CBSE Class 10 Maths Formulas

Maths Class 10 Chapterwise Formulas presented by GeeksforGeeks is a combination of a list of the chapter-wise formulae along with the chapter summary and important definitions. As it is known that, Class 10 is an important grade for every student in various higher education fields like engineering, medical, commerce, finance, computer science, hardware, etc. In almost every industry, the most common formulas introduced in class 10 are used. These CBSE Maths Class 10 Chapterwise Formulas includes all the formulas related to Number system, Polynomials, Trigonometry, Algebra, Mensuration, Probability, and Statistics. 

Hence, this article is very much useful for Class 10 which will help the candidates in scoring good marks in Maths for the upcoming CBSE Board Exams.

Chapter 1: Real Numbers

The first chapter of mathematics for class 10th will introduce you to a variety of concepts such as natural numbers, whole numbers, and real numbers, and others. Let’s look at some concepts and formulas for Chapter 1 Real numbers for Class 10 as:

• Types of Numbers:
  • Natural Numbers – It is the counting numbers that can be expressed as: N = {1, 2, 3, 4, 5 >.
  • Whole number – It is the counting numbers along with zero. Therefore, they are written as: W= {0, 1, 2, 3, 4, 5 >
  • Integers – These are the numbers that include all the numbers, positive numbers, zero, and negative numbers also i.e.  ……-4,-3,-2,-1,0,1,2,3,4,5… so on.
    • Positive integers – These are: Z₊ = 1, 2, 3, 4, 5, ……
    • Negative integers – These are: Z_– = -1, -2, -3, -4, -5, ……
  • Rational Number – The number which is expressed in the form p/q where p and q are integers and q is a positive integer. For example 3/7 etc.
  • Irrational Number – The number which can not be expressed in the form p/q. For example π, √5, etc.
  • Real Numbers – A number that can be found on the number line is referred to as a real number. The numbers we use and use in real-world applications are known as real numbers. Natural Numbers, Whole Numbers, Integers, Fractions, Rational Numbers, and Irrational Numbers are all examples of real numbers.
• Euclid’s Division Algorithm (lemma): According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r such that a = bq + r, where 0 ≤ r ≤ b. (Here, a is the dividend, b is the divisor, q is the quotient, and r is the remainder.)
• The fundamental theorem of arithmetic said that the Composite Numbers are equal to the Product of Primes.
• HCF and LCM by prime factorization method:
  • HCF = Product of the smallest power of each common factor in the numbers
  • LCM = Product of the greatest power of each prime factor involved in the number
  • HCF (a,b) × LCM (a,b) = a × b

Chapter 2: Polynomials

The study of mathematical expressions that describe concepts that are equal is known as algebra. Polynomial equations, for example, are among the most common algebraic equations involving polynomials. Learning algebra formulae in class 10 will assist you in turning diverse word problems into mathematical forms. Students may quickly learn to recognize the types of equations and use rules to solve them once they’ve memorized these formulae. These algebraic formulae feature a variety of inputs and outputs that may be interpreted in a variety of ways. Here are all of the key Algebra Formulas and properties for Class 10:

• The general Polynomial Formula is, F (x) = a_nxⁿ + bx^n-1 + a_n-2x^n-2 + …….. + rx + s
  • When n is a natural number: aⁿ – bⁿ = (a – b)(a^n-1 + a^n-2b +…+ b^n-2a + b^n-1)
  • When n is even (n = 2a): xⁿ + yⁿ = (x + y)(x^n-1 – x^n-2y +…+ y^n-2x – y^n-1)
  • When n is odd number: xⁿ + yⁿ = (x + y)(x^n-1 – x^n-2y +…- y^n-2x + y^n-1)
• Types of Polynomials: Here are some important concepts and properties are mentioned in the below table for each type of polynomials-

