Maths Formulas for Class 10 CBSE
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. These Formulae include chapters such as, Number system, Polynomials, Trigonometry, Algebra, Mensuration, Probability, and Statistics.
In this article, we are going to learn all the Math formulas in Class 10 CBSE syllabus.
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:
Concepts | Description | Examples/Formula |
---|---|---|
Natural Numbers | Counting numbers starting from 1. | N = {1, 2, 3, 4, 5, …} |
Whole Numbers | Counting numbers including zero. | W = {0, 1, 2, 3, 4, 5, …} |
Integers | All positive numbers, zero, and negative numbers. | …, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … |
Positive Integers | All positive whole numbers. | Z+ = 1, 2, 3, 4, 5, … |
Negative Integers | All negative whole numbers. | Z– = -1, -2, -3, -4, -5, … |
Rational Number | Numbers expressed as a fraction where both numerator and denominator are integers and the denominator is not zero. | Examples: 3/7, -5/4 |
Irrational Number | Numbers that cannot be expressed as a simple fraction. | Examples: π, √5 |
Real Numbers | All numbers that can be found on the number line, including rational and irrational numbers. | Includes Natural, Whole, Integers, Rational, Irrational |
Euclid’s Division Algorithm | A method for finding the HCF of two numbers. | a = bq + r, where 0 ≤ r < b |
Fundamental Theorem of Arithmetic | States that every composite number can be expressed as a product of prime numbers. | Composite Numbers = Product of Primes |
HCF and LCM by Prime Factorization | Method to find the highest common factor and least common multiple. | HCF = Product of smallest powers of common factors; LCM = Product of greatest powers of each prime factor; HCF(a,b) × LCM(a,b) = a × b |
Learn More:
- Natural numbers
- Whole Numbers
- Real Numbers
- Integers
- Rational Number
- Irrational Number
- HCF and LCM
- Fundamental Theorem of Arithmetic
- Factorization
Chapter 2 Polynomials
Polynomial equations 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.
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:
Category | Description | Formula/Identity |
---|---|---|
General Polynomial Formula | Standard form of a polynomial | F (x) = a_{n}x^{n} + bx^{n-1} + a_{n-2}x^{n-2} + …….. + rx + s |
Special Case: Natural Number n | Difference of powers formula | a^{n} – b^{n} = (a – b)(a^{n-1} + a^{n-2}b +…+ b^{n-2}a + b^{n-1}) |
Special Case: Even n (n = 2a) | Sum of even powers formula | x^{n} + y^{n} = (x + y)(x^{n-1} – x^{n-2}y +…+ y^{n-2}x – y^{n-1}) |
Special Case: Odd Number n | Sum of odd powers formula | x^{n} + y^{n} = (x + y)(x^{n-1} – x^{n-2}y +…- y^{n-2}x + y^{n-1}) |
Division Algorithm for Polynomials | Division of one polynomial by another | 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, g(x) ≠ 0 and r(x) is remainder. |
Types of Polynomials: Here are some important concepts and properties are mentioned in the below table for each type of polynomials-
Types of Polynomials | General Form | Zeroes | Formation of Polynomial | Relationship Between Zeroes and Coefficients |
---|---|---|---|---|
Linear | ax+b | 1 | f(x)=a (x−α) | α=−b/a |
Quadratic | ax^{2}+bx+c | 2 | f(x)=a (x−α)(x−β) | Sum of zeroes α+β=−b/a ; Product of zeroes, αβ= c/a |
Cubic | ax^{3}+bx^{2}+cx+d | 3 | f(x)=a (x−α)(x−β)(x−γ) | Sum of zeroes, α+β+γ=−b/a; Sum of product of zeroes taken two at a time, αβ+βγ+γα = c/a; Product of zeroes, αβγ= −ad |
Quartic | ax^{4}+bx^{3}+cx^{2}+dx+e | 4 | f(x)=a(x−α)(x−β)(x−γ)(x−δ) | Relationships become more complex; involves sums and products of zeroes in various combinations. |
Algebraic Polynomial Identities
- (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
- (a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)
- (a – b)^{3} = a^{3} – b^{3} – 3ab(a – b)
- (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
- 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}]
- (x + a) (x + b) (x + c) = x^{3} + (a + b +c)x^{2} + (ab + bc + ca)x + abc
- x^{3} + y^{3}= (x + y) (x^{2} – xy + y^{2})
- x^{3} – y^{3} = (x – y) (x^{2} + xy + y^{2})
- x^{2} + y^{2} + z^{2} -xy – yz – zx = ½ [(x-y)^{2} + (y-z)^{2} + (z-x)^{2}]
Learn More :
- Mathematical Expressions
- Types of Polynomials
- Polynomial equations
- Algebraic Polynomial Identities
- Algebraic Polynomial Identities
- Division Algorithm for Polynomials
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.
