CBSE Class 10 Mathematics: Important Formulas for Maths (All Chapters) 2025-2026

Mastering CBSE Class 10 Mathematics formulas is the most reliable strategy to score 90+ in your 2025-2026 board examinations. Having a consolidated sheet of CBSE Class 10 Important Formulas for Maths (All Chapters) allows you to perform swift revisions, avoid algebraic errors, and solve multi-step problems with confidence. This guide provides a complete, chapter-wise breakdown of all NCERT-aligned mathematical formulas, properties, solved exemplar problems, and targeted preparation strategies designed for the CBSE Class 10 board exam 10 pattern.

Key Concepts: Chapter-Wise Formula Sheet

Below is the comprehensive formula repository organized according to the latest CBSE Class 10 Mathematics syllabus.

1. Real Numbers

• Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed as a product of prime numbers, apart from the order in which the prime factors occur.
• HCF & LCM Relationship: For any two positive integers a and b:
  HCF(a, b) × LCM(a, b) = a × b
• Decimal Expansions: A rational number p/q (in simplest form) has a terminating decimal expansion if the prime factorization of q is of the form 2n × 5m, where n and m are non-negative integers.

2. Polynomials

• Standard Form of a Quadratic Polynomial: p(x) = ax2 + bx + c (where a ≠ 0).
• Relationship Between Zeros and Coefficients: If α and β are zeros of ax2 + bx + c:
  • Sum of Zeros: α + β = -b / a = -(Coefficient of x) / (Coefficient of x2)
  • Product of Zeros: α × β = c / a = (Constant term) / (Coefficient of x2)
• Forming a Quadratic Polynomial: If α and β are given:
  p(x) = k [x2 - (α + β)x + αβ] (where k is any non-zero real constant).

3. Pair of Linear Equations in Two Variables

For two linear equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0:

• Intersecting Lines (Unique Solution / Consistent): a1 / a2 ≠ b1 / b2
• Coincident Lines (Infinitely Many Solutions / Dependent & Consistent): a1 / a2 = b1 / b2 = c1 / c2
• Parallel Lines (No Solution / Inconsistent): a1 / a2 = b1 / b2 ≠ c1 / c2

4. Quadratic Equations

• Standard Form: ax2 + bx + c = 0 (a ≠ 0).
• Discriminant (D): D = b2 - 4ac
• Quadratic Formula: x = (-b ± √(b2 - 4ac)) / (2a)
• Nature of Roots:
  • If D > 0: Two distinct real roots.
  • If D = 0: Two equal real roots (x = -b / 2a).
  • If D < 0: No real roots.

5. Arithmetic Progressions (AP)

For an AP with first term a and common difference d (where d = ak - ak-1):

• n^th Term (General Term): an = a + (n - 1)d
• n^th Term from the End: l - (n - 1)d (where l is the last term).
• Sum of First n Terms (S_n):
  Sn = (n / 2) [2a + (n - 1)d] or Sn = (n / 2) [a + l]
• Finding a_n from Sum: an = Sn - Sn-1

6. Coordinate Geometry

• Distance Formula: Distance between P(x1, y1) and Q(x2, y2):
  PQ = √((x2 - x1)2 + (y2 - y1)2)
• Distance from Origin: OP = √(x2 + y2)
• Section Formula (Internal Division): Coordinates of point P(x, y) dividing the segment joining A(x₁, y₁) and B(x₂, y₂) in ratio m1 : m2:
  x = (m1x2 + m2x1) / (m1 + m2),   y = (m1y2 + m2y1) / (m1 + m2)
• Midpoint Formula: ((x1 + x2) / 2, (y1 + y2) / 2)

7. Introduction to Trigonometry & Applications

• Trigonometric Ratios (in a right-angled triangle ΔABC at B):
  • sin θ = Opposite / Hypotenuse,   cosec θ = 1 / sin θ
  • cos θ = Adjacent / Hypotenuse,   sec θ = 1 / cos θ
  • tan θ = Opposite / Adjacent = sin θ / cos θ,   cot θ = 1 / tan θ
• Standard Trigonometric Values:
  • sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1
  • cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, cos 90° = 0
  • tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° = Not Defined
• Fundamental Trigonometric Identities:
  1. sin2θ + cos2θ = 1
  2. 1 + tan2θ = sec2θ ⇒ sec2θ - tan2θ = 1
  3. 1 + cot2θ = cosec2θ ⇒ cosec2θ - cot2θ = 1
• Applications of Trigonometry:
  • Angle of Elevation: Angle formed by the line of sight with the horizontal when viewing an object above the horizontal level.
  • Angle of Depression: Angle formed by the line of sight with the horizontal when viewing an object below the horizontal level.

8. Circles and Areas Related to Circles

• Circle Theorems:
  • The tangent at any point of a circle is perpendicular to the radius through the point of contact.
  • The lengths of tangents drawn from an external point to a circle are equal.
• Circumference and Area:
  • Circumference = 2πr = πd
  • Area of Circle = πr2
• Sector and Segment Formulas (Angle θ in degrees):
  • Length of Arc: l = (θ / 360°) × 2πr
  • Area of Sector: A = (θ / 360°) × πr2 = (1/2) × l × r
  • Area of Minor Segment: Area of Sector - Area of Corresponding Triangle = (θ / 360°) × πr2 - (1/2)r2 sin θ
  • Area of Major Segment: πr2 - Area of Minor Segment

