in the other equation, and reduce it to an equation in one variable, i.e., in terms of x, which can be solved.
- If this statement is true, you can conclude that the pair of linear equations has infinitely many solutions. If the statement is false, then the pair of linear equations is inconsistent.

- Substitute the value of x (or y) obtained in Step 2 in the equation used in Step 1 to obtain the value of the other variable.

**Elimination Method**- Multiply both the equations by a number(non-zero) to make the coefficient of one variable equal. Let us say is y.

- Subtract the equation to eliminate variable y, and find the value of x

- Substitute the value of x in one of the equations to get the value of y.

**Cross-Multiplication Method**- Multiply Equation (1) by b
_{2}and Equation (2) by b_{1}, to get new equations.- b
_{2}a_{1}x + b_{2}b_{1}y + b_{2}c_{1}= 0 eq. 3

- b
_{1}a_{2}x + b_{1}b_{2}y + b_{1}c_{2}= 0 eq. 4

- b

- Subtract eq.4 from eq. 3 to get
- (b
_{2}a_{1}– b_{1}a_{2}) x + (b_{2}b_{1}– b_{1}b_{2})y + (b_{2}c_{1}– b_{1}c_{2}) = 0

- (b

- Find the value of x from there and put it in eq. 1 or eq. 2 to find value of y.

- Multiply Equation (1) by b

## Quadratic Equations Formulas for Class 10

Solve quadratic equation problems by using **Quadratic Equations Maths Formulas** for class 10. Learn about finding solutions and roots of the equations.

Quadratic equation in the form of variable x is of the form ax^{2}+bx+c=0, where a,b,c are real numbers and a≠0.

### Solution of Quadratic Equation by Factorisation

- For any quadratic equation, split the middle term and factorize the equation into two linear factors
- Equate each factor to zero

e.g., 2x^{2}-5x+3=0 can be factorize to linear terms as 2x^{2}-2x-3x+3=0 or 2x(x-1)-3(x-1)=0

Thus, (2x-3)(x-1)=0 or x=1,3/2. Hence, 1 and ½ are roots of the equation.

- a real number α is called a root of the quadratic equation ax
^{2}+bx+c =0, a ≠ 0 if aα^{2}+ bα + c =0.

### Roots of Quadratic Equation:

For an equation ax^{2}+bx+c=0,

**Two distinct real roots**: if b^{2}– 4ac > 0,**Two equal roots**: if b^{2}– 4ac=0**No real roots**: if b^{2}– 4ac <0

## Algebraic Maths Formulas For Class 10

Algebraic maths formulas for class 10 are helpful to solve quadratic equations and cubical equations easily. You have already learned about some of these formulas in previous classes.

Expression | Expansion |

(a+b)^{2} | a^{2}+b^{2}+2ab |

(a-b)^{2} | a^{2}+b^{2}-2ab |

(a-b)(a+b) | a^{2 }– b^{2} |

(x + a)(x + b) | x^{2} + (a + b)x +ab |

(x + a)(x – b) | x^{2} + (a–b)x –ab |

(x – a)(x + b) | x^{2} + (b–a)x –ab |

(x – a)(x – b) | x^{2} – (a+b)x +ab |

(x + y + z)^{2} | x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2xz |

(x + y – z)^{2} | x^{2} + y^{2} + z^{2} + 2xy – 2yz – 2xz |

(x – y + z)^{2} | x^{2} + y^{2} + z^{2} – 2xy – 2yz + 2xz |

(x – y – z)^{2} | x^{2} + y^{2} + z^{2} – 2xy + 2yz – 2xz |

(a + b)^{3} | a^{3} + b^{3} +3ab(a + b) |

(a – b)^{3} | a^{3} – b^{3} –3ab(a – b) |

a^{3} + b^{3} | (a + b) (a^{2} – ab + b^{2}) |

a^{3} – b^{3} | (a – b) (a^{2} + ab + b^{2}) |

(x + a) (x + b) (x + c) | x^{3} + (a + b +c)x^{2} + (ab + bc + ca)x + abc |

x^{3} + y^{3} + z^{3} – 3xyz | (x + y + z)(x^{2} + y^{2} + z^{2} – xy – yz -xz) |