• Algebraic Polynomial Identities:
  1. (a+b)² = a² + b² + 2ab
  2. (a-b)² = a² + b² – 2ab
  3. (a+b) (a-b) = a² – b²
  4. (x + a)(x + b) = x² + (a + b)x + ab
  5. (x + a)(x – b) = x² + (a – b)x – ab
  6. (x – a)(x + b) = x² + (b – a)x – ab
  7. (x – a)(x – b) = x² – (a + b)x + ab
  8. (a + b)³ = a³ + b³ + 3ab(a + b)
  9. (a – b)³ = a³ – b³ – 3ab(a – b)
  10. (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2xz
  11. (x + y – z)² = x² + y² + z² + 2xy – 2yz – 2xz
  12. (x – y + z)² = x² + y² + z² – 2xy – 2yz + 2xz
  13. (x – y – z)² = x² + y² + z² – 2xy + 2yz – 2xz
  14. x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz -xz)
  15. x² + y² =½ [(x + y)² + (x – y)²]
  16. (x + a) (x + b) (x + c) = x³ + (a + b +c)x² + (ab + bc + ca)x + abc
  17. x³ + y³= (x + y) (x² – xy + y²)
  18. x³ – y³ = (x – y) (x² + xy + y²)
  19. x² + y² + z² -xy – yz – zx = ½ [(x-y)² + (y-z)² + (z-x)²]
• Division algorithm for polynomials: 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) = q(x) × g(x) + r(x)

where r(x) = 0 or degree of r(x) < degree of g(x). Here p(x) is divided, g(x) is divisor, q(x) is quotient and r(x) is remainder.

Chapter 3: Pair of Linear Equations in Two Variables

Pair of Linear Equations in Two Variables is a crucial chapter that contains a range of significant Maths formulas for class 10, particularly for competitive examinations. Some of the important concepts from this chapter are included below:

• Linear Equations: An equation which can be put in the form ax + by + c = 0, where a, b and c are Pair of Linear Equations in Two Variables, and a and b are not both zero, is called a linear equation in two variables x and y
• Solution of a system of linear equations: The solution of the above system is the value of x and y that satisfies each of the equations in the provided pair of linear equations.
  1. Consistent system of linear equations: If a system of linear equations has at least one solution, it is considered to be consistent.
  2. Inconsistent system of linear equation: If a system of linear equations has no solution, it is said to be inconsistent.

S. No.	Types of Linear Equation	General form	Description	Solutions
1.	Linear Equation in one Variable	ax + b=0	Where a ≠ 0 and a & b are real numbers	One Solution
2.	Linear Equation in Two Variables	ax + by + c = 0	Where a ≠ 0 & b ≠ 0 and a, b & c are real numbers	Infinite Solutions possible
3.	Linear Equation in Three Variables	ax + by + cz + d = 0	Where a ≠ 0, b ≠ 0, c ≠ 0 and a, b, c, d are real numbers	Infinite Solutions possible

• Simultaneous Pair of Linear Equations: The pair of equations of the form:

a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

• Graphically represented by two straight lines on the cartesian plane as discussed below:

Chapter 4: Quadratic Equations

Quadratic equations are the polynomial equations of degree two in one variable of type: 

f(x) = ax² + bx + c 

where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where ‘a’ is called the leading coefficient and ‘c’ is called the absolute term of f (x). 

The values of x satisfying the quadratic equation are the roots of the quadratic equation (α,β). The quadratic equation will always have two roots. The nature of roots may be either real or imaginary. 

• Solution or roots of a quadratic equation are given by the quadratic formula:

(α, β) = [-b ± √(b2 – 4ac)]/2ac

• Roots of the quadratic equation: x = (-b ± √D)/2a, where D = b² – 4ac is known as the Discriminant of a quadratic equation. The discriminant of a quadratic equation decides the nature of roots.
• Nature of Roots of Quadratic Equation
  • D > 0, roots are real and distinct (unequal).
  • D = 0, roots are real and equal (coincident) i.e. α = β = -b/2a.
  • D < 0, roots are imaginary and unequal i.e α = (p + iq) and β = (p – iq). Where ‘iq’ is the imaginary part of a complex number.
• Sum of roots: S = α+β= -b/a = coefficient of x/coefficient of x²
• Product of roots: P = αβ = c/a = constant term/coefficient of x²
• Quadratic equation in the form of roots: x² – (α+β)x + (αβ) = 0
• The quadratic equations a₁x₂ + b₁x + c₁ = 0 and a₂x₂ + b₂x + c₂ = 0 have;
  • One common root if (b₁c₂ – b₂c₁)/(c₁a₂ – c₂a₁) = (c₁a₂ – c₂a₁)/(a₁b₂ – a₂b₁)
  • Both roots common if a₁/a₂ = b₁/b₂ = c₁/c₂
• In quadratic equation ax² + bx + c = 0 or [(x + b/2a)² – D/4a²]
  • If a > 0, minimum value = 4ac – b²/4a at x = -b/2a.
  • If a < 0, maximum value 4ac – b²/4a at x= -b/2a.
• If α, β, γ are roots of cubic equation ax³ + bx² + cx + d = 0, then, α + β + γ = -b/a, αβ + βγ + λα = c/a, and αβγ = -d/a

