The Master Formula List — Chapter by Chapter
Every core formula a Class 10 student needs, grouped by NCERT chapter. Each cluster links to its full topic article for the derivation and deeper practice.
Chapter 1 — Real Numbers
Fundamental Theorem of Arithmetic: every integer $> 1$ factors uniquely into primes.
HCF and LCM relation: for two numbers $a$ and $b$, $\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$.
Chapter 2 — Polynomials
For a quadratic $ax^2 + bx + c$ with zeroes $\alpha$ and $\beta$:
$$\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}.$$
Chapter 3 — Pair of Linear Equations in Two Variables
For $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, the system has a unique solution when
$$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}.$$
Chapter 4 — Quadratic Equations
The quadratic formula solves $ax^2 + bx + c = 0$:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$
The discriminant $D = b^2 - 4ac$ decides the roots: $D > 0$ gives two real roots, $D = 0$ one repeated root, $D < 0$ no real roots.
Chapter 5 — Arithmetic Progressions
$$a_n = a + (n-1)d, \qquad S_n = \frac{n}{2}\big(2a + (n-1)d\big) = \frac{n}{2}(a + l).$$
The sum formula is derived in full in the sum of arithmetic sequence formula article, which builds on the broader idea of sequences in algebra.
Chapter 6 — Triangles
Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio.
Area ratio of similar triangles: equals the square of the ratio of corresponding sides.
Pythagoras: $a^2 + b^2 = c^2$ for a right triangle with hypotenuse $c$.
Chapter 7 — Coordinate Geometry
For points $(x_1, y_1)$ and $(x_2, y_2)$:
$$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
$$\text{Section formula: } \left(\frac{m x_2 + n x_1}{m + n}, ; \frac{m y_2 + n y_1}{m + n}\right)$$
The midpoint is the section formula with $m = n = 1$.
Chapter 8 — Introduction to Trigonometry
$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\sin\theta}{\cos\theta}$$
$$\sin^2\theta + \cos^2\theta = 1$$
Chapter 10 — Circles and Chapter 12 — Areas Related to Circles
Circumference: $2\pi r$. Area: $\pi r^2$.
Arc length (angle $\theta$ in degrees): $\frac{\theta}{360} \times 2\pi r$.
Sector area: $\frac{\theta}{360} \times \pi r^2$.
Chapter 13 — Surface Areas and Volumes
Solid | Volume | Total surface area |
|---|---|---|
Cube | $s^3$ | $6s^2$ |
Cuboid | $l \cdot b \cdot h$ | $2(lb + bh + hl)$ |
Cylinder | $\pi r^2 h$ | $2\pi r(r + h)$ |
Cone | $\tfrac{1}{3}\pi r^2 h$ | $\pi r(r + l)$ |
Sphere | $\tfrac{4}{3}\pi r^3$ | $4\pi r^2$ |
Chapter 14 — Statistics
$$\text{Mean (assumed-mean): } \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}, \qquad \text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) h$$
Chapter 15 — Probability
$$P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}, \qquad P(E) + P(\text{not } E) = 1.$$
Why the Class 10 Formulas Matter — Built to Be Reused
Every cluster above is the simplest version of an idea that returns, scaled up, in later years.
The quadratic formula and discriminant become the foundation of Class 11 complex numbers (when $D < 0$) and the analysis of conic sections.
Arithmetic progressions generalise to geometric series, then to the infinite series and limits of calculus.
Coordinate geometry's distance formula is Pythagoras in disguise, and it grows into the Class 11 study of straight lines, circles, and three-dimensional geometry.
Trigonometry's $\sin^2\theta + \cos^2\theta = 1$ is the seed identity behind every later trig identity and the unit circle.
Mensuration is where integration is heading — the volume formulas are the discrete answers that calculus later derives from scratch.
The blitzkrieg view — the part most formula sheets skip — is that none of these formulas is a dead end. Each is a stepping stone, and the student who learns why each one holds is already a few steps into a Class 12 chapter without knowing it.
Examples of Maths Formulas for Class 10
Example 1
(Quadratic) Solve $x^2 - 5x + 6 = 0$.
Here $a = 1$, $b = -5$, $c = 6$, so $D = 25 - 24 = 1$:
$$x = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2} = 3 \text{ or } 2.$$
Final answer: $x = 2$ or $x = 3$.
