CBSE Class 10 Maths Formulas 2025: All Concepts and Chapters
For CBSE Class 10 students preparing for the 2025 board exams, mastering mathematics requires a solid grasp of key formulas across all chapters. The CBSE Class 10 Mathematics syllabus, aligned with the NCERT curriculum, covers essential topics under the Standard and Basic options, with a focus on conceptual understanding and application. This blog provides a comprehensive list of formulas for all chapters, organized by unit, to help students revise efficiently and excel in their exams. As of September 5, 2025, the syllabus remains consistent with the 2024-25 academic year, and these formulas are tailored for the CBSE Class 10 Maths (Code No. 041 and 241) exams scheduled for March 2025.
Why Mastering Formulas Matters
Mathematics is a high-scoring subject in CBSE Class 10, with a total of 80 marks for the theory exam and 20 marks for internal assessments. The syllabus includes seven units: Number Systems, Algebra, Coordinate Geometry, Geometry, Trigonometry, Mensuration, and Statistics and Probability. Memorizing and understanding formulas is crucial for solving numerical problems, proving theorems, and tackling application-based questions. This guide covers all essential formulas, ensuring students are well-prepared for both Standard and Basic Maths papers.
Unit-Wise CBSE Class 10 Maths Formulas for 2025
Unit I: Number Systems
Chapter 1: Real Numbers
- Euclid’s Division Lemma: For positive integers (a) and (b), there exist unique integers (q) and (r) such that (a = bq + r), where (0 \leq r < b).
- Fundamental Theorem of Arithmetic: Every composite number can be expressed as a unique product of prime factors.
- HCF and LCM (Prime Factorization):
- HCF(a, b) = Product of the smallest powers of common prime factors.
- LCM(a, b) = Product of the highest powers of all prime factors.
- HCF(a, b) × LCM(a, b) = (a \times b).
- Rational Numbers: A number of the form (p/q), where (q \neq 0), and (p, q) are integers.
- Irrational Numbers: Non-terminating, non-repeating decimals (e.g., (\sqrt{2}), (\pi)).
Unit II: Algebra
Chapter 2: Polynomials
- General Form: (p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0).
- Zeroes of a Polynomial: For a polynomial (p(x)), if (p(a) = 0), then (a) is a zero.
- Quadratic Polynomial: (ax^2 + bx + c), with roots given by:
- Quadratic Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}), where discriminant (D = b^2 - 4ac).
- Sum and Product of Roots (for (ax^2 + bx + c = 0)):
- Sum of roots = (-\frac{b}{a}).
- Product of roots = (\frac{c}{a}).
- Relationship between Zeroes and Coefficients:
- For a quadratic polynomial: If (\alpha, \beta) are roots, then (\alpha + \beta = -\frac{b}{a}), (\alpha \beta = \frac{c}{a}).
Chapter 3: Pair of Linear Equations in Two Variables
- General Form: (a_1x + b_1y + c_1 = 0), (a_2x + b_2y + c_2 = 0).
- Conditions for Consistency:
- Unique solution: (\frac{a_1}{a_2} \neq \frac{b_1}{b_2}).
- No solution: (\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}).
- Infinitely many solutions: (\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}).
- Substitution Method: Solve one equation for one variable and substitute in the other.
- Elimination Method: Add or subtract equations to eliminate one variable.
- Cross-Multiplication Method:
[
\frac{x}{b_1 c_2 - b_2 c_1} = \frac{y}{c_1 a_2 - c_2 a_1} = \frac{1}{a_1 b_2 - a_2 b_1}
]
Chapter 4: Quadratic Equations
- Standard Form: (ax^2 + bx + c = 0), where (a \neq 0).
- Roots: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
- Nature of Roots (based on discriminant (D = b^2 - 4ac)):
- (D > 0): Two distinct real roots.
- (D = 0): Two equal real roots.
- (D < 0): No real roots.
- Sum and Product of Roots:
- Sum = (-\frac{b}{a}).
- Product = (\frac{c}{a}).
Chapter 5: Arithmetic Progressions (AP)
- General Form: (a, a + d, a + 2d, \dots), where (a) is the first term, (d) is the common difference.
- (n)-th Term: (a_n = a + (n-1)d).
- Sum of First (n) Terms: (S_n = \frac{n}{2} [2a + (n-1)d]) or (S_n = \frac{n}{2} (a + l)), where (l) is the last term.
- Number of Terms: (n = \frac{l - a}{d} + 1).
Unit III: Coordinate Geometry
Chapter 6: Coordinate Geometry
- Distance Formula: For points ((x_1, y_1)) and ((x_2, y_2)):
[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
] - Section Formula: For a point dividing the line joining ((x_1, y_1)) and ((x_2, y_2)) in ratio (m:n):
[
\left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
] - Midpoint Formula:
[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
] - Area of a Triangle: For vertices ((x_1, y_1)), ((x_2, y_2)), ((x_3, y_3)):
[
\text{Area} = \frac{1}{2} \left| x_1 (y_2 - y_3) + x_2 (y_3 - y_1) + x_3 (y_1 - y_2) \right|
]
Unit IV: Geometry
Chapter 7: Triangles
- Basic Proportionality Theorem (Thales’ Theorem): If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally: (\frac{AD}{DB} = \frac{AE}{EC}).
