ICSE Class 10 Maths: Every Formula in One Cheat Sheet (2027)

One Page. Every Formula. Zero Panic.

ICSE Class 10 Mathematics is a 100-mark paper where knowing the right formula at the right time is the difference between 70% and 95%. The syllabus spans nine major areas — Algebra, Geometry, Trigonometry, Mensuration, Statistics, Probability, Coordinate Geometry, Quadratic Equations, and Arithmetic & Geometric Progressions — and each comes loaded with formulas you must recall under exam pressure. This cheat sheet organises every single formula chapter-wise so you can revise in one sitting, print it out, and keep it beside you during practice. No fluff, no theory — just the formulas you need.

In This Cheat Sheet

Quadratic Equations
Arithmetic & Geometric Progressions
Ratio, Proportion & Factorisation
Matrices
Coordinate Geometry
Trigonometry
Circles & Tangents
Similarity & Loci
Mensuration (Area & Volume)
Statistics
Probability
Formula Memorisation Tips
Frequently Asked Questions

Quadratic Equations

Quadratic equations carry 10-12 marks and appear in both Section A and Section B every year. Master the standard form, the three methods of solving, and the discriminant conditions.

Formula / Concept	Expression
Standard Form	ax² + bx + c = 0, where a ≠ 0
Quadratic Formula	x = (-b ± √(b² - 4ac)) / 2a
Discriminant (D)	D = b² - 4ac
Nature of Roots	D > 0 → two distinct real roots; D = 0 → two equal real roots; D < 0 → no real roots
Sum of Roots	α + β = -b/a
Product of Roots	αβ = c/a
Forming Equation from Roots	x² - (α + β)x + αβ = 0

Arithmetic & Geometric Progressions

AP and GP together carry 8-10 marks. AP is tested more frequently with numericals on finding terms, sums, and real-life applications. GP appears as 1-2 questions, usually in Section A.

Formula / Concept	Expression
Arithmetic Progression (AP)
n-th Term	aₙ = a + (n - 1)d
Sum of n Terms	Sₙ = n/2 [2a + (n - 1)d] or Sₙ = n/2 (a + l), where l = last term
Common Difference	d = a₂ - a₁ = a₃ - a₂
Condition for AP	2b = a + c (three terms a, b, c in AP)
Geometric Progression (GP)
n-th Term	aₙ = arⁿ¹, where r = common ratio
Sum of n Terms (r ≠ 1)	Sₙ = a(rⁿ - 1)/(r - 1) when r > 1; Sₙ = a(1 - rⁿ)/(1 - r) when r < 1
Sum to Infinity (\|r\| < 1)	S∞ = a / (1 - r)
Condition for GP	b² = ac (three terms a, b, c in GP)

Ratio, Proportion & Factorisation

These algebraic identities and proportion rules are the building blocks used throughout the paper. They appear directly in Section A and are essential for simplifying expressions across all chapters.

Formula / Identity	Expression
Componendo	If a/b = c/d, then (a + b)/b = (c + d)/d
Dividendo	If a/b = c/d, then (a - b)/b = (c - d)/d
Componendo-Dividendo	If a/b = c/d, then (a + b)/(a - b) = (c + d)/(c - d)
Alternendo	If a/b = c/d, then a/c = b/d
(a + b)²	a² + 2ab + b²
(a - b)²	a² - 2ab + b²
a² - b²	(a + b)(a - b)
(a + b)³	a³ + 3a²b + 3ab² + b³
a³ + b³	(a + b)(a² - ab + b²)
a³ - b³	(a - b)(a² + ab + b²)
Remainder Theorem	If f(x) is divided by (x - a), remainder = f(a)
Factor Theorem	If f(a) = 0, then (x - a) is a factor of f(x)

Matrices

Matrices carry 4-6 marks in ICSE. Questions focus on addition, multiplication, and solving matrix equations. Know the order rules and the identity/null matrix definitions.

Concept	Rule / Formula
Order of Matrix	m × n (rows × columns)
Addition/Subtraction	Only possible when both matrices have the same order; add/subtract corresponding elements
Multiplication Rule	A(m×n) × B(n×p) = C(m×p); columns of A must equal rows of B
Scalar Multiplication	k × [a b; c d] = [ka kb; kc kd]
Identity Matrix (I)	AI = IA = A; I = [1 0; 0 1] for 2×2
Null Matrix (O)	A + O = A; all elements are zero

Coordinate Geometry

Coordinate Geometry is one of the highest-scoring chapters, carrying 10-14 marks. It covers straight lines, the section formula, and the equation of a line. Every formula here is directly used in numericals — memorise them precisely.

