Revision reference
Every formula in the CBSE Class 9 syllabus, unit by unit — and next to each one, the short reason it is true. Formulas you understand are the ones you still remember in the exam hall.
Every real number lives somewhere on one line — rationals as fractions, irrationals as the gaps in between.
aᵐ × aⁿ = aᵐ⁺ⁿ
Counting the factors: m of them plus n more.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
m factors, then n of them cancel out.
(aᵐ)ⁿ = aᵐⁿ
n groups of m factors each.
a⁻ᵐ = 1/aᵐ
A negative power just moves the number below the fraction bar.
a^(1/n) = ⁿ√a
The n-th root is the power that undoes aⁿ.
1/√a → multiply by √a/√a
Rationalising: √a × √a = a clears the root from the denominator.
1/(a + √b) → multiply by (a − √b)/(a − √b)
(a + √b)(a − √b) = a² − b, no roots left.
√(a·b) = √a × √b
Why you can pull perfect squares out from under a root.
You never need to divide a long polynomial to find its remainder — just evaluate it at one point.
Remainder of p(x) ÷ (x − a) is p(a)
The Remainder Theorem — substitution replaces long division.
(x − a) is a factor of p(x) ⟺ p(a) = 0
The Factor Theorem — zero remainder means it divides exactly.
(a + b)² = a² + 2ab + b²
A square of side a + b splits into four regions.
(a − b)² = a² − 2ab + b²
Same picture, but the ab strip overlaps and is counted twice.
a² − b² = (a + b)(a − b)
Cut the b-square out of the a-square and rearrange into a rectangle.
(x + a)(x + b) = x² + (a + b)x + ab
The middle coefficient is the sum, the constant is the product.
(a + b)³ = a³ + b³ + 3ab(a + b)
A cube of side a + b cut into pieces — the 3ab slab is the awkward one.
(a − b)³ = a³ − b³ − 3ab(a − b)
Replace b with −b in the expansion above.
a³ + b³ = (a + b)(a² − ab + b²)
Factorising a sum of cubes.
a³ − b³ = (a − b)(a² + ab + b²)
Factorising a difference of cubes.
a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)
When a + b + c = 0, the left side is simply 0.
Degree of p(x) × q(x) = deg p + deg q
The highest powers multiply together.
Two numbers pin down a point, and the signs of those numbers tell you which quadrant it sits in.
Point = (x, y)
x = horizontal distance from the origin, y = vertical.
Quadrant I: (+, +), II: (−, +), III: (−, −), IV: (+, −)
Anticlockwise from top-right.
Point on x-axis → (x, 0)
No vertical distance means y is zero.
Point on y-axis → (0, y)
No horizontal distance means x is zero.
Origin = (0, 0)
Where both axes cross.
One equation with two unknowns is a whole line of answers — every point on the graph is a solution.
ax + by + c = 0
The standard form; a and b can't both be zero.
Infinitely many solutions
Pick any x, the equation hands you the matching y.
x = k → vertical line
Every point has the same x — a line parallel to the y-axis.
y = k → horizontal line
Every point has the same y — a line parallel to the x-axis.
Line through the origin → c = 0
(0, 0) satisfies the equation.
Two points determine the line
Plot any two solutions and join them.
Everything in geometry is built from a handful of self-evident starting points, applied by pure logic.
Things equal to the same thing are equal
Euclid's first axiom — the basis of every 'hence' in a proof.
A straight line may be drawn from any point to any point
First postulate.
A terminated line can be extended indefinitely
Second postulate — lines have no ends.
A circle can be drawn with any centre and any radius
Third postulate.
All right angles are equal
Fourth postulate.
Through a point outside a line, exactly one parallel line exists
The fifth (parallel) postulate — the one that defines flat geometry.
Angles come in fixed partnerships — once one is known, its partners follow automatically.
Linear pair: ∠A + ∠B = 180°
Two angles forming a straight line.
Vertically opposite angles are equal
Formed by two crossing lines.
Corresponding angles are equal (parallel lines)
Transversal cutting parallel lines.
