Math Formulas for Class 10 – Complete List with Trigonometry and Algebra Formulas
Mathematics in Class 10 is often viewed as the gateway to higher education. Whether you are planning to pursue engineering, commerce, or the pure sciences, the concepts introduced at this level form the structural foundation of your academic journey. However, the sheer volume of math formulas for class 10 can be overwhelming for many students.
Success in mathematics isn't just about rote memorization; it's about understanding the application of important formulas for class 10 maths. When you understand the "why" behind a formula, the "how" becomes second nature. This comprehensive guide provides an exhaustive list of all essential formulas across Algebra, Trigonometry, Coordinate Geometry, and Mensuration, accompanied by worked examples and expert tips to help you ace your board exams.
1. Algebra Formulas: The Building Blocks
Algebra in Class 10 focuses heavily on Polynomials and Pairs of Linear Equations in Two Variables. Mastery of these class 10 maths formulas is crucial because algebra accounts for a significant portion of the marking scheme.
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Polynomials and Algebraic Identities
| Identity Name | Formula |
|---|---|
| Square of a Sum | (a + b)² = a² + 2ab + b² |
| Square of a Difference | (a – b)² = a² – 2ab + b² |
| Difference of Squares | a² – b² = (a – b)(a + b) |
| Cube of a Sum | (a + b)³ = a³ + b³ + 3ab(a + b) |
| Cube of a Difference | (a – b)³ = a³ – b³ – 3ab(a – b) |
| Sum of Cubes | a³ + b³ = (a + b)(a² – ab + b²) |
| Difference of Cubes | a³ – b³ = (a – b)(a² + ab + b²) |
Relationship Between Zeroes and Coefficients
For quadratic polynomial p(x) = ax² + bx + c:
- Sum of zeroes (α+β): –b/a
- Product of zeroes (αβ): c/a
For cubic polynomial p(x) = ax³ + bx² + cx + d:
- α+β+γ = –b/a
- αβ+βγ+γα = c/a
- αβγ = –d/a
2. Quadratic Equation Formulas
A quadratic equation in standard form: ax² + bx + c = 0 (a ≠ 0).
The Quadratic Formula
The Discriminant (D)
D = b² – 4ac determines the nature of roots:
- D > 0 → two distinct real roots
- D = 0 → two equal real roots
- D < 0 → no real roots (imaginary)
Find roots of 2x² – 7x + 3 = 0.
a = 2, b = –7, c = 3.
D = (–7)² – 4·2·3 = 49 – 24 = 25.
x = (7 ± √25) / (4) = (7 ± 5)/4 → x = 3, x = 0.5.
3. Trigonometry Formulas Class 10
Trigonometric Ratios (Right-angled triangle)
- sin θ = Opposite/Hypotenuse
- cos θ = Adjacent/Hypotenuse
- tan θ = Opposite/Adjacent = sinθ/cosθ
Reciprocal Relations
- csc θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ
Trigonometric Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
Trigonometry Table (Standard Angles)
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | Undefined |
4. Coordinate Geometry Formulas
Distance Formula
Section Formula (internal division)
Midpoint Formula
Distance between P(2,3) and Q(4,1):
d = √[(4–2)² + (1–3)²] = √[4 + 4] = √8 = 2√2 units.
5. Surface Area and Volume Formulas
| Shape | Lateral/Curved SA | Total SA | Volume |
|---|---|---|---|
| Cube | 4a² | 6a² | a³ |
| Cuboid | 2h(l+b) | 2(lb+bh+hl) | l×b×h |
| Cylinder | 2πrh | 2πr(r+h) | πr²h |
| Cone | πrl | πr(r+l) | ⅓πr²h |
| Sphere | 4πr² | 4πr² | ⁴⁄₃πr³ |
| Hemisphere | 2πr² | 3πr² | ⅔πr³ |
Note: For cone, slant height l = √(r² + h²).
Radius 7 cm, height 10 cm.
Volume = πr²h = (22/7)×7×7×10 = 1540 cm³.
6. Arithmetic Progression (AP) Formulas
- nᵗʰ term: aₙ = a + (n – 1)d
- Sum of first n terms: Sₙ = n/2 [2a + (n – 1)d]
- Sₙ = n/2 (a + l), where l = last term.
7. Statistics and Probability
Statistics (Grouped Data)
- Mean (Direct method): x̄ = Σfᵢxᵢ / Σfᵢ
- Mode: l + ( (f₁ – f₀)/(2f₁ – f₀ – f₂) ) × h
- Median: l + ( (n/2 – cf)/f ) × h
Probability
- P(E) = (Number of favourable outcomes) / (Total outcomes)
- 0 ≤ P(E) ≤ 1 and P(E) + P(not E) = 1
Engineering uses trigonometry & algebra, Data Science relies on statistics & probability, and Gaming/Animation uses coordinate geometry. Mastering these formulas builds logical reasoning and spatial intelligence.
Tips to Memorize Class 10 Maths Formulas
- Create a Formula Sheet: Paste a large chart near your study desk for visual reinforcement.
- Understand the Derivation: Knowing the "why" ensures you never forget the "how".
- Practice Daily: Apply at least five different formulas to new problems every day.
- Use Mnemonics: "Some People Have, Curly Brown Hair, Through Proper Brushing" → sin = P/H, cos = B/H, tan = P/B.
Frequently Asked Questions (FAQs)
Q1. Which chapter has the most formulas for class 10 maths?
Trigonometry, Algebra, and Surface Area & Volume contain the highest density of formulas frequently tested in board exams.
Q2. What is the Quadratic Formula?
x = [–b ± √(b² – 4ac)] / (2a). Use it when factorisation is difficult.
Q3. How can I remember the Trigonometry Table?
For sin: 0, 1/2, 1/√2, √3/2, 1. Cos is reverse of sin. Tan = sin/cos.
Q4. Difference between Lateral and Total Surface Area?
Lateral (curved) includes only the sides; Total includes sides + bases.
Q5. Is frustum of a cone included in class 10 formulas?
Optional, but volume formula: V = ⅓πh(r₁² + r₂² + r₁r₂).
Q6. What is the Section Formula used for?
To find coordinates of a point dividing a line segment in a given ratio.
Q7. What does the Discriminant tell us?
It tells whether roots are real, equal, or imaginary.
Q8. Why is sin²θ + cos²θ = 1?
It is a Pythagorean identity derived from the unit circle (a² + b² = c²).
Q9. What are zeroes of a polynomial?
The values of x that make the polynomial equal zero.
Q10. How to calculate mean for grouped data?
Use direct method: x̄ = Σfᵢxᵢ / Σfᵢ where xᵢ = class mark.
Mastering class 10 maths formulas is your first major step toward academic excellence. This list serves as a comprehensive toolkit for your preparation. Remember, the key to scoring high is not just knowing the formulas but knowing which formula to apply and when. Keep this guide bookmarked, practice the worked examples, and approach every problem with confidence.