Formula / Concept	Expression
Distance Formula	d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Section Formula (Internal)	P = ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂))
Midpoint Formula	M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Slope (Gradient)	m = (y₂ - y₁) / (x₂ - x₁) = tan θ
Slope-Intercept Form	y = mx + c (m = slope, c = y-intercept)
Point-Slope Form	y - y₁ = m(x - x₁)
Two-Point Form	(y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)
Parallel Lines	m₁ = m₂ (slopes are equal)
Perpendicular Lines	m₁ × m₂ = -1
Centroid of Triangle	G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)

Trigonometry

Trigonometry carries 10-14 marks across identities, proving questions, and height-and-distance problems. The ratios, identities, and standard angle values must be memorised cold.

Formula / Identity	Expression
Basic Ratios (Right Triangle)
sin θ	Opposite / Hypotenuse
cos θ	Adjacent / Hypotenuse
tan θ	Opposite / Adjacent = sin θ / cos θ
cosec θ, sec θ, cot θ	1/sin θ, 1/cos θ, 1/tan θ
Fundamental Identities
Pythagorean Identity 1	sin²θ + cos²θ = 1
Pythagorean Identity 2	1 + tan²θ = sec²θ
Pythagorean Identity 3	1 + cot²θ = cosec²θ
Complementary Angle Relations
sin(90° - θ)	cos θ
cos(90° - θ)	sin θ
tan(90° - θ)	cot θ

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

Heights & Distances

Angle of Elevation: tan θ = Height of object / Horizontal distance from observer
Angle of Depression: Angle below horizontal line of sight (equals angle of elevation from ground by alternate interior angles)
Two-observation problems: Set up two equations using tan for two different angles, then solve simultaneously

Circles & Tangents

Circle theorems and tangent properties carry 8-10 marks. Questions involve angle calculations using theorems and tangent-length proofs. Know each theorem statement precisely.

Theorem / Property	Statement
Angle at Centre	Angle subtended at centre = 2 × angle subtended at circumference (by same arc)
Angle in Semicircle	Angle in a semicircle = 90°
Angles in Same Segment	Angles subtended by the same arc in the same segment are equal
Cyclic Quadrilateral	Opposite angles of a cyclic quadrilateral are supplementary (sum = 180°)
Tangent-Radius	Tangent is perpendicular to radius at point of contact
Tangent Lengths	Tangents from an external point are equal in length
Alternate Segment Theorem	Angle between tangent and chord = angle in the alternate segment
Intersecting Chords	If two chords intersect: PA × PB = PC × PD
Tangent-Secant	PT² = PA × PB (tangent from external point P, secant through A and B)
Arc Length	l = (θ/360) × 2πr
Area of Sector	A = (θ/360) × πr²
Area of Segment	Area of sector - Area of triangle formed by radii and chord

Similarity & Loci

Similarity of triangles and locus form the core of the Geometry section, carrying 8-10 marks. BPT, criteria for similarity, and area-ratio theorems are tested every year.

Theorem / Concept	Statement / Formula
BPT (Basic Proportionality)	If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally: AD/DB = AE/EC
Criteria for Similarity	AA (Angle-Angle), SAS (Side-Angle-Side with proportional sides), SSS (all sides in proportion)
Corresponding Sides	If ▵ABC ~ ▵DEF, then AB/DE = BC/EF = AC/DF
Area Ratio	Area(▵ABC) / Area(▵DEF) = (AB/DE)² = (BC/EF)² = (AC/DF)²
Pythagoras Theorem	In a right triangle: Hypotenuse² = Base² + Perpendicular²
Mid-Point Theorem	Line joining midpoints of two sides of a triangle is parallel to the third side and half its length
Locus (Equidistant from 2 points)	Perpendicular bisector of the line segment joining the two points
Locus (Equidistant from 2 lines)	Bisector of the angle between the two intersecting lines

Mensuration (Area & Volume)

Mensuration carries 12-15 marks and is one of the most formula-dense chapters. Questions involve calculating surface area and volume of solids, and conversion between shapes (e.g., a cone melted into a sphere). Precision with formulas is everything here.

