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Class 10 Mathematics
Formula Sheet

A comprehensive quick-revision resource covering the essential formulas, identities, theorems and results required for Class 10 Mathematics.

CBSE Class X • Mathematics • 2026–27

Real Numbers Polynomials Linear Equations Quadratic AP Coordinate Triangles Circles Trigonometry Mensuration Statistics Probability

Unit I • Number Systems

Real Numbers

Prime factorisation, HCF, LCM and irrationality results for quick examination revision.

№

Fundamental Theorem

Prime factorisation

Prime Factorisation Every composite number can be expressed as a product of primes, uniquely apart from the order of factors.

HCF HCF is obtained by taking the smallest powers of common prime factors.

LCM LCM is obtained by taking the greatest powers of all prime factors.

HCF & LCM Relationship

Two positive integers

Important Relationship HCF(a,b) × LCM(a,b) = a × b

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

Irrational Numbers Numbers that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.

Exam Tip

When proving irrationality, assume the number is rational, write it in lowest form, and use divisibility to obtain a contradiction.

Unit II • Algebra

Polynomials

Zeros of quadratic polynomials and their relationship with coefficients.

Quadratic Polynomial p(x) = ax² + bx + c, a ≠ 0

Sum of Zeros α + β = −b/a

Product of Zeros αβ = c/a

Polynomial from Zeros p(x) = a(x − α)(x − β)

Monic Quadratic x² − (α+β)x + αβ

Zero α is a zero if p(α) = 0.

Algebra

Pair of Linear Equations

Methods and conditions for solving simultaneous linear equations in two variables.

General Form a₁x + b₁y + c₁ = 0

Second Equation a₂x + b₂y + c₂ = 0

Unique Solution a₁/a₂ ≠ b₁/b₂

No Solution a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Infinitely Many Solutions a₁/a₂ = b₁/b₂ = c₁/c₂

Methods Graphical, substitution and elimination methods.

Algebra

Quadratic Equations

Standard form, discriminant and quadratic formula.

Standard Form ax² + bx + c = 0, a ≠ 0

Quadratic Formula x = [−b ± √(b²−4ac)] / 2a

Discriminant D = b² − 4ac

D > 0 Two distinct real roots.

D = 0 Two equal real roots.

D < 0 No real roots.

Remember

For ax² + bx + c = 0, the sum of roots is −b/a and the product of roots is c/a.

Algebra • Arithmetic Progressions

Arithmetic Progressions

Essential formulas for nth term and sum of first n terms.

General AP a, a+d, a+2d, a+3d, ...

Common Difference d = a₂ − a₁

nth Term aₙ = a + (n−1)d

Sum of n Terms Sₙ = n/2 [2a + (n−1)d]

Alternative Sum Formula Sₙ = n/2 (a + l)

Last Term l = a + (n−1)d

Unit III • Coordinate Geometry

Coordinate Geometry

Distance, section, midpoint and area formulas.

Distance Formula d = √[(x₂−x₁)² + (y₂−y₁)²]

Midpoint Formula M = ((x₁+x₂)/2, (y₁+y₂)/2)

Section Formula If P divides AB internally in m:n: P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))

Area of Triangle Area = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

Collinear Points Three points are collinear if area of the triangle formed by them is 0.

Unit IV • Geometry

Triangles

Similarity criteria, proportionality and Pythagoras theorem.

AAA Similarity

If corresponding angles are equal, the triangles are similar.

SAS Similarity

Two corresponding sides are proportional and the included angle is equal.

SSS Similarity

All corresponding sides are proportional.

Corresponding Sides

For similar triangles, corresponding sides are proportional.

Area Relationship

Ratio of areas of similar triangles equals the square of the ratio of corresponding sides.

Pythagoras Theorem

In a right triangle, hypotenuse² = perpendicular² + base².

Pythagorean Relation c² = a² + b²

Converse If c² = a² + b², then the triangle is right-angled.

Geometry • Circles

Circles

Important tangent and circle results for examination problems.

Radius & Diameter d = 2r

Tangent Tangent at any point of a circle is perpendicular to the radius through that point.

Equal Tangents Tangents drawn from an external point to a circle are equal.