Alternate interior angles are equal (parallel lines)
The Z-pattern pair.
Co-interior angles: sum = 180° (parallel lines)
The C-pattern pair — same-side interior.
Angle sum of a triangle = 180°
Proved by drawing a line parallel to one side through the third vertex.
Exterior angle = sum of the two opposite interior angles
The exterior and its neighbour make a linear pair.
Five rules guarantee two triangles are identical copies — and SSA is not one of them.
SSS — all three sides equal
Rigid frame: sides fix the angles.
SAS — two sides and the included angle
The angle must sit between the two sides.
ASA / AAS — two angles and a side
The third angle is forced by the 180° rule anyway.
RHS — right angle, hypotenuse, one side
Right triangles only.
a + b > c (for any two sides)
Triangle inequality — a side can't reach past the other two.
Larger side ⟷ larger opposite angle
Sides and their opposite angles grow together.
One diagonal cuts a quadrilateral into two triangles — so every quadrilateral fact is a triangle fact in disguise.
Sum of interior angles = 360°
Two triangles' worth of 180°.
Parallelogram: opposite sides equal and parallel
Proved by the alternate-angles rule.
Parallelogram: opposite angles equal
And each diagonal creates equal alternate angles.
Diagonals of a parallelogram bisect each other
The two triangles formed are congruent.
Rhombus: diagonals bisect at 90°
All four sides equal forces the perpendicular crossing.
Rectangle: diagonals are equal
Right angles make the two crossing diagonals congruent.
Square = rhombus + rectangle
Equal sides and right angles together.
Mid-point theorem: segment joining mid-points ∥ third side and half its length
Extend and double to build a parallelogram.
Converse: line through one mid-point, ∥ to a side, bisects the other side
Same parallelogram trick, read backwards.
Almost every circle proof is one of three facts: equal chords, the double angle at the centre, or the cyclic quadrilateral rule.
Equal chords are equidistant from the centre
And the converse holds too.
Perpendicular from centre bisects the chord
A radius standing on a chord makes two congruent halves.
Angle at centre = 2 × angle at circumference (same arc)
The exterior angle of an isosceles triangle, twice.
Angles in the same segment are equal
Each is half the same central angle.
Angle in a semicircle = 90°
Half of the 180° angle a diameter makes at the centre.
Cyclic quadrilateral: opposite angles sum to 180°
Interior + exterior at the opposite vertex is a linear pair.
Area without any height or angle — three sides are enough, because they fix the triangle's shape completely.
s = (a + b + c) / 2
The semi-perimeter.
Area = √(s(s−a)(s−b)(s−c))
Heron's formula — no height needed.
Equilateral triangle: Area = (√3/4)a²
What Heron's formula collapses to when all sides are equal.
Perimeter of a triangle = a + b + c
Needed to compute s first.
A cone is one-third of its cylinder, and a sphere is two-thirds of its cylinder — the formulas remember that for you.
Cuboid: V = lbh, SA = 2(lb + bh + hl)
Six rectangular faces, in matching pairs.
Cube: V = a³, SA = 6a²
A cuboid with l = b = h = a.
Cylinder: CSA = 2πrh, TSA = 2πr(r + h), V = πr²h
A rolled rectangle plus two circular ends.
Cone: CSA = πrl, TSA = πr(l + r), V = ⅓πr²h
l = √(r² + h²) is the slant height; three cones fill the cylinder.
Sphere: SA = 4πr², V = 4/3·πr³
Exactly four great circles wrap a sphere.
Hemisphere: CSA = 2πr², TSA = 3πr², V = ⅔πr³
Half the sphere plus one flat circular lid.
The mean is the fair share: total value divided among total count.
Mean = Σfᵢxᵢ / Σfᵢ
Weighted average for a frequency table.
Mean = Σxᵢ / n
Simple average of n raw values.
Range = maximum − minimum
The spread of the data.
Median = middle value (odd n), mean of two middle values (even n)
Sort the data first, always.
Mode = the most frequent value
A dataset can have more than one — or none.
Each Class 9 chapter has an animated lesson, a worked example and a quiz that explains every answer.