Class 10 – All Maths Formulas – Complete List

Sunil Kumar — Thu, 03 Jul 2025 06:49:25 +0000

Perfect for quick revision and last-minute prep!

This is an interactive, colorful and extremely engaging 💕 comprehensive ✅ list of formulas of Class 10 Mathematics. This is a one-stop 🎯 solution for the students to quickly review / recall all formulas that would be very helpful especially for last minute revision before their math exam.

Click a link displayed below to read the corresponding chapter’s formulas.

Ch 01 – Real Numbers
Ch 02 – Polynomials
Ch 03 – Pair of Linear Equations in Two Variables
Ch 04 – Quadratic Equations
Ch 05 – Arithmetic Progressions
Ch 06 – Triangles
Ch 07 – Coordinate Geometry
Ch 08 – Introduction to Trigonometry
Ch 09 – Some Applications of Trigonometry
Ch 10 – Circles
Ch 11 – Areas Related to Circles
Ch 12 – Surface Areas and Volumes
Ch 13 – Statistics
Ch 14 – Probability

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Chapter 1 - Real Numbers

Square root of any prime number is irrational.

E.g. – \( \begin{aligned} &\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7} \end{aligned}\)

Click here to watch proof of √3

\( \begin{aligned} LCM (a, b) \times HCF (a, b) = a \times b \end{aligned}\)

Fundamental Theorem of Arithmetic

Every composite number can be factorised as a product of primes, and this factorisation is unique.

Click here to watch its proof

Theorem

If p is a prime and p divides a², then p divides a, where a is a positive integer.

Click here to watch its proof

Chapter 2 - Polynomials

Linear Polynomial

Degree = 1

No. of real zeros: At most 1

E.g: x

Quadratic Polynomial

Degree = 2

No. of real zeros: At most 2

E.g: x²

Cubic Polynomial

Degree = 3

No. of real zeros: At most 3

E.g: x³

Monomial

No: of terms = 1

E.g: 2x

Binomial

No: of terms = 2

E.g: 2x + 3

Trinomial

No: of terms = 3

E.g: 2x² + 3x + 2

Quadratic Polynomials

Standard form of Quadratic Polynomial

\( \begin{aligned} &ax^2+bx+c, \ \ a\ne0\\ \end{aligned}\)

Zeros of Quadratic Polynomial

If α is a zero of polynomial

\( \begin{aligned} &p(x) = ax^2+bx+c, \ \ a\ne0\\ &then \ \ p(α) = 0 \end{aligned}\)

Sum of Zeros

\( \begin{aligned} &α + β = -\frac{b}{a} \end{aligned}\)

Product of Zeros

\( \begin{aligned} &α \times β = \frac{c}{a} \end{aligned}\)

Add Graphs – Later XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

Chapter 3 - Pair of Linear Equations in Two Variables

Standard form of Linear Equation in Two Variables

\( \begin{aligned} &ax+by+c =0,\\ &\text{where a and b are not both 0.} \end{aligned}\)

\( \begin{aligned} &ax+by+c =0, \ \ where \ \ a^2+b^2\ne 0 \end{aligned}\)

Standard form of Pair of Linear Equation in Two Variables

\( \begin{aligned} &a_{1}x+b_{1}y+c_{1} =0, \ \ where \ \ a_{1}^2+b_{1}^2\ne 0 \\&a_{2}x+b_{2}y+c_{2} =0, \ \ where \ \ a_{2}^2+b_{2}^2\ne 0 \end{aligned}\)

Solution of Pair of LE in 2 Variables

Intersecting Lines
➤ Consistent Pair of LE

➺ Lines Intersect at 1 point
🎯 No. of Solutions = 1

\( \begin{aligned} &\frac{a_{1}}{a_{2}} \ne \frac{b_{1}}{b_{2}} \end{aligned}\)

Coincident Lines

➤ Consistent Pair of LE
➺ Lines overlap each other

🎯 No. of Solutions = Infinite

\( \begin{aligned} &\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \end{aligned}\)

Parallel Lines
➤ inconsistent Pair of LE

➺ No common point

🎯 No. of Solutions = 0

\( \begin{aligned} &\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \ne \frac{c_{1}}{c_{2}} \end{aligned}\)

Click here to watch Introduction to pair of Linear equations

Chapter 4 - Quadratic Equations

Standard form of Quadratic Equation

\( \begin{aligned} &ax^2+bx+c=0, \ \ a\ne0\\ \end{aligned}\)

Roots of Quadratic Equation

Zeros of quadratic polynomial \( \begin{aligned} &ax^2+bx+c \end{aligned}\)

➤ Roots of the quadratic equation \( \begin{aligned} &ax^2+bx+c=0 \end{aligned}\)

are the same.

