Complete Class 10 Mathematics Formulas – All Chapters (CBSE 2026-27)

This is the only Class 10 Maths formula guide you need for CBSE 2025–26 board exams. Every chapter is covered not just the formula list, but the context, application, and exam strategy behind each formula. Bookmark this page and return to it every revision session.

Why Formula Command is Non-Negotiable in Class 10 Maths

Class 10 Mathematics is structured around approximately 80 core formulas spread across 15 chapters. Unlike subjects where marks come from memory, Maths rewards strategic formula application under timed conditions. A student who understands when to apply a formula not just what it says consistently outperforms a student who has merely memorised it.

CBSE board papers are designed to test three things simultaneously formula recall, contextual application, and accuracy under pressure. This means rote memorisation is necessary but not sufficient. Every formula in this guide is presented alongside its exam context the type of question it is most likely to appear in and the mistake that most commonly costs marks.

• Real Numbers
• Polynomials
• Linear Equations
• Quadratic Equations
• AP
• Triangles
• Coordinate Geometry
• Trigonometry
• Circles
• Surface Areas & Volumes
• Statistics
• Probability

Real Numbers

Real Numbers is typically the first chapter and sets the tone for the year. It introduces two foundational tools Euclid’s Division Algorithm for finding HCF, and the Fundamental Theorem of Arithmetic for prime factorisation. Questions from this chapter tend to be direct and procedural, making it a reliable source of early marks in the exam.

Euclid’s Division Algorithm

The algorithm expresses any integer as a combination of another integer, a quotient, and a remainder. It is the systematic procedure behind finding the Highest Common Factor (HCF) of two numbers.

Euclid’s Division Lemma a = bq + r, where 0 ≤ r < b

To find HCF(a, b), apply the lemma repeatedly: divide the larger number by the smaller, then replace the larger with the smaller and the smaller with the remainder. Continue until the remainder is zero. The last non-zero divisor is the HCF.

Exam tip: Always show every step of the division process when applying Euclid’s Algorithm each step carries method marks. Skipping to the answer directly risks losing 1–2 marks even if the final answer is correct.

HCF and LCM Formulas

Fundamental Theorem of Arithmetic

Every composite number greater than 1 can be expressed as a unique product of prime numbers, regardless of the order in which those primes are written. This theorem underpins prime factorisation and is used to prove the irrationality of numbers like √2, √3, and √5.

Application: To prove that a number like √2 is irrational, assume it is rational (p/q in lowest terms), then show that 2 divides both p and q — a contradiction. This is a standard 3-mark proof question in CBSE exams.

Polynomials

The Polynomials chapter tests your understanding of the geometric meaning of zeros (the x-intercepts of the polynomial’s graph) and the algebraic relationship between zeros and coefficients. These relationships are powerful shortcuts they allow you to find unknown zeros without solving the full equation.

Types of Polynomials

Relationship Between Zeros and Coefficients

For Quadratic Polynomial ax² + bx + c (zeros: α and β)

α + β = −b/a (Sum of zeros) αβ = c/a (Product of zeros)

For Cubic Polynomial ax³ + bx² + cx + d (zeros: α, β, γ)

α + β + γ = −b/a (Sum of zeros) αβ + βγ + αγ = c/a (Sum of products of pairs) αβγ = −d/a (Product of all zeros)

Common mistake: Students often write the product of cubic zeros as +d/a. The correct formula is −d/a. The negative sign applies to the product of all three zeros in a cubic, just as −b/a applies to the sum. Always double-check the sign.

Division Algorithm for Polynomials

p(x) = g(x) × q(x) + r(x) where degree of r(x) < degree of g(x), or r(x) = 0

This mirrors Euclid’s Division Lemma but for polynomials. It is used to verify zeros, find missing coefficients, or determine what polynomial was divided.

Pair of Linear Equations in Two Variables

This chapter combines algebra with geometry. Every pair of linear equations in two variables represents two straight lines in a coordinate plane. The nature of their intersection or lack thereof determines the type of solution. Understanding the ratio conditions is what separates conceptual understanding from mechanical solving.

Standard Form

a₁x + b₁y + c₁ = 0 a₂x + b₂y + c₂ = 0

Conditions for Type of Solution

Methods of Solving

Substitution Method

Express one variable in terms of the other from either equation, then substitute into the second equation. Best when one equation has a variable with coefficient 1 or −1.

Elimination Method

Multiply one or both equations by suitable constants so that the coefficient of one variable becomes equal in both. Then add or subtract the equations to eliminate that variable. Most efficient when coefficients are larger numbers.

Cross-Multiplication (Cramer’s Rule)

x = (b₁c₂ − b₂c₁) / (a₁b₂ − a₂b₁) y = (a₂c₁ − a₁c₂) / (a₁b₂ − a₂b₁)

Fastest method for direct substitution. Ideal for word problems where the equations are already clearly defined and speed matters.

Exam tip: When a question asks you to find k such that the system has “no solution” or “infinitely many solutions”, set up the ratio conditions and solve for k directly. No actual solving of the system is needed this is a pure concept question worth 2 marks.

