Mathematics formulas for 10th class board exam Pakistan — algebra, geometry, trigonometry, and more in one complete reference sheet for SSC students.

Complete Mathematics Formula Sheet for 10th Class Board Exam Pakistan

Quick Answer: The most important 10th class maths formulas include the quadratic formula x = (−b ± √(b²−4ac)) / 2a, distance formula d = √((x₂−x₁)² + (y₂−y₁)²), trigonometric ratios, circle theorems, and algebraic identities. Memorizing these 40+ core formulas gives you a strong foundation for the board exam.

Mathematics in 10th class covers a wide range of topics — from algebra and trigonometry to geometry and coordinate geometry. The good news is that most of the exam questions are formula-driven. If you have the right formulas memorized and know when to apply them, you can solve the majority of board paper questions confidently.

This complete formula sheet is organized by chapter so you can use it for quick revision before your exam.

Algebra and Quadratic Equations

Algebra carries significant weight in the 10th class board paper. Here are the essential formulas you must know.

Algebraic Identities

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)

Quadratic Formula

For any equation of the form ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) / 2a

The discriminant (b² − 4ac) tells you the nature of roots:

If b² − 4ac > 0: two distinct real roots
If b² − 4ac = 0: two equal real roots
If b² − 4ac < 0: no real roots (complex roots)

Sum and Product of Roots

If α and β are roots of ax² + bx + c = 0:

Sum of roots: α + β = −b/a
Product of roots: αβ = c/a

Polynomials

Remainder Theorem: When p(x) is divided by (x − a), remainder = p(a)
Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0

These formulas appear in MCQs, short questions, and long questions. Practice applying the quadratic formula to at least 20 different equations before the exam.

Trigonometry Formulas

Trigonometry is one of the highest-scoring chapters in 10th class maths if you get the formulas right. Many students lose marks here simply because they mix up ratios or forget identities.

Basic Trigonometric Ratios

For a right-angled triangle with angle θ:

sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
cosec θ = 1 / sin θ
sec θ = 1 / cos θ
cot θ = 1 / tan θ

Pythagorean Identities

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

Standard Angle Values

| Angle | sin | cos | tan | |-------|-----|-----|-----| | 0° | 0 | 1 | 0 | | 30° | 1/2 | √3/2 | 1/√3 | | 45° | 1/√2 | 1/√2 | 1 | | 60° | √3/2 | 1/2 | √3 | | 90° | 1 | 0 | undefined |

Allied Angles

sin(90° − θ) = cos θ
cos(90° − θ) = sin θ
tan(90° − θ) = cot θ
sin(180° − θ) = sin θ
cos(180° − θ) = −cos θ

Trigonometry questions in board exams often ask you to prove identities. The key strategy is to work on the more complex side and simplify it to match the simpler side.

Geometry and Circle Theorems

Geometry in 10th class focuses on circles and triangles. The theorems must be stated precisely when asked to prove them.

Key Circle Theorems

The angle subtended at the centre is double the angle subtended at any point on the remaining part of the circle
Angles in the same segment of a circle are equal
The angle in a semicircle is 90°
Opposite angles of a cyclic quadrilateral sum to 180°
The tangent to a circle is perpendicular to the radius at the point of tangency
Tangents drawn from an external point to a circle are equal in length
The angle between a tangent and a chord equals the inscribed angle on the opposite side (Tangent-Chord Angle)

Triangle Properties

For a triangle with sides a, b, c and semi-perimeter s = (a + b + c)/2:
- Area = √(s(s−a)(s−b)(s−c)) — Heron's Formula
Sum of angles in a triangle = 180°
Exterior angle = sum of two non-adjacent interior angles

Pythagoras Theorem

In a right triangle: a² + b² = c² where c is the hypotenuse

Converse: If a² + b² = c², then the triangle has a right angle.

Coordinate Geometry Formulas

Coordinate geometry questions are highly predictable in board papers. Master these formulas and you will score full marks on these questions.

Distance Between Two Points

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Midpoint Formula

Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Slope of a Line

m = (y₂ − y₁) / (x₂ − x₁)

Equation of a Line

Slope-intercept form: y = mx + c
Point-slope form: y − y₁ = m(x − x₁)
Two-point form: (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁)

Condition for Parallel and Perpendicular Lines

Parallel lines: m₁ = m₂
Perpendicular lines: m₁ × m₂ = −1

Collinear Points

Three points (x₁,y₁), (x₂,y₂), (x₃,y₃) are collinear if: x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) = 0

Practice 5-6 coordinate geometry questions daily for two weeks before the exam. These formulas have consistent applications across all BISE boards.

Sets, Functions, and Mensuration

Sets Formulas

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
De Morgan's Laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'

Mensuration — Area and Volume

2D Shapes:

Circle: Area = πr², Circumference = 2πr
Rectangle: Area = l × w, Perimeter = 2(l + w)
Triangle: Area = ½ × base × height
Trapezium: Area = ½(a + b) × h

3D Shapes:

Cylinder: Volume = πr²h, Curved Surface Area = 2πrh
Cone: Volume = ⅓πr²h, Slant height l = √(r² + h²)
Sphere: Volume = (4/3)πr³, Surface Area = 4πr²
Cube: Volume = a³, Surface Area = 6a²
Cuboid: Volume = l × w × h

Use π = 22/7 or 3.14 based on what the question specifies. Most BISE papers specify which value to use.

How to Use This Formula Sheet Effectively

Simply reading formulas is not enough. Here is a revision method that actually works for exam preparation:

Week 3 before exam: Write each formula by hand once. Do not just read — write.

Week 2 before exam: For every formula, solve one past-paper question that uses it. Link formula to application.

Week 1 before exam: Do timed practice using this sheet as a reference. Then try without the sheet.

Day before exam: Read through the sheet once. Do not solve new problems. Trust your preparation.

Check your result and performance at /10th-class-result-2026 and use the GPA Calculator to understand how your maths marks affect your overall grade.

Frequently Asked Questions

Q: How many formulas should I memorize for 10th class maths board exam? A: Focus on the 40-50 core formulas across algebra, trigonometry, coordinate geometry, and mensuration. You do not need to memorize every variation — understand the core formula and derive others from it.

Q: Which chapter has the most formulas in 10th class maths? A: Trigonometry has the most formulas, followed by coordinate geometry. Algebra has fewer formulas but more identity applications. All three chapters usually appear in Section C (long questions).

Q: Are these formulas provided in the exam paper? A: No. BISE Pakistan board exams do not provide formula sheets. You must memorize all formulas before the exam. This is why starting early matters.

Q: How do I remember trigonometric identities? A: Start with sin²θ + cos²θ = 1 and derive the other two Pythagorean identities by dividing both sides by sin²θ or cos²θ. This way you only need to memorize one and derive the rest.

Q: What is the most common mistake students make with the quadratic formula? A: Forgetting the ± sign and calculating only one root, or making arithmetic errors under the square root. Write out every step clearly in the exam to avoid sign errors.

Q: Do I need to memorize proofs for geometry theorems? A: Yes. Board exams regularly ask for proofs, especially in Section C. For each circle theorem, understand the logical flow of the proof, not just the statement.

Q: How much time should I spend on maths revision daily? A: For the final two weeks, 2-3 hours per day focused on practice (not reading). Maths is learned by doing, not by reading notes.

Q: Is coordinate geometry important for scoring in board exams? A: Very important. Coordinate geometry questions are predictable, formula-based, and usually offer full marks if you know the formulas. It is one of the highest-return areas to prepare.