Sum of Roots

\( \begin{aligned} &α + β = -\frac{b}{a} \end{aligned}\)

Product of Roots

\( \begin{aligned} &α \times β = \frac{c}{a} \end{aligned}\)

Nature of Roots

Discriminant

\( \begin{aligned} &D = b^2-4ac \end{aligned}\)

D > 0 2 Distinct, Real Roots

D = 0

2 Equal, Real Roots

(coincident roots)

D < 0

No Real Roots

➤ Roots of the QE are

\( \begin{aligned} &x=\frac{-b ± \sqrt{D}}{2a} \end{aligned}\)

Click here to watch the Derivation of Quadratic Formula

Chapter 5 - Arithmetic Progression

General Form of AP

a, a + d, a + 2d, a + 3d, . . .

First Term: a

Common Difference: d

➺ d can be +ve, -ve or 0

\( \begin{aligned} &a_{2} -a_{1}=a_{3} -a_{2}=a_{4} -a_{3}=…=a_{n} -a_{n-1}=d \end{aligned}\)

nth Term of AP

\( \begin{aligned} &a_{n} = a +(n-1)d \end{aligned}\)

Sum of first n Terms

\( \begin{aligned} &S_{n} =\frac{n}{2}[2a+(n-1)d] \end{aligned}\)

Sum of first n Terms

\( \begin{aligned} &S_{n} =\frac{n}{2}(a+l) \\&\text{where }l \text{ is the last term}\end{aligned}\)

📌 Super important 🔥

Total terms in AP: 25

11th term from the end = 25 – 11 + 1 = 15th Term

If a, b, c are in AP, then

\( \begin{aligned} &b=\frac{a+c}{2} \end{aligned}\)

Chapter 6 - Triangles

Similar Figures

➺ Same shape

➺ Not necessarily the same size

🦆 🦆 🔺🔺

➺ All Circles are similar 🔵

➺ All Squares are similar 🟩

➺ All Equilateral Triangles are similar 🔺

➺ All Congruent figures are similar ✅

❌ All similar figures are not congruent

🔺 🔺 Similar Triangles 🔺🔺

Two Triangles ABC and DEF are similar if

➺ their corresponding angles are equal

\( \begin{aligned} &\angle A = \angle D, \ \ \angle B = \angle E, \ \ \angle C = \angle F\end{aligned}\)

➺ & their corresponding sides are in the same ratio (proportion).

\( \begin{aligned} &\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}\end{aligned}\)

Basic Proportionality Theorem (Thales Theorem)

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Click here to watch its proof

Converse of Basic Proportionality Theorem

If a line divides any two sides of a triangle in the same ratio then the lines is parallel to the third side.

Click here to watch its proof

Triangles Similarity Criteria

SSS

If Corresponding Sides are in the same ratio, then the ▲s are similar.

\( \begin{aligned} &\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}\end{aligned}\)

AAA

If Corresponding Angles are equal, then the ▲s are similar.

\( \begin{aligned} \angle A &= \angle D,\\ \angle B &= \angle E,\\ \ \ \angle C &= \angle F\end{aligned}\)

SAS

If one angle of a triangle is equal to one angle of another triangle and the sides including this angle are in the same ratio, then the ▲s are similar. For e.g.

\( \begin{aligned} \frac{AB}{DE}&=\frac{BC}{EF} \ \ \\ \angle B &= \angle E \end{aligned}\)

RHS

If in two right angled ▲s, hypotenuse and one side of a ▲ are proportional to hypotenuse and one side of the other ▲, then the ▲s are similar.

If two angles of one ▲ are respectively equal to the two angles of the other ▲, then the two ▲s are similar.

\( \begin{aligned} \angle A &= \angle D,\\ \angle B &= \angle E\end{aligned}\)

This case is same as AAA because 3rd angle will be equal as well.