x^{2 }+ y^{2} | ½ [(x + y)^{2} + (x – y)^{2}] |

a^{m+n} | a^{m} x a^{n} |

a^{m-n} | (a^{m})/(a^{n}) |

a^{mn} | (a^{m})^{n} |

a^{-m} | 1/a^{m} |

a^{1} | a |

a^{0} | 1 |

## Arithmetic Progression

list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

e.g., a_{1}, a_{2}, a_{3}, . . ., a_{n }or 2,4,6,8,….., ∞

**Term:**Every number in the list is a term**Common Difference**: The fixed number which is added to the terms to make the list. It can either be positive, negative or zero.- Common Difference (d) = a
_{n}– a_{n-1}

- Common Difference (d) = a

#### General Form of AP

a, a+d, a+2d, a+3d,…..

where d is the common difference.

**Finite AP**: AP having limited number of terms**Infinite AP**: AP having unlimited number of terms

#### nth Term of AP

a_{n} = a+(n-1)d, where a = first term, d = common difference, n = number of term

e.g., finding 10^{th} terms will be, a_{10} = a+(10-1)d

#### Sum of n Terms of AP

S = n/2 [2a+(n-1)d] or S = n/2 (a+a_{n}) or, S = n/2(a+l), where l is the last term.

Sum of first n Positive Integers, S_{n} = n(n+1)/2

## Coordinate Geometry

Distance between two points P(x_{1},y_{1}) and Q(x_{2},y_{2}) is,

Distance of P(x,y) from Origin O(0,0) is,

The coordinates of the point P(x, y) which divides the line segment joining the points A(x_{1},y_{1}) and B(x_{2}, y_{2}) internally in the ratio m_{1} : m_{2} are

The mid-point of the line segment joining the points P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) is

The area of the triangle formed by the points (x1, y1), (x2, y2) and (x3, y3) is the numerical value of the expression

## Trigonometry Formulas for Class 10

This section includes all maths formulas for class 10 trigonometry. Studying trigonometry in class 10 is really necessary as it will get more complex in upcoming classes (class 11 and class 12). Make sure you practice well with all trigonometry formulas for class 10.

For a right angled triangle ABC, which is right angled at B. and have ∠A = ∠θ

### Trigonometry Ratio of Angles

Angle | 0° | 30° | 45° | 60° | 90° |

Sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |

Cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |

Tan θ | 0 | 1/√3 | 1 | √3 | ∞ |

Cot θ | ∞ | √3 | 1 | 1/√3 | |

Sec θ | 1 | 2/√3 | √2 | 2 | ∞ |

Cosec θ | ∞ | 2 | √2 | 2/√3 | 1 |

### Other Trigonometry Formulas and Identities

sin(90° – θ) = cos θ | cos(90° – θ) = sin θ |

tan(90° – θ) = cot θ | cot(90° – θ) = tan θ |

sec(90° – θ) = cosec θ | cosec(90° – θ) = sec θ |

sin^{2} θ + cos^{2} θ = 1 or cos^{2} θ = 1- sin^{2} θ | sec^{2} θ – tan^{2} θ = 1 for 0° ≤ θ < 90°, |

cosec^{2} θ – cot^{2} θ = 1 for 0° < θ ≤ 90° |

**Line of Sight**: the line drawn from the eye of an observer to the point in the object viewed by the observer.**Angle of Elevation:**angle formed by the line of sight with the horizontal when it is above the horizontal level. (e.g., when we raise our head to look any object)**Angle of Depression:**the angle formed by the line of sight with the horizontal when it is below the horizontal level. (e.g., when we look down at any object)

## Circles Formulas of Class 10

**Area of a circle**= πr^{2}**Circumference (perimeter) of the circle**= πD or 2πr**Radius**= D/2, where D is the diameter- Area of a sector of a circle with radius r and angle θ = (θ/360) x πr
^{2} - Length of an arc formed by the sector of angle θ = (θ/360) x 2πr
- Area of a segment of a circle =
*Area of the corresponding sector – Area of the corresponding triangle.*

π = 22/7 or 3.14159265

- The tangent to a circle is perpendicular to the radius through the point of contact.
- The lengths of the two tangents from an external point to a circle are equal.