9. Surface Areas and Volumes

• Cuboid: TSA = 2(lb + bh + hl),   CSA / LSA = 2h(l + b),   Volume = l × b × h
• Cube: TSA = 6a2,   CSA = 4a2,   Volume = a3,   Diagonal = a√3
• Right Circular Cylinder: CSA = 2πrh,   TSA = 2πr(r + h),   Volume = πr2h
• Right Circular Cone:
  Slant Height (l) = √(r2 + h2),   CSA = πrl,   TSA = πr(l + r),   Volume = (1/3)πr2h
• Sphere: Surface Area = 4πr2,   Volume = (4/3)πr3
• Hemisphere: CSA = 2πr2,   TSA = 3πr2,   Volume = (2/3)πr3

10. Statistics & Probability

• Mean (¯x):
  • Direct Method: ¯x = Σ(fixi) / Σfi
  • Assumed Mean Method: ¯x = a + (Σ(fidi) / Σfi) (where di = xi - a)
• Mode:
  Mode = l + [ (f1 - f0) / (2f1 - f0 - f2) ] × h
  Where l = lower limit of modal class, f₁ = frequency of modal class, f₀ = frequency of preceding class, f₂ = frequency of succeeding class, h = class width.
• Median:
  Median = l + [ ( (n / 2) - cf ) / f ] × h
  Where l = lower limit of median class, n = Σf_i, cf = cumulative frequency of class preceding median class, f = frequency of median class, h = class size.
• Empirical Relationship: 3 × Median = Mode + 2 × Mean
• Theoretical Probability: P(E) = (Number of outcomes favourable to E) / (Total number of possible outcomes)
0 ≤ P(E) ≤ 1 and P(E) + P(not E) = 1

Important CBSE Questions with Answers

Here are high-yield questions directly illustrating formula execution for the CBSE Class 10 Mathematics board exam:

Question 1 (Arithmetic Progression)

Problem: Find the sum of the first 20 terms of an AP whose 7^th term is 34 and 12^th term is 64.

Solution:

1. Using an = a + (n - 1)d:
   • a7 = a + 6d = 34   --- (Equation 1)
   • a12 = a + 11d = 64   --- (Equation 2)
2. Subtracting (1) from (2): 5d = 30 ⇒ d = 6.
3. Substitute d into (1): a + 6(6) = 34 ⇒ a = 34 - 36 = -2.
4. Apply sum formula Sn = (n / 2)[2a + (n - 1)d] for n = 20:
   S20 = (20 / 2) [2(-2) + (20 - 1)(6)] = 10 [-4 + 114] = 10 × 110 = 1100.

Answer: The sum of the first 20 terms is 1100.

Question 2 (Trigonometric Identities)

Problem: Prove that (sin θ - 2 sin3θ) / (2 cos3θ - cos θ) = tan θ.

Solution:

1. Factor numerator and denominator:
   LHS = [sin θ (1 - 2 sin2θ)] / [cos θ (2 cos2θ - 1)]
2. Express numerator using 1 = sin2θ + cos2θ:
   1 - 2 sin2θ = (sin2θ + cos2θ) - 2 sin2θ = cos2θ - sin2θ
3. Express denominator using 1 = sin2θ + cos2θ:
   2 cos2θ - (sin2θ + cos2θ) = cos2θ - sin2θ
4. Substitute back into the expression:
   LHS = (sin θ / cos θ) × [(cos2θ - sin2θ) / (cos2θ - sin2θ)] = tan θ × 1 = tan θ = RHS.

Conclusion: Hence proved.

Question 3 (Surface Area of Combined Solids)

Problem: A solid toy is in the form of a hemisphere surmounted by a right circular cone. The radius of both is 3.5 cm and the total height of the toy is 15.5 cm. Find the total surface area of the toy. (Take π = 22/7)

Solution:

1. Radius r = 3.5 cm = 7/2 cm.
2. Height of hemisphere = radius = 3.5 cm.
3. Height of conical part h = Total Height - Radius = 15.5 - 3.5 = 12 cm.
4. Slant height of cone l = √(r2 + h2) = √((3.5)2 + 122) = √(12.25 + 144) = √(156.25) = 12.5 cm.
5. Total Surface Area (TSA) of toy = CSA of Cone + CSA of Hemisphere:
   TSA = πrl + 2πr2 = πr (l + 2r)
6. Substitute numerical values:
   TSA = (22/7) × (7/2) × [12.5 + 2(3.5)] = 11 × [12.5 + 7] = 11 × 19.5 = 214.5 cm2.

Answer: The total surface area of the toy is 214.5 cm².

How to Prepare for This Topic

• Construct a Dedicated Formula Notebook: Write out formulas by hand rather than relying solely on digital PDFs. Categorize them into Algebra, Geometry, Trigonometry, Mensuration, and Statistics.
• Pay Attention to Dimensional Units: In Mensuration, always convert all dimensions to uniform units (e.g., cm or m) before applying volume or area formulas. Surface areas use square units (cm²), while volumes use cubic units (cm³).
• Master Sign Conventions in Coordinate Geometry: Watch out for negative coordinates when applying the distance formula (x2 - x1)2 to prevent sign inversion mistakes.
• Daily 15-Minute Formula Drills: Dedicate 15 minutes every morning to flashcard revision of trigonometric identities and statistics formulas.

Where to Practice More

Applying formulas to diverse problem formats is essential for securing a top percentile in the CBSE Class 10 Mathematics board exam. Generate personalized, chapter-wise question papers, NCERT exemplar problem sets, and full-length simulated board exams on Theorify QPTool to test your speed and accuracy today.