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Chapter 7 - Coordinate Geometry

Abscissa / x-coordinate

The distance of a point from the y-axis

Ordinate / y-coordinate

The distance of a point from the x-axis

Distance Formula

Distance between points A (x₁, y₁) and B (x₂, y₂)

Distance from origin

Distance of point A (x, y) from origin O (0, 0)

Section Formula

Mid-Point Formula

Chapter 8 - Introduction to Trigonometry

For Angle A

BC – Perpendicular

AB – Base

AC – Hypotenuse

For Angle C

AB – Perpendicular

BC – Base

AC – Hypotenuse

Trigonometric Ratios

\(\begin{align*}&\color{red}{sin} \ \ \color{green}{cos} \ \ \color{blue}{tan}\\ &\boxed{\color{red}{\frac{P}{H}}}\boxed{\color{green}{\frac{B}{H}}}\boxed{\color{blue}{\frac{P}{B}}}\end{align*}\)

P B P – Pandit Badri Prasad

H H B – Har Har Bole

\( \begin{aligned} sinθ = \frac{P}{H}\end{aligned}\)

\( \begin{aligned} &cosθ = \frac{B}{H}\end{aligned}\)

\( \begin{aligned} &tanθ = \frac{P}{B}\end{aligned}\)

\( \begin{aligned} &cosecθ = \frac{H}{P}\end{aligned}\)

\( \begin{aligned} &secθ = \frac{H}{B}\end{aligned}\)

\( \begin{aligned} &cotθ = \frac{B}{P}\end{aligned}\)

Imp Relationships b/w ratios

\( \begin{aligned} &sin θ= \frac{1}{cosecθ}\end{aligned}\)

\( \begin{aligned} &cos θ= \frac{1}{secθ}\end{aligned}\)

\( \begin{aligned} &tan θ= \frac{1}{cotθ}\end{aligned}\)

Trigonometric ratios of standard angles

∠A	0°	30°	45°	60°	90°
sin A	0	\(\begin{aligned} \frac {1}{2}\end{aligned} \)	\(\begin{aligned} \frac {1}{\sqrt{2}}\end{aligned} \)	\(\begin{aligned} \frac {\sqrt{3}}{2}\end{aligned} \)	1
cos A	1	\(\begin{aligned} \frac {\sqrt{3}}{2}\end{aligned} \)	\(\begin{aligned} \frac {1}{\sqrt{2}}\end{aligned} \)	\(\begin{aligned} \frac {1}{2}\end{aligned} \)	0
tan A	0	\(\begin{aligned} \frac {1}{\sqrt {3}}\end{aligned} \)	1	\(\begin{aligned} \sqrt {3} \end{aligned} \)	N.D.
cosec A	N.D.	2	\(\begin{aligned} \sqrt {2} \end{aligned} \)	\(\begin{aligned} \frac {2}{\sqrt{3}}\end{aligned} \)	1
sec A	1	\(\begin{aligned} \frac {2}{\sqrt{3}}\end{aligned} \)	\(\begin{aligned} \sqrt{2}\end{aligned} \)	2	N.D.
cot A	N.D.	\(\begin{aligned} \sqrt{3}\end{aligned} \)	1	\(\begin{aligned} \frac {1}{\sqrt{3}}\end{aligned} \)	1

N.D. – Not Defined

Trigonometric Identities

\( \begin{aligned} &sin^2θ+cos^2θ =1\end{aligned}\)

\( \begin{aligned} &1+tan^2θ =sec^2θ\end{aligned}\)

\( \begin{aligned} &1+cot^2θ =cosec^2θ\end{aligned}\)

Chapter 9 - Some Applications of Trigonometry

This chapter is about the applications of the formulas that we learnt in the previous chapter. You Students are recommended to practice different types of questions to master this chapter.

Chapter 10 - Circles

Tangent

Line that intersects the circle at only one point

Important

➺ Diameter is the longest chord

➺ Tangent is perpendicular to the radius through the point of contact

➺ A circle has infinitely many tangents

Theorem

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Click here to watch its proof

Theorem

The lengths of tangents drawn from an external point to a circle are equal.