## Surface Areas and Volumes formulas

You have already studied about surface areas and volumes in previous classes. These problems get a bit complex here in class 10. Download all maths formulas for class 10 for this chapter. It would be best if you learn these formulas as soon as possible so that you can solve the problems easily in no-time.

### Cylinder

r = radius of circular base, h = height of cylinder

Name | Formula |

Curved surface area of Cylinder | 2πrh |

Total surface area of Cylinder | 2πrh + 2πr^{2} |

Volume of Cylinder | πr^{2}h |

Area of two circular bases | πr^{2 }+ πr^{2} = 2πr^{2} |

### Cone Formula

r = radius of circular base, l = slant height

Name | Formula |

Curved surface area of Cone | πrl |

Total surface area of Cone | πr(r + l) |

Volume of Cone | (1/3)πr^{2}h |

Slant height of Cone (l) | √(r^{2 }+ h^{2}) |

Volume of frustum of the cone | (1/3)πh (r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}) (r_{1}-r_{2})^{2} |

Curved surface area of the frustum of the cone | π(r_{1} + r_{2})l where l = √(h^{2} + (r_{1} – r_{2})) |

Total surface area of the frustum of cone | πl(r_{1} + r_{2}) + πr_{1}^{2}+ πr_{2}^{2}, where l = √(h^{2} + (r_{1} – r_{2})) |

### Sphere Formulas

r = radius, Semi-sphere = half of sphere

Name | Formula |

Surface area of a sphere | 4πr^{2} |

Volume of a sphere | (4/3)πr^{3} |

Volume of hemisphere | (2/3)πr^{3} |

The curved surface area of the hemisphere | 2πr^{2} |

The total surface area of the hemisphere | 3πr^{2} |

Diameter (same as a circle) | 2r |

### Formulas for Cuboid

l = length, b = breadth (width), h = height

Name | Formula |

Perimeter of a cuboid | 4(l+b+h) |

Volume of a cuboid | lxbxh |

Total surface area of cuboid shape | 2(lxb + bxh + hxl) |

Longest diagonal length in a cuboid | √(l^{2 }+ b^{2}+h^{2}) |

### Cube Formulas

Length = breadth = height = a

Name | Formula |

Perimeter of a cube | 12a |

Volume of a cube | a^{3} |

Surface area of cube | 6a^{2} |

Diagonal of cube | √(3a) |

## Statistics formulas for Class 10

### Formulas to find mean of grouped data

#### Direct Method of finding mean of grouped data

Or we can also write it as,

#### Assumed mean method

#### Step-deviation method

### Mode of grouped data

### Median of Grouped data

## Probability Formulas for Class 10

You are studying probability since class 7. Here you will find more probability maths formulas for class 10. Make sure to practice these questions well.

### Experimental Probability

### Theoretical Probability

Where, E is an event.

For an event E,

Important Points:

- The sum of the probabilities of all the elementary events of an experiment is 1
- The probability of any impossible event is 0
- Probability of an event that is sure to occur is 1. Such event is called a
or*sure event*.*certain event* - A playing die has six outcomes (1,2,3,4,5,6)

### Playing Cards – total 52 in sets of four

Types | Cards |

Diamonds (♦) | 13 (Ace, King, Queen, Jack, 10,9,8,7,6,5,4,3,2) |

Hearts (♥) | 13 (Ace, King, Queen, Jack, 10,9,8,7,6,5,4,3,2) |

Spades (♠) | 13 (Ace, King, Queen, Jack, 10,9,8,7,6,5,4,3,2) |

Clubs (♣) | 13 (Ace, King, Queen, Jack, 10,9,8,7,6,5,4,3,2) |

Those were all of the maths formulas for class 10. You can download the maths formulas PDF by clicking on the download link shared below. We will soon upload more study materials including notes and solutions. Till then practice the maths problems and memorize the formulas quickly.

Download Maths Formulas for Class 10 PDF File Here**password:** *vt123*

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