- Similarity Criteria:
- AA (Angle-Angle) Similarity.
- SSS (Side-Side-Side) Similarity.
- SAS (Side-Angle-Side) Similarity.
- Pythagoras Theorem: In a right triangle, (a^2 + b^2 = c^2), where (c) is the hypotenuse.
- Area of Similar Triangles: If (\triangle ABC \sim \triangle DEF), then:
[
\frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = \left( \frac{AB}{DE} \right)^2
]
Chapter 8: Circles
- Tangent Theorem: The tangent at any point on a circle is perpendicular to the radius through the point of contact.
- Equal Tangents: Tangents from an external point to a circle are equal in length.
- Alternate Segment Theorem: The angle between a tangent and a chord equals the angle subtended by the chord in the alternate segment.
Unit V: Trigonometry
Chapter 9: Introduction to Trigonometry
- Trigonometric Ratios (for acute angle (\theta) in a right triangle):
- (\sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}})
- (\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}})
- (\tan \theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{\sin \theta}{\cos \theta})
- (\cot \theta = \frac{1}{\tan \theta}), (\sec \theta = \frac{1}{\cos \theta}), (\cosec \theta = \frac{1}{\sin \theta})
- Trigonometric Identities:
- (\sin^2 \theta + \cos^2 \theta = 1)
- (1 + \tan^2 \theta = \sec^2 \theta)
- (1 + \cot^2 \theta = \cosec^2 \theta)
- Standard Values (e.g., (\sin 30^\circ = \frac{1}{2}), (\cos 45^\circ = \frac{1}{\sqrt{2}}), (\tan 60^\circ = \sqrt{3})).
Chapter 10: Some Applications of Trigonometry
- Angle of Elevation/Depression: Use trigonometric ratios to find heights or distances.
- Line of Sight Formula: For height (h), distance (d), and angle (\theta):
- (\tan \theta = \frac{h}{d})
- (h = d \tan \theta)
Unit VI: Mensuration
Chapter 11: Areas Related to Circles
- Circumference of a Circle: (C = 2\pi r) or (C = \pi d).
- Area of a Circle: (A = \pi r^2).
- Area of a Sector: For angle (\theta):
[
A = \frac{\theta}{360^\circ} \times \pi r^2
] - Length of an Arc: For angle (\theta):
[
l = \frac{\theta}{360^\circ} \times 2\pi r
] - Area of a Segment: Area of sector − Area of triangle formed by radii and chord.
Chapter 12: Surface Areas and Volumes
- Cube:
- Surface Area: (6a^2)
- Volume: (a^3)
- Cuboid:
- Surface Area: (2(lb + bh + hl))
- Volume: (l \times b \times h)
- Cylinder:
- Curved Surface Area: (2\pi rh)
- Total Surface Area: (2\pi r (r + h))
- Volume: (\pi r^2 h)
- Cone:
- Curved Surface Area: (\pi rl)
- Total Surface Area: (\pi r (r + l))
- Volume: (\frac{1}{3} \pi r^2 h)
- Slant Height: (l = \sqrt{r^2 + h^2})
- Sphere:
- Surface Area: (4\pi r^2)
- Volume: (\frac{4}{3} \pi r^3)
- Hemisphere:
- Curved Surface Area: (2\pi r^2)
- Total Surface Area: (3\pi r^2)
- Volume: (\frac{2}{3} \pi r^3)
- Frustum of a Cone:
- Curved Surface Area: (\pi (r_1 + r_2) l)
- Total Surface Area: (\pi [r_1^2 + r_2^2 + (r_1 + r_2) l])
- Volume: (\frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2))
- Slant Height: (l = \sqrt{h^2 + (r_1 - r_2)^2})
Unit VII: Statistics and Probability
Chapter 13: Statistics
- Mean:
- Direct Method: (\text{Mean} = \frac{\sum f_i x_i}{\sum f_i})
- Assumed Mean Method: (\text{Mean} = a + \frac{\sum f_i d_i}{\sum f_i}), where (d_i = x_i - a).
- Step Deviation Method: (\text{Mean} = a + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h), where (u_i = \frac{x_i - a}{h}).
- Median:
- For odd (n): Median = (\left( \frac{n+1}{2} \right))-th term.
- For even (n): Median = Average of (\left( \frac{n}{2} \right))-th and (\left( \frac{n}{2} + 1 \right))-th terms.
- For grouped data: (\text{Median} = l + \left( \frac{\frac{n}{2} - cf}{f} \right) \times h), where (l) = lower limit of median class, (cf) = cumulative frequency, (f) = frequency of median class, (h) = class size.
- Mode:
- For grouped data: (\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h), where (l) = lower limit of modal class, (f_1) = frequency of modal class, (f_0) = frequency of preceding class, (f_2) = frequency of succeeding class, (h) =