Click here to watch its proof

Chapter 11 - Areas Related to Circles

Area of Circle = πr²

Circumference of Circle = 2πr

Area of a Sector

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{\frac{θ}{360}} \color{black}\times \color{blue}{\pi r^2}}\end{aligned}\)

Length of Arc

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{\frac{θ}{360}} \color{black}\times \color{blue}{2\pi r}}\end{aligned}\)

Area of a Segment

Area of the corresponding sector

– Area of the corresponding triangle

Area of Major Sector = Area of Circle – Area of Minor Sector

Area of Major Segment = Area of Circle – Area of Minor Segment

Chapter 12 - Surface Areas and Volumes

HS – Hemisphere
SA – Surface Area
CSA – Curved Surface Area
TSA – Total Surface Area
LSA – Lateral Surface Area

Surface Areas

TSA of a Cube

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{6} \color{blue}{a^2}}\end{aligned}\)

LSA of a Cube

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{4} \color{blue}{a^2}}\end{aligned}\)

TSA of a Cuboid

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{2} \color{blue}{(lb + bh + hl)}}\end{aligned}\)

LSA of a Cuboid

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{2h} \color{blue}{(l + b)}}\end{aligned}\)

SA of a Sphere

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{4} \color{blue}{\pi r^2}}\end{aligned}\)

CSA of a Solid HS

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{2} \color{blue}{\pi r^2}}\end{aligned}\)

TSA of a Solid HS

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{2} \color{blue}{\pi r^2} + \color{magenta}{\pi r^2}} \end{aligned}\)

TSA of a Hollow HS

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{2} \color{blue}{\pi r^2} + \color{red}{2} \color{blue}{\pi r^2} } \end{aligned}\)

CSA of a Solid Cylinder

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{ \color{blue}{2 \pi r} \color{red}{h} } \end{aligned}\)

TSA of a Solid Cylinder

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{ \color{blue}{2 \pi r} \color{red}{h} + \color{red}{2}\color{blue}{\pi r^2} } \end{aligned}\)

CSA of a Hollow Cylinder

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{2} \color{blue}{ \pi r} \color{red}{h} + \color{red}{2} \color{blue}{ \pi r} \color{red}{h}} \end{aligned}\)

TSA of a Hollow Cylinder

with Base

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{ \color{red}{4} \color{blue}{\pi r} \color{red}{h} +\color{red}{2} \color{blue}{\pi r^2} } \end{aligned}\)

CSA of a Solid Cone

\(\begin{aligned}&\bbox[15px, border: 3px solid green;] {\color{blue}{\pi rl}} \end{aligned}\)

TSA of a Solid Cone

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{blue}{\pi rl} + \color{red}{\pi r^2}} \end{aligned}\)

CSA of a Hollow Cone

\(\begin{aligned}&\bbox[15px, border: 3px solid green;] {\color{red}{2} \color{blue}{\pi rl}} \end{aligned}\)

TSA of a Hollow Cone

\(\begin{aligned}&\bbox[15px, border: 3px solid green;] {\color{red}{2} \color{blue}{\pi rl}} \end{aligned}\)

Volumes

Volume of a Cube

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{blue}{a^3} } \end{aligned}\)

Volume of a Cuboid

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{blue}l \color{black} \times \color{green}b \color{black}\times \color{red}h} \end{aligned}\)

Volume of Sphere

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{\frac{4}{3}}\color{blue}{\pi r^3}} \end{aligned}\)

Volume of HS

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{\frac{2}{3}}\color{blue}{\pi r^3}} \end{aligned}\)

Volume of Cylinder

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{blue}{\pi r^2} \color{red}{h} } \end{aligned}\)

Volume of Cone

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\color{red}{\frac{1}{3}}\color{blue}{\pi r^2} \color{red}{h} } \end{aligned}\)

Chapter 13 - Statistics

Direct Method

\(\begin{aligned} &\bbox[15px, border: 3px solid green;]{\overline{x} = \frac{\sum {f_ix_i}}{\sum{f_i}}} \end{aligned}\)

Assumed Mean Method

\(\begin{aligned} &\bbox[15px, border: 3px solid green;]{\overline{x} = a +\frac{\sum {f_id_i}}{\sum{f_i}}} \end{aligned}\)

Step Deviation Method

\(\begin{aligned} &\bbox[15px, border: 3px solid green;]{\overline{x} = a +\frac{\sum {f_iu_i}}{\sum{f_i}}\times h }\end{aligned}\)

Mode of Grouped Data

\(\begin{aligned} &\bbox[15px, border: 3px solid green;]{l +\frac{f_1-f_0}{2f_1-f_0-f_2}\times h }\end{aligned}\)

Median of Grouped Data

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{l +\frac{\frac{n}{2}-cf}{f}\times h }\end{aligned}\)

