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Maths Formulas for Class 10 CBSE

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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.

Maths Formulas for Class 10 (Chapterwise)

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:

ConceptsDescriptionExamples/Formula
Natural NumbersCounting numbers starting from 1.N = {1, 2, 3, 4, 5, …}
Whole NumbersCounting numbers including zero.W = {0, 1, 2, 3, 4, 5, …}
IntegersAll positive numbers, zero, and negative numbers.…, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …
Positive IntegersAll positive whole numbers.Z+ = 1, 2, 3, 4, 5, …
Negative IntegersAll negative whole numbers.Z– = -1, -2, -3, -4, -5, …
Rational NumberNumbers expressed as a fraction where both numerator and denominator are integers and the denominator is not zero.Examples: 3/7, -5/4
Irrational NumberNumbers that cannot be expressed as a simple fraction.Examples: π, √5
Real NumbersAll numbers that can be found on the number line, including rational and irrational numbers.Includes Natural, Whole, Integers, Rational, Irrational
Euclid’s Division AlgorithmA method for finding the HCF of two numbers.a = bq + r, where 0 ≤ r < b
Fundamental Theorem of ArithmeticStates that every composite number can be expressed as a product of prime numbers.Composite Numbers = Product of Primes
HCF and LCM by Prime FactorizationMethod 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

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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:

CategoryDescriptionFormula/Identity
General Polynomial FormulaStandard form of a polynomialF (x) = anxn + bxn-1 + an-2xn-2 + …….. + rx + s
Special Case: Natural Number nDifference of powers formulaan – bn = (a – b)(an-1 + an-2b +…+ bn-2a + bn-1)
Special Case: Even n (n = 2a)Sum of even powers formulaxn + yn = (x + y)(xn-1 – xn-2y +…+ yn-2x – yn-1)
Special Case: Odd Number nSum of odd powers formulaxn + yn = (x + y)(xn-1 – xn-2y +…- yn-2x + yn-1)
Division Algorithm for PolynomialsDivision 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 PolynomialsGeneral FormZeroesFormation of PolynomialRelationship Between Zeroes and Coefficients
Linearax+b1f(x)=a (xα)α=−b/a
Quadraticax2+bx+c2f(x)=a (xα)(xβ)Sum of zeroes α+β=−b/a ​; Product of zeroes, αβ= c/a
Cubicax3+bx2+cx+d3f(x)=a (xα)(xβ)(xγ)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
Quarticax4+bx3+cx2+dx+e4f(x)=a(xα)(xβ)(xγ)(xδ)Relationships become more complex; involves sums and products of zeroes in various combinations.

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]

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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 Variableax + b=0Where a ≠ 0 and a & b are real numbersOne Solution 
2.Linear Equation in Two Variablesax + by + c = 0Where a ≠ 0 & b ≠ 0 and a, b & c are real numbersInfinite Solutions possible
3.Linear Equation in Three Variablesax + by + cz + d = 0Where a ≠ 0, b ≠ 0, c ≠ 0 and a, b, c, d are real numbersInfinite 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:

Graphical representation of linear equations

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Chapter 4 Quadratic Equations

ConceptDescription
Quadratic EquationA 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 EquationThe 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 FormulaThe 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.
DiscriminantThe discriminant ‘D’ of a quadratic equation is given by D = b² – 4ac. It determines the nature of the roots of the equation.
Nature of RootsDepending 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 RootsThe 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 FormA quadratic equation can be expressed in the form of its roots as x² – (α + β)x + (αβ) = 0.
Common Roots of Quadratic EquationsTwo 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 ValuesFor 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.

ConceptDescription
Arithmetic Progressions (AP)A sequence of terms where the difference between consecutive terms is constant.
Common DifferenceThe constant difference between any two consecutive terms in an AP. It is denoted as ‘d’. d=a2– a1 = a3 – a2 = …
nth Term of APan = 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 Sn= 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 

ConceptDescription
Similar TrianglesTriangles with equal corresponding angles and proportional corresponding sides.
Equiangular TrianglesTriangles with all corresponding angles equal. The ratio of any two corresponding sides is constant.
Criteria for Triangle Similarity
Angle-Angle-Angle (AAA) SimilarityTwo triangles are similar if their corresponding angles are equal.
Side-Angle-Side (SAS) SimilarityTwo triangles are similar if two sides are in proportion and the included angles are equal.
Side-Side-Side (SSS) SimilarityTwo triangles are similar if all three corresponding sides are in proportion.
Basic Proportionality TheoremIf 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 TheoremIf 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

DescriptionFormula
Distance FormulaDistance between two points A(x1, y1) and B(x2, y2)AB= √[(x− x1)+ (y− y1)2]
Section FormulaCoordinates of a point P dividing line AB in ratio m : nP={[(mx2 + nx1) / (m + n)] , [(my2 + ny1) / (m + n)]}
Midpoint FormulaCoordinates of the midpoint of line ABP = {(x1 + x2)/ 2, (y1+y2) / 2}
Area of a TriangleArea of triangle formed by points A(x1, y1), B(x2, y2) and C(x3, y3)(∆ABC = ½ |x1(y2 − y3) + x2(y3 – y1) + x3(y1 – y2)|

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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:

