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 a², 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 2ⁿ5^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., 3x²-4x+7
Cubic Polynomials: Polynomials of degree 3, i.e., 4x³+3x²+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 ax²+ 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 ax² + bx + c, then,

If α, β, γ are the zeroes of the cubic polynomial ax³+ bx²+ 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

a₁x+b₁y+c₁=0 a₂x+b₂y+c₂=0

a₁,b₁,c₁ & a₂,b₂,c₁ are real numbers and a₁² + b₁² ≠ 0, a₂² + b₂² ≠ 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 b₂ and Equation (2) by b₁, to get new equations.
  - b₂a₁x + b₂b₁y + b₂c₁ = 0 eq. 3
  - b₁a₂x + b₁b₂y + b₁c₂= 0 eq. 4
- Subtract eq.4 from eq. 3 to get
  - (b₂a₁– b₁a₂) x + (b₂b₁– b₁b₂)y + (b₂c₁– b₁c₂) = 0
- Find the value of x from there and put it in eq. 1 or eq. 2 to find value of y.

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 ax²+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., 2x²-5x+3=0 can be factorize to linear terms as 2x²-2x-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 ax²+bx+c =0, a ≠ 0 if aα²+ bα + c =0.

Roots of Quadratic Equation:

For an equation ax²+bx+c=0,

Two distinct real roots: if b²– 4ac > 0,
Two equal roots: if b²– 4ac=0
No real roots: if b²– 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	a²+b²+2ab
(a-b)²	a²+b²-2ab
(a-b)(a+b)	a²– b²
(x + a)(x + b)	x² + (a + b)x +ab
(x + a)(x – b)	x² + (a–b)x –ab
(x – a)(x + b)	x² + (b–a)x –ab
(x – a)(x – b)	x² – (a+b)x +ab
(x + y + z)²	x² + y² + z² + 2xy + 2yz + 2xz
(x + y – z)²	x² + y² + z² + 2xy – 2yz – 2xz
(x – y + z)²	x² + y² + z² – 2xy – 2yz + 2xz
(x – y – z)²	x² + y² + z² – 2xy + 2yz – 2xz
(a + b)³	a³ + b³ +3ab(a + b)
(a – b)³	a³ – b³ –3ab(a – b)
a³ + b³	(a + b) (a² – ab + b²)
a³ – b³	(a – b) (a² + ab + b²)
(x + a) (x + b) (x + c)	x³ + (a + b +c)x² + (ab + bc + ca)x + abc
x³ + y³ + z³ – 3xyz	(x + y + z)(x² + y² + z² – xy – yz -xz)
x²+ y²	½ [(x + y)² + (x – y)²]
a^m+n	a^m x aⁿ
a^m-n	(a^m)/(aⁿ)
a^mn	(a^m)ⁿ
a^-m	1/a^m
a¹	a
a⁰	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., a₁, a₂, a₃, . . ., a_nor 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

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, a₁₀ = 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(x₁,y₁) and Q(x₂,y₂) is,

\(\sqrt{\left( x_{2} – x_{1} \right)^{2} + \left( y_{2} – y_{1} \right)^{2}}\)

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

\(\sqrt{x^{2} + y^{2}}\)

The coordinates of the point P(x, y) which divides the line segment joining the points A(x₁,y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂ are

and x = \(\frac{m_{1}x_{2} + m_{2}x_{1}}{m_{1} + m_{2}}\) , y = \(\frac{m_{1}y_{2} + m_{2}y_{1}}{m_{1} + m_{2}}\)

The mid-point of the line segment joining the points P(x₁, y₁) and Q(x₂, y₂) 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

Sin θ = \(\frac{\text{Perpendicular}}{\text{Hypotenus}}\) = \(\frac{\text{Side opposite to angle }\theta\ }{\text{Hypotenus}}\) = \(\frac{P\ }{H}\)

Cos θ = \(\frac{\text{Base}}{\text{Hypotenus}}\) = \(\frac{\text{Side }\text{adjacent}\text{ to angle }\theta\ }{\text{Hypotenus}}\) = \(\frac{B\ }{H}\)

Cot θ = \(\frac{\text{Base}}{\text{Perpendicular}}\ \)=\(\ \frac{\text{Side adjacent to angle}\ \theta}{\text{Side opposite to angle}\ \theta}\) = \(\frac{B\ }{P}\) = \(\frac{1\ }{\tan\ \theta}\) = \(\frac{\cos\ \theta\ }{\sin\ \theta}\)

Tan θ = \(\frac{\text{Perpendicular}}{\text{Base}}\ \)=\(\ \frac{\text{Side }\text{Opposite}\text{ to angle }\theta}{\text{Side }\text{Adjacent}\text{ to angle }\theta}\) = \(\frac{P\ }{B}\) = \(\frac{1\ }{\cot\ \theta}\) = \(\frac{\text{sing}\ \theta\ }{\cos\ \theta}\)