Class Marks

\(\begin{aligned}&\bbox[15px, border: 3px solid green;]{\frac{\text{Upper Class Limit}- \text{Lower Class Limit}}{2}}\end{aligned}\)

Chapter 14 - Probability

Experiment: Throwing a dice 🎲

Outcomes: { 1, 2, 3, 4, 5, 6 }

Total no. of outcomes = 6

P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6

➺ Outcomes are Equally Likely

Examples of Events

E1: Getting 3

E2: Getting a number > 4

E3: Getting a number < 7

E4: Getting a prime number

E5: Getting a number < 1

E – An Event 0 ≤ P(E) ≤ 1

\( \begin{aligned} &P(E) = \frac{\text{No. of favourable outcomes}}{\text{Total no. of outcomes}} \end{aligned}\)

Probability of Events (E1 – E5)

P(E1)

E1: Getting 3

Favourable Outcome: { 3 }

No. of favourable outcomes = 1

\( \begin{aligned} P(E1)=\frac{1}{6} \end{aligned}\)

P(E2)

E2: Getting a number > 4

Numbers > 4: { 5, 6 }

No. of favourable outcomes = 2

\( \begin{aligned} P(E2)=\frac{2}{6} =\frac{1}{3} \end{aligned}\)

P(E3)

E3: Getting a number < 7

Numbers < 7: { 1, 2, 3, 4, 5, 6 }

No. of favourable outcomes = 6

\( \begin{aligned} P(E3)=\frac{6}{6} =1 \end{aligned}\)

Sure Event

P(E4)

E4: Getting a prime number

Prime numbers: { 2, 3, 5 }

No. of favourable outcomes = 3

\( \begin{aligned} P(E4)=\frac{3}{6} =\frac{1}{2} \end{aligned}\)

P(E5)

E5: Getting a number < 1

Number < 1: None

No. of favourable outcomes = 0

\( \begin{aligned} P(E5)=\frac{0}{6} =0 \end{aligned}\)

Impossible Event

Sure Event 👍

E = It will rain OR it will not rain today

P(E) = 1

Impossible Event 👎

E = It will rain AND it will not rain today

P(E) = 0

Complementary Events 🎭

E = It will rain today

not E = It will not rain today

P(E) + P(not E) = 1

Equally Likely & Not Equally Likely Outcomes

Randomly take a ball out of a bag of same sized 4 red and 2 blue balls 🔴 🔴 🔴 🔴 🔵 🔵

Experiment 1

Outcomes: Red Ball 🔴, Blue Ball 🔵

Total possible outcomes = 2

P(Red Ball) = 4/6 = 2/3

P(Blue Ball) = 2/6 = 1/3

➺ Outcomes are Not Equally Likely

Experiment 2

Outcomes: A Ball of any color

(Color doesn’t matter)

Total possible outcomes = 6

P(Ball 1) = P(Ball 2) = P(Ball 3) = P(Ball 4) = P(Ball 5) = P(Ball 6) = 1/6

➺ Outcomes are Equally Likely

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class 10 Archives – Mathemafia

Class 10 – All Maths Formulas – Complete List

Perfect for quick revision and last-minute prep!

Mathemafia

Creative Hub of Mathematics

Chapter 1 - Real Numbers

Chapter 2 - Polynomials

Quadratic Polynomials

Chapter 3 - Pair of Linear Equations in Two Variables

Solution of Pair of LE in 2 Variables

Chapter 4 - Quadratic Equations

Nature of Roots

Chapter 5 - Arithmetic Progression

Chapter 6 - Triangles

Triangles Similarity Criteria

Mathemafia

Creative Hub of Mathematics

Chapter 7 - Coordinate Geometry

Distance Formula

Section Formula

Mid-Point Formula

Chapter 8 - Introduction to Trigonometry

Trigonometric Ratios

Trigonometric ratios of standard angles

Trigonometric Identities

Chapter 9 - Some Applications of Trigonometry

Chapter 10 - Circles

Chapter 11 - Areas Related to Circles

Chapter 12 - Surface Areas and Volumes

Surface Areas

Volumes

Chapter 13 - Statistics

Chapter 14 - Probability

Probability of Events (E1 – E5)

Equally Likely & Not Equally Likely Outcomes

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CBSE Board – Sample Papers & Previous Years Question Papers

CBSE Board - Sample Papers

CBSE Board - Previous Years' Question Papers

CBSE Board - Case Studies (Question Bank)