Quadratic Equations

Quadratic equations are among the most heavily tested topics in Class 10. The chapter appears in both short-answer (2-mark) and long-answer (4-mark) questions, and almost always contributes to word problems. Mastering the quadratic formula and discriminant is essential — not optional.

Standard Form

ax² + bx + c = 0, where a ≠ 0

The Quadratic Formula

x = [−b ± √(b² − 4ac)] / 2a

This formula gives both roots in a single expression. The ± means you compute two values: one with addition and one with subtraction. When the discriminant is zero, both roots are identical and you only get one value.

The Discriminant

The discriminant (Δ) is the expression under the square root in the quadratic formula. It reveals the nature of the roots without solving the full equation — making it a powerful 1-step tool for conceptual questions.

Δ = b² − 4ac

Sum and Product of Roots

Forming a quadratic from roots: If you are told the roots are 3 and −5, the quadratic is x² − (3 + (−5))x + (3 × −5) = x² + 2x − 15 = 0. This reverse process is asked in at least one board question every year.

Arithmetic Progressions (AP)

Arithmetic Progressions reward students who understand the structure of the chapter. The nth term formula and sum formula are the two core tools. Every single question in this chapter from finding a specific term to summing a range of terms — can be solved using these two formulas in combination.

General Form of an AP

a, a+d, a+2d, a+3d, … where a = first term, d = common difference

Core AP Formulas

Sum of Natural Numbers (Special Cases)

Common mistake: When finding the number of terms n in an AP, students often get a non-integer answer and assume it is correct. If n is not a positive integer, there is an error somewhere. Check whether the common difference d has the correct sign, and verify that you are solving for n, not for a term value.

Triangles

The Triangles chapter is concept-heavy but formula-light. The marks come from knowing which similarity criterion applies to a given figure and correctly applying the theorems in proof questions. The Pythagoras theorem and its converse are the two most practically tested results.

Similarity Criteria

Important Theorems

Insight: The Areas of Similar Triangles result is frequently tested in 3-mark questions. If two triangles are similar in ratio 3:4, their areas are in ratio 9:16 — the ratio is squared, not doubled. This distinction catches many students off guard.

Coordinate Geometry

Coordinate geometry requires integrating three formulas distance, section, and area often within a single problem. The key skill is recognising which formula is the entry point for a given question and sequencing the steps correctly.

Core Formulas

Collinearity shortcut: If the area of a triangle formed by three points evaluates to zero, the three points are collinear. This is faster than the slope method and avoids the need to compute gradients.

Introduction to Trigonometry

Trigonometry is consistently among the top three highest-scoring chapters in Class 10 for well-prepared students. The formulas are fixed and finite — once you have mastered the six ratios, three Pythagorean identities, and the standard angle table, you have the tools to solve every question in this chapter.

Six Trigonometric Ratios (for angle θ in a right triangle)

Three Pythagorean Identities

sin²θ + cos²θ = 1 (Primary identity) 1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ

These identities are derived from each other by dividing the primary identity by cos²θ or sin²θ. They are used to simplify expressions and prove other identities a staple of the 4-mark proof question.

Standard Angle Values

indicates the value is undefined

Memory trick for sin values: Think √0/2, √1/2, √2/2, √3/2, √4/2 for angles 0°, 30°, 45°, 60°, 90°. Simplify: 0, 1/2, 1/√2, √3/2, 1. For cos, reverse the order.

Complementary Angle Formulas

Some Applications of Trigonometry (Height and Distance)

This chapter applies trigonometric ratios to real-world problems involving heights of buildings, towers, trees, and distances across rivers or between two points. Every problem in this chapter reduces to a right triangle - your job is to correctly identify the angle, the sides, and the appropriate ratio.

Important Terms

Working Formulas

tan θ = Height / Horizontal Distance Height = Horizontal Distance × tan θ Horizontal Distance = Height / tan θ

Alternate angles: The angle of depression from point A to point B is equal to the angle of elevation from point B to point A (alternate interior angles formed by the horizontal and the line of sight). Use this to simplify two-angle problems it converts the depression angle into an elevation angle in the lower triangle.

Circles

Circle geometry in Class 10 is theorem-driven. The questions test whether you can correctly identify the geometric situation (tangent from an external point, tangent-secant relationship) and apply the right theorem. Proof questions from this chapter carry up to 4 marks.

Basic Circle Formulas

Circle Theorems

• Tangent-Radius Perpendicularity: A tangent drawn to a circle at any point is perpendicular to the radius at that point. This is the most commonly used theorem in proofs.
• Equal Tangents from External Point: The two tangents drawn from an external point to a circle are always equal in length. This is used to find unknown lengths in complex figures.
• Angle Subtended at Centre: The angle subtended by an arc at the centre is twice the angle it subtends at any point on the remaining part of the circle.

Areas Related to Circles

This chapter extends basic circle area to composite figures involving sectors, segments, and combinations of shapes. The key is knowing which area to add and which to subtract.