- Consistent system of linear equations: If a system of linear equations has at least one solution, it is considered to be consistent.
- 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:
a_{1}x + b_{1}y + c_{1 }= 0
a_{2}x + b_{2}y + c_{2 }= 0
- Graphically represented by two straight lines on the cartesian plane as discussed below:
Read More :
- Pair of Linear Equations in Two Variables
- Linear Equations
- Linear Pair of Equations in Two Variables Graph Method
Chapter 4 Quadratic Equations
Concept | Description |
---|---|
Quadratic Equation | A polynomial equation of degree two in one variable, typically written as f(x) = ax² + bx + c, where ‘a,’ ‘b,’ and ‘c’ are real numbers, and ‘a’ is not equal to zero. |
Roots of Quadratic Equation | The values of ‘x’ that satisfy the quadratic equation f(x) = 0 are the roots (α, β) of the equation. Quadratic equations always have two roots. |
Quadratic Formula | The formula to find the roots (α, β) of a quadratic equation is given by: (α, β) = [-b ± √(b² – 4ac)] / (2a), where ‘a,’ ‘b,’ and ‘c’ are coefficients of the equation. |
Discriminant | The discriminant ‘D’ of a quadratic equation is given by D = b² – 4ac. It determines the nature of the roots of the equation. |
Nature of Roots | Depending on the value of the discriminant ‘D,’ the nature of the roots can be categorized as follows: |
– D > 0: Real and distinct roots (unequal). | |
– D = 0: Real and equal roots (coincident). | |
– D < 0: Imaginary roots (unequal, in the form of complex numbers). | |
Sum and Product of Roots | The sum of the roots (α + β) is equal to -b/a, and the product of the roots (αβ) is equal to c/a. |
Quadratic Equation in Root Form | A quadratic equation can be expressed in the form of its roots as x² – (α + β)x + (αβ) = 0. |
Common Roots of Quadratic Equations | Two quadratic equations have one common root if (b₁c₂ – b₂c₁) / (c₁a₂ – c₂a₁) = (c₁a₂ – c₂a₁) / (a₁b₂ – a₂b₁). |
Both equations have both roots in common if a₁/a₂ = b₁/b₂ = c₁/c₂. | |
Maximum and Minimum Values | For a quadratic equation ax² + bx + c = 0: |
Roots of Cubic Equation | – If ‘a’ is greater than zero (a > 0), it has a minimum value at x = -b/(2a). |
– If ‘a’ is less than zero (a < 0), it has a maximum value at x = -b/(2a). | |
If α, β, γ are roots of the cubic equation ax³ + bx² + cx + d = 0, then: | |
– α + β + γ = -b/a | |
– αβ + βγ + λα = c/a | |
– αβγ = -d/a |
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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.
Concept | Description |
---|---|
Arithmetic Progressions (AP) | A sequence of terms where the difference between consecutive terms is constant. |
Common Difference | The constant difference between any two consecutive terms in an AP. It is denoted as ‘d’. d=a_{2}– a_{1} = a_{3 }– a_{2} = … |
nth Term of AP | a_{n} = a + (n – 1) d,, where ‘a’ is the first term, ‘n’ is the term number, and ‘d’ is the common difference. |
Sum of nth Terms of AP | S_{n}= n/2 [2a + (n – 1)d], where ‘a’ is the first term, ‘n’ is the number of terms, and ‘d’ is the common difference. |
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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:
Chapter 6: Triangles | |
---|---|
Concept | Description |
Similar Triangles | Triangles with equal corresponding angles and proportional corresponding sides. |
Equiangular Triangles | Triangles with all corresponding angles equal. The ratio of any two corresponding sides is constant. |
Criteria for Triangle Similarity | |
Angle-Angle-Angle (AAA) Similarity | Two triangles are similar if their corresponding angles are equal. |
Side-Angle-Side (SAS) Similarity | Two triangles are similar if two sides are in proportion and the included angles are equal. |