CategoryFormula/IdentityDescription/Equivalent
Arc Length in a Circlel =r × θl is arc length, r is radius, θ is angle in radians
Radian and Degree ConversionRadian Measure = π/180 × Degree MeasureConversion from degrees to radians
 Degree Measure= 180/π × Radian MeasureConversion from radians to degrees

Trigonometric Ratios

Trigonometric RatioFormulaDescription
sin θP / HPerpendicular (P) / Hypotenuse (H)
cos θB / HBase (B) / Hypotenuse (H)
tan θP / BPerpendicular (P) / Base (B)
cosec θH / PHypotenuse (H) / Perpendicular (P)
sec θH / BHypotenuse (H) / Base (B)
cot θB / PBase (B) / Perpendicular (P)

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Reciprocal of Trigonometric Ratios

Reciprocal RatioFormulaEquivalent to
sin θ1 / (cosec θ)Reciprocal of cosecant
cosec θ1 / (sin θ)Reciprocal of sine
cos θ1 / (sec θ)Reciprocal of secant
sec θ1 / (cos θ)Reciprocal of cosine
tan θ1 / (cot θ)Reciprocal of cotangent
cot θ1 / (tan θ)Reciprocal of tangent

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Trigonometric Identities

IdentityFormula
Pythagorean Identitysin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 – cos2 θ ⇒ cos2 θ = 1 – sin2 θ
Cosecant-Cotangent Identitycosec2 θ – cot2 θ = 1 ⇒ sin2 θ = 1 – cos2 θ ⇒ cos2 θ = 1 – sin2 θ
Secant-Tangent Identitysec2 θ – tan2 θ = 1 ⇒ sec2 θ = 1 + tan2 θ ⇒ tan2 θ = sec2 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 SightThe line formed by our vision as it passes through an item when we look at it.
Horizontal LineA line representing the distance between the observer and the object, parallel to the horizon.
Angle of ElevationThe angle formed above the horizontal line by the line of sight when an observer looks up at an object.
Angle of DepressionThe 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.

ConceptDescription
CircleA circle is a closed figure consisting of all points in a plane that are equidistant from a fixed point called the center.
RadiusThe radius of a circle is the distance from the center to any point on the circle’s circumference.
DiameterThe 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.
ChordA 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.
ArcAn arc is a part of the circumference of a circle, typically measured in degrees. A semicircle is an arc that measures 180 degrees.
SectorA sector is a region enclosed by two radii of a circle and an arc between them. Sectors can be measured in degrees or radians.
SegmentA segment is a region enclosed by a chord and the arc subtended by the chord.
CircumferenceThe 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 CircleThe 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 AngleA 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 AngleAn inscribed angle is an angle formed by two chords in a circle with its vertex on the circle’s circumference.
Tangent LineA tangent line to a circle is a straight line that touches the circle at only one point, known as the point of tangency.
Secant LineA secant line is a straight line that intersects a circle at two distinct points.
Concentric CirclesConcentric circles are circles that share the same center but have different radii.
Circumcircle and IncircleThe 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 :

  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.

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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

ConceptDescriptionFormula
Area of a CircleThe space enclosed by the circle’s circumferenceArea=πr2
Circumference of a CircleThe perimeter or boundary line of a circleCircumference=2πr or πd
Area of a SectorThe area of a ‘pie-slice’ part of a circleArea of Sector= (θ/360​) × πr2 (θ in degrees)
Length of an ArcThe length of the curved line forming the sectorLength of Arc= (θ/360​) ​× 2πr (θ in degrees)
Area of a SegmentArea of a sector minus the area of the triangle formed by the sectorArea 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 FigureTotal Surface Area (TSA)Lateral/Curved Surface Area (CSA/LSA)Volume
Cuboid2(lb + bh + hl)2h(l + b)l × b × h
Cube6a²4a²
Right Circular Cylinder2πr(h + r)2πrhπr²h
Right Circular Coneπr(l + r)πrl1/3πr²h
Sphere4πr²2πr²4/3πr³
Right PyramidLSA + Area of the base½ × p × l1/3 × Area of the base × h
PrismLSA × 2Bp × hB × h
Hemisphere3πr²2πr²2/3πr³
  • 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 MeasureMethod/DescriptionFormula
Mean Direct methodX = ∑fi xi / ∑fi​​
Assumed Mean Method X = a + ∑fi di / ∑fi
,(where di = xi – a)
Step Deviation Method: X = a + ∑fi ui / ∑fi × h
MedianMiddlemost TermFor even number of observations: Middle term
For odd number of observations: (n+1/2) th term
ModeFrequency Distribution \text{Mode} = 1 + \left[\dfrac{f_1-f_0}{2f_1-f_0-f_2}\right]\times h


where l = lower limit of the modal class,

f1 =frequency of the modal class,

f0 = frequency of the preceding class of the modal class,

f2 = 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 ProbabilityDescriptionFormula
Empirical ProbabilityProbability based on actual experiments or observations.Empirical Probability = Number of Trials with expected outcome / Total Number of Trials
Theoretical ProbabilityProbability based on theoretical reasoning rather than actual experiments.Theoretical Probability = Number of favorable outcomes to E / Total Number of possible outcomes

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Last Updated : 28 Feb, 2024
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