Chapter 5: Arithmetic Progressions

Many things in our everyday lives have a pattern to them. Sequences are the name given to these patterns. Arithmetic and geometric sequences are two examples of such sequences. The terms of a sequence are the various numbers that appear in it.

• Arithmetic Progressions: a₁ a₂, a₃,………….. a_n are sequence terms. An arithmetic progression is a set of integers in which the difference between the terms is the same.
• Common difference: The difference between two consecutive term is the common difference of an AP. If a₁, a₂, a_3, a₄ a₅, a₆ are terms in an AP, then the common difference D = a₂– a₁ = a₃ – a₂ = …
• nth term of AP: a_n = a + (n – 1) d, where an is the nth term.
• Sum of nth terms of AP: S_n= n/2 [2a + (n – 1)d]

Chapter 6: Triangles 

Triangle is a three-side closed figure made up of three straight lines close together. In CBSE Class 10 curriculum, chapter 6 majorly discusses the similarity criteria between two triangles and some important theorems which may help to understand the problems of triangles. The main points of the chapter triangle’s summary are listed as:

• Similar Triangles: The term is given to a pair of triangles that have equal corresponding angles and proportional corresponding sides.
• Equiangular Triangles: The term is given to a pair of triangles that have their corresponding angles equal, also the ratio of any two corresponding sides in two equiangular triangles is always the same.
• Criteria for Triangle Similarity
  1. Angle angle angle (AAA Similarity)
  2. Side angle Side (SAS) Similarity
  3. Side-side side (SSS) Similarity
• Basic Proportionality Theorem: According to this theorem, when a line is drawn parallel to one side of a triangle to intersect the other sides in distinct points, the other two sides are divided in the same ratio.
• Converse of Basic Proportionality Theorem: According to this theorem, in a pair of triangles when the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.

Chapter 7: Coordinate Geometry

Coordinate geometry is a part of mathematics that helps in the presentation of geometric forms on a two-dimensional plane and the learning of its properties. To gain an initial understanding of Coordinate geometry, we will learn about the coordinate plane and the coordinates of a point, as discussed in the below-mentioned points:

• Distance Formulae: For a line having two-point A(x₁, y₁) and B(x₂, y₂), then the distance of these points is given as:

AB= √[(x₂ − x₁)² + (y₂ − y₁)²]

• Section Formula: For any point p divides a line AB with coordinates A(x₁, y₁) and B(x₂, y₂), in ratio m:n, then the coordinates of the point p are given as:

P={[(mx₂ + nx₁) / (m + n)] , [(my₂ + ny₁) / (m + n)]}

• Midpoint Formula: The coordinates of the mid-point of a line AB with coordinates A(x₁, y₁) and B(x₂, y₂), are given as:

P = {(x₁ + x₂)/ 2, (y₁+y₂) / 2}

• Area of a Triangle: Consider the triangle formed by the points A(x₁, y₁) and B(x₂, y₂) and C(x₃, y₃) then the area of a triangle is given as-

∆ABC = ½ |x₁(y₂ − y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)|

Chapter 8: Introduction to Trigonometry

Trigonometry is the science of relationships between the sides and angles of a right-angled triangle. Trigonometric ratios are ratios of sides of the right triangle. Here are some important trigonometric formulas related to trigonometric ratios:

