# 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, HCF(a,b)×LCM(a,b)=a×b.

### Chapter 2 — Polynomials
For a quadratic ax² + bx + c with zeroes α and β:
\[
α + β = -\frac{b}{a},\quad αβ = \frac{c}{a}.
\]

### Chapter 3 — Pair of Linear Equations in Two Variables
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 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 ax² + bx + c = 0:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
\]
The **discriminant** D=b²−4ac decides the roots: D>0 gives two real roots, D=0 one repeated root, D<0 no real roots.

### Chapter 5 — Arithmetic Progressions
\[
a_n = a + (n-1)d,\qquad S_n = \frac{n}{2}\big(2a + (n-1)d\big) = \frac{n}{2}(a + l).
\]

### 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:** a² + b² = c² for a right triangle with hypotenuse c.

### Chapter 7 — Coordinate Geometry
For points (x₁,y₁) and (x₂,y₂):
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
\]
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:** 2πr. **Area:** πr².
- **Arc length:** \[
\frac{θ}{360} \times 2πr.
\]
- **Sector area:** \[
\frac{θ}{360} \times πr².
\]

### Chapter 13 — Surface Areas and Volumes
| Solid | Volume | Total surface area |
| --- | --- | --- |
| Cube | s³ | 6s² |
| Cuboid | l⋅b⋅h | 2(lb+bh+hl) |
| Cylinder | πr²h | 2πr(r+h) |
| Cone | \(\frac{1}{3}πr²h\) | πr(r+l) |
| Sphere | \(\frac{4}{3}πr³\) | 4πr² |

### Chapter 14 — Statistics
Mean (assumed-mean): \[
\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i},\qquad Mode = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) h.
\]

### Chapter 15 — Probability
\[
P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}, \qquad P(E) + P(\text{not }E) = 1.
\]

## Examples of Maths Formulas for Class 10

### Example 1
**(Quadratic) Solve x² - 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).**
\[
\text{Distance} = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{25} = 5.
\]
**Final answer:** 5 units.

### Example 3
**(Arithmetic progression) Find the sum of the first 20 terms of 3,7,11,…**
\[
S_{20} = \frac{20}{2}\big(2(3) + 19(4)\big) = 820.
\]
**Final answer:** 820.

### Example 4
**(Trigonometry) If sinθ=\(\frac{3}{5}\), find cosθ.**
Using sin²θ + cos²θ = 1:
\[
cos²θ = 1 - \frac{9}{25} = \frac{16}{25} → cosθ = \frac{4}{5}.
\]
**Final answer:** cosθ=\(\frac{4}{5}\).

### Example 5
**(Mensuration) Find the volume of a cylinder with radius 7 cm and height 10 cm.**
\[
V = \pi r^2 h = \frac{22}{7} \times 7^2 \times 10 = 1540 cm³.
\]
**Final answer:** 1540 cm³.

### Example 6
**(Polynomials) The zeroes of x² - 7x + 12 — find their sum and product without solving.**
Sum = -\(\frac{b}{a}\) = 7 and product = \(\frac{c}{a}\) = 12.
**Final answer:** Sum 7, product 12.

## 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 (x₂−x₁)+(y₂−y₁) 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, 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 2πrh 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 sin²θ + cos²θ = 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 2x² - 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 π=\(\frac{22}{7}\)).