Side-Side-Side (SSS) Similarity | Two triangles are similar if all three corresponding sides are in proportion. |
Basic Proportionality Theorem | If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides proportionally. |
Converse of Basic Proportionality Theorem | If in two triangles, corresponding angles are equal, then their corresponding sides are proportional and the triangles are similar. |
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Chapter 7 Coordinate Geometry
Coordinate geometry 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:
Formulas Related to Coordinate Geometry | ||
---|---|---|
Description | Formula | |
Distance Formula | Distance between two points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) | AB= √[(x_{2 }− x_{1})^{2 }+ (y_{2 }− y_{1})^{2}] |
Section Formula | Coordinates of a point P dividing line AB in ratio m : n | P={[(mx_{2 }+ nx_{1}) / (m + n)] , [(my_{2 }+ ny_{1}) / (m + n)]} |
Midpoint Formula | Coordinates of the midpoint of line AB | P = {(x_{1 }+ x_{2})/ 2, (y_{1}+y_{2}) / 2} |
Area of a Triangle | Area of triangle formed by points A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) | (∆ABC = ½ |x_{1}(y_{2 }− y_{3}) + x_{2}(y_{3 }– y_{1}) + x_{3}(y_{1 }– y_{2})| |
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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:
Category | Formula/Identity | Description/Equivalent |
---|---|---|
Arc Length in a Circle | l =r × θ | l is arc length, r is radius, θ is angle in radians |
Radian and Degree Conversion | Radian Measure = π/180 × Degree Measure | Conversion from degrees to radians |
Degree Measure= 180/π × Radian Measure | Conversion from radians to degrees |
Trigonometric Ratios
Trigonometric Ratio | Formula | Description |
---|---|---|
sin θ | P / H | Perpendicular (P) / Hypotenuse (H) |
cos θ | B / H | Base (B) / Hypotenuse (H) |
tan θ | P / B | Perpendicular (P) / Base (B) |
cosec θ | H / P | Hypotenuse (H) / Perpendicular (P) |
sec θ | H / B | Hypotenuse (H) / Base (B) |
cot θ | B / P | Base (B) / Perpendicular (P) |
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Reciprocal of Trigonometric Ratios
Reciprocal Ratio | Formula | Equivalent to |
---|---|---|
sin θ | 1 / (cosec θ) | Reciprocal of cosecant |
cosec θ | 1 / (sin θ) | Reciprocal of sine |
cos θ | 1 / (sec θ) | Reciprocal of secant |
sec θ | 1 / (cos θ) | Reciprocal of cosine |
tan θ | 1 / (cot θ) | Reciprocal of cotangent |
cot θ | 1 / (tan θ) | Reciprocal of tangent |
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Trigonometric Identities
Identity | Formula |
---|---|
Pythagorean Identity | sin^{2} θ + cos^{2} θ = 1 ⇒ sin^{2} θ = 1 – cos^{2} θ ⇒ cos^{2} θ = 1 – sin^{2} θ |
Cosecant-Cotangent Identity | cosec^{2} θ – cot^{2} θ = 1 ⇒ sin^{2} θ = 1 – cos^{2} θ ⇒ cos^{2} θ = 1 – sin^{2} θ |
Secant-Tangent Identity | sec^{2} θ – tan^{2} θ = 1 ⇒ sec^{2} θ = 1 + tan^{2} θ ⇒ tan^{2} θ = sec^{2} θ – 1 |
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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:
Important Concepts in Chapter 9 Trigonometry | |
---|---|
Line of Sight | The line formed by our vision as it passes through an item when we look at it. |
Horizontal Line | A line representing the distance between the observer and the object, parallel to the horizon. |
Angle of Elevation | The angle formed above the horizontal line by the line of sight when an observer looks up at an object. |
Angle of Depression | The angle formed below the horizontal line by the line of sight when an observer looks down at an object. |
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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.
Let’s learn some important concepts discussed in Chapter 10 Circles of your NCERT textbook.