• If in a circle of radius r, an arc of length l subtends an angle of θ radians, then l = r × θ.
  • Radian Measure = π/180 × Degree Measure
  • Degree Measure = 180/π × Radian Measure
• Trigonometric ratios:
  • sin θ = (Perpendicular (P)) / (Hypotenuse (H)).
  • cos θ = (Base (B)) / ( Hypotenuse (H)).
  • tan θ = (Perpendicular (P)) / (Base (B)).
  • cosec θ = (Hypotenuse (H)) / (Perpendicular (P)).
  • sec θ = (Hypotenuse (H)) / (Base (B)).
  • cot θ = (Base (B)) / (Perpendicular (P)).
• Reciprocal Trigonometric Ratios:
  • sin θ = 1 / (cosec θ)
  • cosec θ = 1 / (sin θ)
  • cos θ = 1 / (sec θ)
  • sec θ = 1 / (cos θ)
  • tan θ =  1 / (cot θ)
  • cot θ = 1 / (tan θ)
• Trigonometric Ratios of Complementary Angles:
  • 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
• Trigonometric Identities
  • sin² θ + cos² θ = 1 ⇒ sin² θ = 1 – cos² θ ⇒ cos² θ = 1 – sin² θ
  • cosec² θ – cot² θ = 1 ⇒ cosec² θ = 1 + cot² θ ⇒ cot² θ = cosec² θ – 1
  • sec² θ – tan² θ = 1 ⇒ sec² θ = 1 + tan² θ ⇒ tan² θ = sec² θ – 1

Chapter 9: Some Applications of Trigonometry

Trigonometry can be used in many ways in the things around us like we can use it for calculating the height and distance of some objects without calculating them actually. Below mentioned is the chapter summary of Some Applications of Trigonometry as:

1. Line of Sight – The Line of Sight is the line formed by our vision as it passes through an item when we look at it.
2. Horizontal Line – The distance between the observer and the object is measured by a horizontal line.
3. Angle of Elevation – The angle formed by the line of sight to the top of the item and the horizontal line is called an angle of elevation. It is above the horizontal line, i.e. when we gaze up at the item, we make an angle of elevation.
4. Angle of Depression – When the spectator must look down to perceive the item, an angle of depression is formed. When the horizontal line is above the angle, the angle of depression is formed between it and the line of sight.

Chapter 10: Circles

A circle is a collection of all points in a plane that are at a constant distance from a fixed point. The fixed point is called the centre of the circle and the constant distance from the centre is called the radius. A line segment joining any two points on a circle is called a chord. A chord passing through the centre of the circle is called the diameter. It is the longest chord. When a line meets the circle at one point the line is known as a tangent. The tangent to a circle is perpendicular to the radius through the point of contact. 

• The tangent to a circle equation x₂ + y₂ = a₂ for a line y = mx + c is given by the equation y = mx ± a √[1+ m₂].
• The tangent to a circle equation x₂ + y₂ = a₂ at (a₁,b₁) is xa₁ + yb₁ = a₂
• Circumference of the circle = 2 π r
• Area of the circle = π r²
• Area of the sector of angle, θ = (θ/360) × π r²
• Length of an arc of a sector of angle, θ = (θ/360) × 2 π r
• Distance moved by a wheel in one revolution = Circumference of the wheel.
• The number of revolutions = Total distance moved / Circumference of the wheel.

Chapter 11: Constructions

Construction helps to understand the approach to construct different types of triangles for different given conditions using a ruler and compass of required measurements. Here the list of important constructions learned in this chapter of class 10 are mentioned as,

1. Determination of a Point Dividing a given Line Segment, Internally in the given Ratio M : N
2. Construction of a Tangent at a Point on a Circle to the Circle when its Centre is Known
3. Construction of a Tangent at a Point on a Circle to the Circle when its Centre is not Known
4. Construction of a Tangents from an External Point to a Circle when its Centre is Known
5. Construction of a Tangents from an External Point to a Circle when its Centre is not Known
6. Construction of a Triangle Similar to a given Triangle as per given Scale Factor m/n, m<n.
7. Construction of a Triangle Similar to a given Triangle as per given Scale Factor m/n, m > n.

Chapter 12: Areas Related to Circles

The fundamentals of area, circumference, segment, sector, angle and length of a circle, and area for a circle’s sector are all covered here. This section also covers the visualization of several planes and solid figure areas. Below mentioned are the major points from the chapter summary of Areas Related to Circles,

• The equal chord of a circle is equidistant from the centre.
• The perpendicular drawn from the centre of a circle bisects the chord of the circle.
• The angle subtended at the centre by an arc = Double the angle at any part of the circumference of the circle.
• Angles subtended by the same arc in the same segment are equal.
• To a circle, if a tangent is drawn and a chord is drawn from the point of contact, then the angle made between the chord and the tangent is equal to the angle made in the alternate segment.
• The sum of the opposite angles of a cyclic quadrilateral is always 180°.
• Area of a Segment of a Circle: If AB is a chord that divides the circle into two parts, then the bigger part is known as the major segment and the smaller one is called the minor segment.