Example 2
(Coordinate geometry) Find the distance between $(2, 3)$ and $(5, 7)$.
Wrong attempt. A student subtracts the coordinates and adds them: $(5 - 2) + (7 - 3) = 3 + 4 = 7$. Picture the points — they sit at the ends of a slanted line, not a straight horizontal or vertical one, so adding the steps overshoots the true straight-line gap. Distance is not the sum of the horizontal and vertical moves.
Correct. Use the distance formula:
$$\sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.$$
Final answer: $5$ units.
Example 3
(Arithmetic progression) Find the sum of the first 20 terms of $3, 7, 11, \ldots$
$a = 3$, $d = 4$, $n = 20$:
$$S_{20} = \frac{20}{2}\big(2(3) + 19(4)\big) = 10(6 + 76) = 820.$$
Final answer: $820$.
Example 4
(Trigonometry) If $\sin\theta = \tfrac{3}{5}$, find $\cos\theta$.
Use $\sin^2\theta + \cos^2\theta = 1$:
$$\cos^2\theta = 1 - \tfrac{9}{25} = \tfrac{16}{25} ;\Rightarrow; \cos\theta = \tfrac{4}{5}.$$
Final answer: $\cos\theta = \tfrac{4}{5}$ (taking the positive value for an acute angle).
Example 5
(Mensuration) Find the volume of a cylinder with radius 7 cm and height 10 cm. (Use $\pi = \tfrac{22}{7}$.)
$$V = \pi r^2 h = \tfrac{22}{7} \times 7^2 \times 10 = \tfrac{22}{7} \times 49 \times 10 = 1540 \text{ cm}^3.$$
Final answer: $1540$ cm³.
Example 6
(Polynomials) The zeroes of $x^2 - 7x + 12$ — find their sum and product without solving.
By the zero relations, sum $= -\frac{b}{a} = 7$ and product $= \frac{c}{a} = 12$.
Final answer: Sum $7$, product $12$ (the zeroes are $3$ and $4$, which check out).
Where Class 10 Formula Problems Go Wrong
Mistake 1: Adding coordinate differences instead of using the distance formula
Where it slips in: Distance and length questions in coordinate geometry.
Don't do this: Compute $(x_2 - x_1) + (y_2 - y_1)$ and call it the distance.
The correct way: Square the differences, add, then take the square root.
Mistake 2: Misreading the discriminant sign
Where it slips in: Deciding how many real roots a quadratic has.
Don't do this: Conclude "no solution" whenever the discriminant is anything other than a perfect square.
The correct way: $D > 0$ means two real roots (whether or not $D$ is a perfect square — a non-perfect-square $D$ just gives irrational roots), $D = 0$ means one repeated root, and only $D < 0$ means no real roots.
Mistake 3: Mixing up curved surface area and total surface area
Where it slips in: Mensuration problems on cylinders, cones, and hemispheres.
Don't do this: Use $2\pi rh$ (curved surface) when the question asks for total surface area, or vice versa.
Conclusion
The maths formulas for class 10 cluster into algebra, sequences, geometry and coordinate geometry, trigonometry, mensuration, and data — about fifteen NCERT chapters in total.
The quadratic formula, the AP sum formula, the distance formula, and $\sin^2\theta + \cos^2\theta = 1$ are the most-tested across boards.
Each formula is the simplest version of an idea that returns in Class 11 and 12 — derive each one once rather than memorising it cold.
The most common slip is adding coordinate differences instead of using the distance formula.
The real destination of the Class 10 formula set is calculus, conics, and the unit circle.
Practice These Before Moving On
Solve $2x^2 - 7x + 3 = 0$ using the quadratic formula.
Find the distance between $(-1, 2)$ and $(2, 6)$.
Find the volume of a sphere of radius $3$ cm (use $\pi = \tfrac{22}{7}$).
Answer to Question 1: $x = 3$ or $x = \tfrac{1}{2}$. Work through Questions 2 and 3 with the formulas above; if Question 2 trips you up, return to Mistake 1.
Want a live Bhanzu trainer to walk your child through the full maths formulas for class 10 set chapter by chapter? Book a free demo class — online globally, or in person at our McKinney, TX center.
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