Maths Formulas for Class 10 — Chapter-Wise List
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
- Solve 2x² - 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 π=(\frac{22}{7})).