Chapter 13: Surface Areas and Volumes

This page explains the concepts of surface area and volume for Class 10. The surface area and volume of several solid shapes such as the cube, cuboid, cone, cylinder, and so on will be discussed in this article. Lateral Surface Area (LSA), Total Surface Area (TSA), and Curved Surface Area are the three types of surface area (CSA). Let’s take a closer look at the surface area and volume formulas for various three-dimensional geometries. The combination of several solid shapes can be examined in this chapter. In addition, the formula for determining the volume and its surface area is mentioned as,

1. Total Surface Area (TSA): The whole area covered by the object’s surface is called the Total Surface area. Following is the list of the total surface areas of some important geometrical figures-
   1. TSA of a Cuboid = 2(l x b) +2(b x h) +2(h x l)
   2. TSA of a Cube = 6a²
   3. TSA of a Right circular Cylinder = 2πr(h+r)
   4. TSA of a Right circular Cone = πr(l+r)
   5. TSA of a Sphere = 4πr²
   6. TSA of a Right Pyramid = LSA + Area of the base
   7. TSA of a Prism = LSA × 2B
   8. TSA of a Hemisphere = 3 × π × r²
2. Lateral/Curved Surface Area: The curved surface area is the area of only the curved component, or in the case of cuboids or cubes, it is the area of only four sides, excluding the base and top. It’s called the lateral surface area for forms like cylinders and cones.
   1. CSA of a Cuboid = 2h(l+b)
   2. CSA of a Cube = 4a²
   3. CSA of a Right circular Cylinder = 2πrh
   4. CSA of a Right circular Cone = πrl
   5. LSA of a Right Pyramid = ½ × p × l
   6. LSA of a Prism = p × h
   7. LSA of a Hemisphere = 2 × π × r²

Volume: The volume of an object or material is the amount of space it takes up, measured in cubic units. There is no volume in a two-dimensional object, only area. A circle’s volume cannot be calculated since it is a 2D figure, while a sphere’s volume can be calculated because it is a 3D figure.

1. Volume of a Cuboid = l x b x h
2. Volume of a Cube = a³
3. Volume of a Right circular Cylinder = πr²h
4. Volume of a Right circular Cone = 1/3πr²h
5. Volume of a Sphere = 4/3πr³
6. Volume of a Right Pyramid = ⅓ × Area of the base × h
7. Volume of a Prism = B × h
8. Volume of a Hemisphere = ⅔ × (πr³)

Here, l is the length, b is the breadth, h is the height, r is the radius, a is the side, p is the perimeter of the base, B is the area of the base of the respective geometrical figure.

Chapter 14: Statistics

Statistics in Class 10 mainly consist of the study of given data b evaluating its mean, mode, median. The statistic formulas are given below:

Mean

• Direct method: Mean, X = ∑f_i x_i / ∑f_i
• Assumed Mean Method: X = a + ∑f_i d_i / ∑f_i                                      (where d_i = x_i – a)
• Step Deviation Method: X = a + ∑f_i u_i / ∑f_i × h

Median is the middlemost term for an even number of observations while (n+1)th/2 observations for an odd number of observations.

Mode

where l is the lower limit of the modal class, f₁ is the frequency of the modal class, f₀ is the frequency of the preceding class of the modal class, f₂ is the frequency of the succeeding class of the modal class and h is the size of the class interval.

Chapter 15: Probability

Probability denotes the likelihood of something happening. It’s a field of mathematics that studies the probability of a random event occurring. From 0 to 1, the value is expressed. Probability is a mathematical concept that predicts how probable occurrences are to occur. The definition of probability is the degree to which something is likely to occur. This is the fundamental probability theory, which is also applied to the probability distribution, where you will learn about the possible results of a random experiment. Let’s discuss some important formulas of Probability in the Class 10 curriculum:

Empirical Probability: The probability of events that depends on the experiments and it is defined as,

Empirical Probability = Number of Trials which expected outcome come / Total Number of Trials

Theoretical Probability: The probability of events that depends on the experiments and it is defined as,

Theoretical Probability = Number of favourable outcomes to E / Total Number of possible outcomes of the experiment