Class 10 board exams test a surprisingly focused set of mathematical ideas. While the CBSE and ICSE syllabi are broad, certain formulas appear again and again — in different forms, different chapters, and different question types. These are the 10 you cannot afford to get wrong.
1. The Quadratic Formula
x = (−b ± √(b²−4ac)) / 2a
Used to find the roots of any quadratic equation ax² + bx + c = 0. The discriminant (b²−4ac) tells you how many real roots exist: positive = two real roots, zero = one real root, negative = no real roots. Memory trick: "Negative b, plus or minus root, b squared minus four ac, all over two a."
2. Pythagorean Theorem
a² + b² = c²
For any right-angled triangle where c is the hypotenuse. Used across circles (chord-distance problems), coordinate geometry (distance formula), and trigonometry (proving identities). Know the triplets: 3-4-5, 5-12-13, 8-15-17.
3. Trigonometric Ratios and Identities
sin²θ + cos²θ = 1 | tan θ = sin θ / cos θ | sin(90°−θ) = cos θ
The foundational identities appear in almost every trigonometry question. The complementary angle identities (sin = cos of complement) are frequently tested in fill-in-the-blank questions.
4. Surface Area of Sphere and Cylinder
Sphere: A = 4πr² | Cylinder: A = 2πr(r+h)
Mensuration is worth 10-12 marks in most Class 10 papers. Surface area and volume combinations (a solid melted into another) are the most common question type. Practice the formula for CSA (curved surface area) vs TSA (total surface area) — they differ by the circular bases.
5. Volume of Cone, Sphere, Cylinder
Cone: V = (1/3)πr²h | Sphere: V = (4/3)πr³ | Cylinder: V = πr²h
Note that Cone = (1/3)Cylinder and Sphere = (2/3)Cylinder (when h=2r). These relationships make it easy to convert between them without memorising three separate formulas.
6. Coordinate Geometry — Distance and Section Formulas
Distance: d = √((x₂−x₁)² + (y₂−y₁)²)
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
This is Pythagoras in disguise — the distance formula is the hypotenuse of a right triangle formed by the coordinate differences. Section formula (m:n ratio) extends this.
7. Arithmetic Progressions
aₙ = a + (n−1)d | Sₙ = n/2 × (2a + (n−1)d)
One of the most consistently tested topics. Know how to find the nth term, the sum of n terms, and how to work backwards from a given term or sum to find a or d.
8. Area of Triangle (Coordinate Version)
Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Used in coordinate geometry to find areas without needing the height. The condition that three points are collinear (area = 0) is a very common question.
9. Statistics — Mean, Median, Mode
Mean (grouped) = Σfᵢxᵢ / Σfᵢ | Empirical relation: Mode = 3Median − 2Mean
Class 10 statistics questions almost always use grouped frequency tables. The empirical relation between mean, median, and mode is tested as a formula fill-in consistently.
10. Simple and Compound Interest
SI = PRT/100 | CI = P(1 + R/100)ⁿ − P
Financial maths appears regularly and is often seen as easy marks. The difference between SI and CI for 2 years = P(R/100)² — useful shortcut for MCQ and 2-mark questions.
How to Memorise These 10 Formulas
Do not try to memorise formulas by staring at them. For each of these 10, open MathVis and do two things: (1) drag the sliders to see the formula change visually, and (2) solve one board-exam problem using the solver. The kinetic act of calculation creates stronger memory than repetitive reading.
Make one A4 revision sheet with just these 10 formulas and one example each. Review it every evening for 2 weeks before the board exam. These formulas alone can account for 30-40 marks in a standard Class 10 paper.