Sector and Segment Formulas

Special Shape Areas

Surface Areas and Volumes

This chapter carries 12–15 marks in most board papers and is a reliable scoring area. The most important category is combination solids shapes formed by joining two standard solids. Knowing exactly which surfaces to include and which are internal (and therefore excluded) is what distinguishes a full-marks answer from a partial one.

Complete Formula Reference

Cube (side = a)

Cuboid (length l, breadth b, height h)

Cylinder (radius r, height h)

Cone (radius r, height h, slant height l)

Sphere and Hemisphere (radius r)

Frustum of a Cone (radii r₁ and r₂, height h, slant height l)

Slant height: l = √[h² + (r₁ − r₂)²] Volume = ⅓πh(r₁² + r₂² + r₁r₂) Curved Surface Area = π(r₁ + r₂)l Total Surface Area = π(r₁ + r₂)l + π(r₁² + r₂²)

Combination solid rule: When two solids are joined (e.g., a cone placed on top of a cylinder), the circular face at the junction is internal. Do NOT include it in the total surface area. Only count the exposed outer surfaces.

Statistics

Statistics is procedural and predictable. Once you understand which formula applies to each method (direct, assumed mean, step deviation for mean; the class-finding process for median and mode), this chapter becomes one of the most reliable for full marks. The empirical relationship between mean, median, and mode ties the three measures together.

Mean - Three Calculation Methods

Median for Grouped Data

Median = l + [(n/2 − cf) / f] × h

Where: l = lower class boundary of the median class, cf = cumulative frequency of the class before the median class, f = frequency of the median class, h = class width, n = total frequency.

To find the median class: compute n/2, then identify the class whose cumulative frequency first exceeds n/2.

Mode for Grouped Data

Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

Where: l = lower boundary of the modal class (highest frequency class), f₁ = frequency of modal class, f₀ = frequency of the class before the modal class, f₂ = frequency of the class after the modal class, h = class width.

Empirical Relationship

Mode = 3 × Median − 2 × Mean

This approximate relationship holds for moderately skewed distributions. It is used to find the third measure when any two are known - a common 2-mark question that requires zero computation beyond substitution.

Common mistake: Students confuse the modal class (the class with the highest frequency) with the class having the highest cumulative frequency. Always scan the frequency column not the cumulative frequency column to identify the modal class.

Probability

Probability is conceptually straightforward but requires careful counting. The questions range from basic single-event probability to complementary events and more complex sample space problems (coins, dice, cards). This chapter is a consistent source of 1-3 mark questions that well-prepared students should secure in full.

Core Probability Formulas

Card problems remember: A standard deck has 52 cards, 4 suits (hearts, diamonds, clubs, spades), 13 cards per suit (A, 2–10, J, Q, K), 26 red cards, 26 black cards, 12 face cards (J, Q, K across all 4 suits), and 4 aces. These numbers come up in every card-based probability question.

Exam Strategy and Revision Plan

Having all the formulas is the starting point, not the finish line. How you revise, practise, and allocate your time in the final weeks determines whether these formulas translate into marks. Here is a structured approach built around CBSE board exam patterns.

Chapter-wise Priority and Time Allocation

Very High Priority

• Trigonometry (20%)
• Quadratic Equations (15%)
• Surface Areas & Volumes (15%)

High Priority

• Coordinate Geometry (12%)
• Arithmetic Progressions (10%)
• Statistics (10%)

Medium Priority

• Polynomials (8%)
• Real Numbers (5%)
• Circles & Areas (5%)

Month-by-Month Revision Plan

3 Months Before Exam

• Weeks 1-2: Complete all chapters and compile formulas chapter-wise in your own handwriting.
• Weeks 3-4: Create a master formula booklet. Include derivations for the Pythagorean identities, AP sum formula, and discriminant.
• Weeks -8: Solve all NCERT exercises and examples using only your formula booklet no textbook formulas while solving.
• Weeks 9-12: Attempt at least five full previous year question papers under timed conditions (3 hours).

Final Month

• Daily 30-minute formula revision: cycle through two chapters per day
• For each formula, solve at least three different problem types to build flexibility
• Maintain an error diary note every formula-related mistake, the chapter it came from, and the correction
• Take one full mock test per week under exam conditions

Final Week

• Two hours of formula revision daily focus entirely on high-priority chapters
• Solve sample papers and previous year papers; do not attempt new topics
• Prepare a one-page quick-reference cheat sheet (for revision only not for the exam)

Smart Study Techniques

• Active recall over passive reading: Cover the formula column and try to reproduce it from the description. This is more effective than re-reading.
• Spaced repetition: Review formulas from earlier chapters every three days, not just once. Forgetting is faster than most students expect.
• Error categorisation: Classify your mistakes as either “formula error” (wrong formula), “sign error”, or “calculation error”. Each type needs a different fix.
• Teach it back: If you can explain a formula’s derivation and application to someone else without notes, you truly understand it.

Important Constants and Values