Number of Tangents From an external point, two tangents can be drawn to a circle.

Angle Between Radius & Tangent Angle = 90°

Power Relationship For tangents PA and PB: PA = PB

Unit V • Trigonometry

Trigonometric Ratios & Identities

The essential Class 10 trigonometry formulas and standard values.

△

Basic Ratios

Right-angled triangle

sin θ Perpendicular / Hypotenuse

cos θ Base / Hypotenuse

tan θ Perpendicular / Base

cosec θ 1 / sin θ

sec θ 1 / cos θ

cot θ 1 / tan θ

√

Trigonometric Identities

Must remember

Identity 1 sin²θ + cos²θ = 1

Identity 2 1 + tan²θ = sec²θ

Identity 3 1 + cot²θ = cosec²θ

Quotient Identity tan θ = sin θ / cos θ

Standard Trigonometric Values

Learn these values by heart

sin θ θ = 0°, 30°, 45°, 60°, 90°
0, 1/2, 1/√2, √3/2, 1

cos θ θ = 0°, 30°, 45°, 60°, 90°
1, √3/2, 1/√2, 1/2, 0

tan θ θ = 0°, 30°, 45°, 60°, 90°
0, 1/√3, 1, √3, not defined

cosec θ Not defined, 2, √2, 2/√3, 1

sec θ 1, 2/√3, √2, 2, not defined

cot θ Not defined, √3, 1, 1/√3, 0

Memory Trick

For sin 0°, 30°, 45°, 60°, 90°: √0/2, √1/2, √2/2, √3/2, √4/2. The cosine values appear in reverse order.

Applications of Trigonometry

Heights & Distances

Use trigonometric ratios to calculate unknown heights, distances and angles.

Angle of Elevation

Angle between the horizontal line and line of sight when the object is above the observer.

Angle of Depression

Angle between the horizontal line and line of sight when the object is below the observer.

Typical Approach

Draw the right triangle, identify opposite, adjacent and hypotenuse, then choose the suitable ratio.

Unit VI • Mensuration

Areas, Surface Areas & Volumes

Essential formulas for circles and three-dimensional solids.

Circle Circumference C = 2πr

Circle Area A = πr²

Arc Length l = (θ/360°) × 2πr

Sector Area A = (θ/360°) × πr²

Cuboid TSA = 2(lb+bh+hl)
Volume = lbh

Cube TSA = 6a²
Volume = a³

Cylinder CSA = 2πrh
TSA = 2πr(h+r)
Volume = πr²h

Cone l = √(r²+h²)
CSA = πrl
TSA = πr(l+r)
Volume = ⅓πr²h

Sphere Surface Area = 4πr²
Volume = ⁴⁄₃πr³

Hemisphere CSA = 2πr²
TSA = 3πr²
Volume = ⅔πr³

Unit VII • Statistics

Statistics

Mean, median and mode formulas for grouped data.

Mean — Direct Method x̄ = Σfᵢxᵢ / Σfᵢ

Assumed Mean Method x̄ = a + Σfᵢdᵢ / Σfᵢ

Step-Deviation Method x̄ = a + h(Σfᵢuᵢ / Σfᵢ)

Median Median = l + [(n/2 − cf)/f]h

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

Empirical Relationship Mode = 3 Median − 2 Mean

Statistics Symbols

l = lower boundary of the relevant class, n = total frequency, cf = cumulative frequency before the class, f = class frequency and h = class size.

Unit VII • Probability

Probability

Simple theoretical probability and complementary events.

Probability P(E) = Number of favourable outcomes / Number of equally likely outcomes

Range 0 ≤ P(E) ≤ 1

Impossible Event P(E) = 0

Certain Event P(E) = 1

Complementary Event P(not E) = 1 − P(E)

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

Final Revision

4-Step Class 10 Mathematics Revision

Use this checklist during your final preparation.

Recall

Read formulas and identities without looking at solutions.

Practise

Solve representative questions from every chapter.

Check

Review calculation errors and incorrect formulas.

Revise

Use this sheet for rapid revision before examinations.

Build Strong Mathematics Foundations

Understand concepts, practise intelligently and prepare with confidence for Class 10 Mathematics.

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Class 10 Mathematics Formula Sheet