Shape	Surface Area	Volume
Cylinder	CSA = 2πrh; TSA = 2πr(r + h)	V = πr²h
Cone	CSA = πrl; TSA = πr(r + l); l = √(r² + h²)	V = (1/3)πr²h
Sphere	SA = 4πr²	V = (4/3)πr³
Hemisphere	CSA = 2πr²; TSA = 3πr²	V = (2/3)πr³
Hollow Cylinder	TSA = 2πh(R + r) + 2π(R² - r²)	V = πh(R² - r²)
Frustum of Cone	CSA = π(R + r)l; l = √[h² + (R - r)²]	V = (πh/3)(R² + r² + Rr)

Combination / Conversion Problems

Melting & Recasting: Volume of original shape = Volume of new shape. Use this to find the unknown dimension.
Combined Solids: Total surface area = Sum of individual CSAs + any exposed flat surfaces. Be careful not to double-count joined faces.
Water Problems: Volume of water = Volume of container - Volume of submerged object. Rise in water level: Δh = Volume of object / Base area of container.

Statistics

Statistics carries 8-12 marks and appears without fail in Section B as a long-answer question. You must know how to calculate mean (three methods), median from grouped data, mode from a histogram, and the inter-quartile range.

Measure	Formula
Mean (Grouped Data)
Direct Method	Mean = Σfᵢxᵢ / Σfᵢ, where xᵢ = class mark (midpoint)
Short-Cut (Assumed Mean)	Mean = A + (Σfᵢdᵢ / Σfᵢ), where dᵢ = xᵢ - A
Step-Deviation Method	Mean = A + (Σfᵢuᵢ / Σfᵢ) × h, where uᵢ = (xᵢ - A)/h
Median & Quartiles (Grouped Data)
Median (from ogive)	Plot cumulative frequency curve; median = value at N/2 on the y-axis
Median (formula)	Median = l + [(N/2 - cf) / f] × h, where l = lower boundary of median class, cf = cumulative frequency before median class, f = frequency of median class, h = class width
Lower Quartile (Q₁)	Value at N/4 on the cumulative frequency curve
Upper Quartile (Q₃)	Value at 3N/4 on the cumulative frequency curve
Inter-Quartile Range	IQR = Q₃ - Q₁
Mode
Mode (from histogram)	Draw histogram → identify tallest bar → draw diagonals from top corners of tallest bar to adjacent bars → x-coordinate of intersection = mode

Probability

Probability carries 5-8 marks and is often the easiest section for students who understand the basic framework. Questions use cards, dice, coins, and bag-of-balls scenarios.

Formula / Concept	Expression
Basic Probability	P(E) = Number of favourable outcomes / Total number of outcomes
Range	0 ≤ P(E) ≤ 1
Complement Rule	P(not E) = 1 - P(E)
Sure Event	P(E) = 1
Impossible Event	P(E) = 0
Sum of Probabilities	P(E) + P(not E) = 1

Common Sample Spaces to Remember

Single die: 6 outcomes (1, 2, 3, 4, 5, 6)
Two dice: 36 outcomes (6 × 6)
Single coin: 2 outcomes (H, T)
Two coins: 4 outcomes (HH, HT, TH, TT)
Pack of cards: 52 cards = 4 suits × 13 cards; 26 red + 26 black; 12 face cards (4 Jacks + 4 Queens + 4 Kings)

Formula Memorisation Tips

Having all the formulas listed in one place is only half the battle. You need to get them into long-term memory so they surface automatically during the exam. Here are five proven techniques.

1. Write, Don't Just Read

Write every formula by hand at least three times daily. The physical act of writing engages motor memory, which is stronger than visual memory alone. Maintain a separate formula notebook and fill one page each morning before breakfast.

2. Group by Pattern

Many formulas share a structure. For example, all three Pythagorean identities follow the pattern: 1 + something² = something-else². All mensuration volumes for pointed shapes (cone, pyramid) have a 1/3 factor. Spotting these patterns reduces what you need to memorise by 40%.

