Maths Formulas for Class 10 — Chapter-Wise List

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

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

1. Solve $2x^2 - 7x + 3 = 0$ using the quadratic formula.

2. Find the distance between $(-1, 2)$ and $(2, 6)$.

3. 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.

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