Concept | Description |
---|---|
Circle | A circle is a closed figure consisting of all points in a plane that are equidistant from a fixed point called the center. |
Radius | The radius of a circle is the distance from the center to any point on the circle’s circumference. |
Diameter | The diameter of a circle is a line segment that passes through the center and has endpoints on the circle’s circumference. It is twice the length of the radius. |
Chord | A chord is a line segment with both endpoints on the circle’s circumference. A diameter is a special type of chord that passes through the center. |
Arc | An arc is a part of the circumference of a circle, typically measured in degrees. A semicircle is an arc that measures 180 degrees. |
Sector | A sector is a region enclosed by two radii of a circle and an arc between them. Sectors can be measured in degrees or radians. |
Segment | A segment is a region enclosed by a chord and the arc subtended by the chord. |
Circumference | The circumference of a circle is the total length around its boundary. It is calculated using the formula: Circumference = 2πr, where ‘r’ is the radius. |
Area of a Circle | The area of a circle is the total space enclosed by its boundary. It is calculated using the formula: Area = πr², where ‘r’ is the radius. |
Central Angle | A central angle is an angle whose vertex is at the center of the circle, and its sides pass through two points on the circle’s circumference. |
Inscribed Angle | An inscribed angle is an angle formed by two chords in a circle with its vertex on the circle’s circumference. |
Tangent Line | A tangent line to a circle is a straight line that touches the circle at only one point, known as the point of tangency. |
Secant Line | A secant line is a straight line that intersects a circle at two distinct points. |
Concentric Circles | Concentric circles are circles that share the same center but have different radii. |
Circumcircle and Incircle | The circumcircle is a circle that passes through all the vertices of a polygon, while the incircle is a circle that is inscribed inside the polygon. |
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 :
- Determination of a Point Dividing a given Line Segment, Internally in the given Ratio M : N
- Construction of a Tangent at a Point on a Circle to the Circle when its Centre is Known
- Construction of a Tangent at a Point on a Circle to the Circle when its Centre is not Known
- Construction of a Tangents from an External Point to a Circle when its Centre is Known
- Construction of a Tangents from an External Point to a Circle when its Centre is not Known
- Construction of a Triangle Similar to a given Triangle as per given Scale Factor m/n, m<n.
- Construction of a Triangle Similar to a given Triangle as per given Scale Factor m/n, m > n.
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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.
Formulas of Areas Related to Circles | ||
---|---|---|
Concept | Description | Formula |
Area of a Circle | The space enclosed by the circle’s circumference | Area=πr^{2} |
Circumference of a Circle | The perimeter or boundary line of a circle | Circumference=2πr or πd |
Area of a Sector | The area of a ‘pie-slice’ part of a circle | Area of Sector= (θ/360) × πr^{2} (θ in degrees) |
Length of an Arc | The length of the curved line forming the sector | Length of Arc= (θ/360) × 2πr (θ in degrees) |
Area of a Segment | Area of a sector minus the area of the triangle formed by the sector | Area of Segment = Area of Sector – Area of Triangle |
- r is the radius of the circle.
- d is the diameter of the circle.
- θ is the angle of the sector or segment in degrees.
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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).
Formulas Related to Surface Areas and Volumes | |||
---|---|---|---|
Geometrical Figure | Total Surface Area (TSA) | Lateral/Curved Surface Area (CSA/LSA) | Volume |
Cuboid | 2(lb + bh + hl) | 2h(l + b) | l × b × h |
Cube | 6a² | 4a² | a³ |
Right Circular Cylinder | 2πr(h + r) | 2πrh | πr²h |
Right Circular Cone | πr(l + r) | πrl | 1/3πr²h |
Sphere | 4πr² | 2πr² | 4/3πr³ |
Right Pyramid | LSA + Area of the base | ½ × p × l | 1/3 × Area of the base × h |
Prism | LSA × 2B | p × h | B × h |
Hemisphere | 3πr² | 2πr² | 2/3πr³ |
- l = length, b = breadth, h = height, r = radius, a = side, p = perimeter of the base, B = area of the base.
- TSA includes all surfaces of the figure, CSA/LSA includes only the curved or lateral surfaces, and
- Volume measures the space occupied by the figure.
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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:
Statistical Measure | Method/Description | Formula |
---|---|---|
Mean | Direct method | 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 | Middlemost Term | For even number of observations: Middle term For odd number of observations: (n+1/2) th term |
Mode | Frequency Distribution | where l = lower limit of the modal class, f_{1} =frequency of the modal class, f_{0} = frequency of the preceding class of the modal class, f_{2} = frequency of the succeeding class of the modal class, h is the size of the class interval. |
Chapter 15 Probability
Probability denotes the likelihood of something happening. Its value is expressed from 0 to 1.
Let’s discuss some important Probability formulas in the Class 10 curriculum:
Type of Probability | Description | Formula |
---|---|---|
Empirical Probability | Probability based on actual experiments or observations. | Empirical Probability = Number of Trials with expected outcome / Total Number of Trials |
Theoretical Probability | Probability based on theoretical reasoning rather than actual experiments. | Theoretical Probability = Number of favorable outcomes to E / Total Number of possible outcomes |
Related :
- CBSE Board Exams
- CBSE Class 10 Syllabus 2023-24
- CBSE Maths Class 10 Notes
- Number system
- Polynomials
- Trigonometry
- Algebra
- Mensuration
- Probability
- Statistics