Sec θ = \(\frac{\text{Hypotenus}}{\text{Base}}\) = \(\frac{\text{Hypotenus}\ }{S\text{ide }\text{adjacent}\text{ to angle }\theta}\) = \(\frac{H\ }{B}\) = \(\frac{1\ }{\text{co}s\ \theta}\)

Cosec θ = \(\frac{\text{Hypotenus}}{\text{Perpendicular}}\) = \(\frac{\text{Hypotenus }}{\text{Side opposite to angle }\theta}\) = \(\frac{\text{H }}{P}\) = \(\frac{1\ }{\sin\ \theta}\)

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

Trigonometry Ratio of Angles

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
sin² θ + cos² θ = 1 or cos² θ = 1- sin² θ	sec² θ – tan² θ = 1 for 0° ≤ θ < 90°,
cosec² θ – cot² θ = 1 for 0° < θ ≤ 90°

Trigonometry Formulas

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 = πr²
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 πr²
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 + 2πr²
Volume of Cylinder	πr²h
Area of two circular bases	πr²+ πr² = 2πr²

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)πr²h
Slant height of Cone (l)	√(r²+ h²)
Volume of frustum of the cone	(1/3)πh (r₁²+r₂²+r₁r₂) (r₁-r₂)²
Curved surface area of the frustum of the cone	π(r₁ + r₂)l where l = √(h² + (r₁ – r₂))
Total surface area of the frustum of cone	πl(r₁ + r₂) + πr₁²+ πr₂², where l = √(h² + (r₁ – r₂))

Sphere Formulas

r = radius, Semi-sphere = half of sphere

Name	Formula
Surface area of a sphere	4πr²
Volume of a sphere	(4/3)πr³
Volume of hemisphere	(2/3)πr³
The curved surface area of the hemisphere	2πr²
The total surface area of the hemisphere	3πr²
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	√(l²+ b²+h²)

Cube Formulas

Length = breadth = height = a

Name	Formula
Perimeter of a cube	12a
Volume of a cube	a³
Surface area of cube	6a²
Diagonal of cube	√(3a)

Statistics formulas for Class 10

Formulas to find mean of grouped data

Direct Method of finding mean of grouped data

\[\overline{x} = \ \frac{f_{1}x_{1} + f_{2}x_{2} + \ldots\ + f_{n}x_{n}}{f_{1} + f_{2} + \ldots\ + f_{n}}\]

Or we can also write it as,

\(\overline{x} = \ \frac{\sum_{i = 1}^{n}{f_{i}x_{i}}}{\sum_{i = 1}^{n}f_{i}}\) or \(\overline{x} = \ \frac{\Sigma f_{i}x_{i}}{\Sigma f_{i}}\) , where i varies from 1 to n

Assumed mean method

\[\overline{x} = \ a + \ \frac{\Sigma f_{i}d_{i}}{\Sigma f_{i}}\]

Step-deviation method

\[\overline{x} = \ a + \left( \frac{\Sigma f_{i}u_{i}}{\Sigma f_{i}} \right)\text{x h}\]

Mode of grouped data

Mode = \(l + \ \frac{f_{1} – f_{0}}{2f_{1} + f_{0} – f_{2}}\)

Median of Grouped data

Median = \(l + \ \left( \frac{\frac{n}{2} – cf}{f} \right)\text{x h}\)

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

P(E) = \(\frac{\text{Number of trials in which the event happened}}{\text{Total Number of trials}}\)

Theoretical Probability

P(E) = \(\frac{\text{Number of outcomes favourable to E}}{\text{Number of all possible outcomes of the experiment}}\)

Where, E is an event.

For an event E,

P (\(\overline{E}\)) = 1 — P(E), where \(\overline{E}\) denotes “not E” or the complement of 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 sure event or 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.

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Maths Formulas for Class 10 PDF Chapterwise

Real Numbers

Euclid’s Division Lemma

HCF using Euclid’s Division Algorithm

Theorem of Arithmetic

Rational Number Theorems

Polynomials

Degree of polynomials

Division Algorithm:

Pair of Linear Equations in two Variables

Linear Equations:

Pair of linear equations:

Graphical Method of solving pair of linear equations

Algebraic Method

Quadratic Equations Formulas for Class 10

Solution of Quadratic Equation by Factorisation

Roots of Quadratic Equation:

Algebraic Maths Formulas For Class 10

Arithmetic Progression

General Form of AP

nth Term of AP

Sum of n Terms of AP

Coordinate Geometry

Trigonometry Formulas for Class 10

Trigonometry Ratio of Angles

Other Trigonometry Formulas and Identities

Circles Formulas of Class 10

Surface Areas and Volumes formulas

Cylinder

Cone Formula

Sphere Formulas

Formulas for Cuboid

Cube Formulas

Statistics formulas for Class 10

Formulas to find mean of grouped data

Direct Method of finding mean of grouped data

Assumed mean method

Step-deviation method

Mode of grouped data

Median of Grouped data

Probability Formulas for Class 10

Experimental Probability

Theoretical Probability

Playing Cards – total 52 in sets of four

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