3. Derive, Don't Just Memorise

If you understand where a formula comes from, you can reconstruct it under pressure. For example, the quadratic formula is derived by completing the square on ax² + bx + c = 0. If you blank on the formula, you can re-derive it in 90 seconds.

4. Use Flashcards

Put the formula name on one side and the expression on the other. Shuffle and test yourself daily. Remove cards you know perfectly and focus on the ones you keep getting wrong. Ten minutes of flashcard practice beats one hour of passive reading.

5. Apply Immediately

After memorising a formula, immediately solve 2-3 problems using it. Application cements the formula in context. Your brain remembers "I used this formula to find the slant height of a cone" far better than "CSA of cone = πrl."

Frequently Asked Questions

Q: How many formulas do I need to know for the ICSE Class 10 Maths exam?

Approximately 80-90 formulas across all chapters. That sounds like a lot, but many are related (e.g., all three trig identities are variations of sin²θ + cos²θ = 1). If you group them by chapter and learn 8-10 formulas per chapter over two weeks, you will have them all covered with time to spare for practice.

Q: Which chapter carries the most marks in ICSE Maths?

Mensuration and Trigonometry typically carry the highest combined weightage (12-15 marks each). Coordinate Geometry and Quadratic Equations follow closely at 10-14 marks each. Together, these four areas account for roughly 55-60% of the entire theory paper. Prioritise them during your revision.

Q: I keep confusing the surface area formulas for cone, sphere, and cylinder. Any trick?

Remember the hierarchy: a cylinder's CSA is 2πrh (the "unwrapped rectangle"), a cone's CSA is πrl (half the cylinder's wrapper since it tapers), and a sphere's SA is 4πr² (exactly 4 times the area of a circle). For TSA, just add the appropriate base areas: two circles for a cylinder, one circle for a cone, and none for a sphere. This logical progression makes them much easier to retain.

Q: Are proofs asked in the ICSE Maths exam, or only numericals?

Both are asked. Trigonometric identity proofs ("prove that LHS = RHS") appear in almost every paper and carry 3-4 marks per question. Circle theorem proofs and similarity proofs also feature regularly. For trig proofs, the strategy is always the same: pick the more complicated side, convert everything to sin and cos, simplify using the Pythagorean identities, and show it equals the other side.

Q: How should I revise Statistics for the board exam?

Statistics is one of the most predictable sections. In Section B, you will almost always get one question asking you to draw an ogive and find the median, quartiles, and inter-quartile range. Practise drawing ogives neatly on graph paper using at least 5 previous year problems. For mean, master the step-deviation method since it is the fastest and is what examiners expect to see in long-answer questions.

Q: Can I use this cheat sheet during the exam?

No, you cannot carry any reference material into the examination hall. The purpose of this cheat sheet is for revision — use it daily during your preparation phase to quickly review all formulas in one sitting. The goal is to internalise every formula so thoroughly that you do not need the sheet by exam day.

Q: What is the best order to revise all chapters in the last 10 days?

Start with the highest-scoring chapters: Mensuration (Day 1-2), Trigonometry (Day 3-4), Coordinate Geometry (Day 5), Quadratic Equations + AP/GP (Day 6), Circles + Similarity (Day 7-8), Statistics + Probability (Day 9), and a full revision of all formulas + one timed paper on Day 10. This order ensures you lock in the marks-heavy chapters first, so even if you run short on time, the high-weightage areas are solid.

Q: Is the quadratic formula enough, or do I need to know factorisation and completing the square too?

You need all three methods. The ICSE paper often specifies which method to use ("solve by factorisation" or "solve using the formula"). Factorisation is faster for simple equations, the quadratic formula works universally, and completing the square is occasionally asked as a standalone question. Practise at least 5 problems with each method so you can switch confidently based on what the question demands.

Formulas Are Your Weapons — Sharpen Them Daily

ICSE Maths rewards students who can recall the right formula instantly and apply it precisely. This cheat sheet gives you every formula in one place — but the work of moving them from this page into your memory is yours. Write them, test yourself, solve problems, and repeat. Students who revise formulas daily for just 15 minutes consistently outscore those who cram for hours the night before.

Need structured help with ICSE Maths preparation? Bright Tutorials offers focused coaching for Class 10 board exams with chapter-wise formula drills, problem sets, and mock tests. Get in touch today.

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