CBSE Math Revisioncbsemathrevision.com
CBSE Class 9 MathematicsAll 15 chapters · one-page formula sheet · 2026–27 syllabus
CBSE gives no formula booklet: learn every result on this sheet. Organised by the chapters of the CBSE curriculum.
1 Number System
\(\mathbb N\subset\mathbb W\subset\mathbb Z\subset\mathbb Q\subset\mathbb R\); \(\frac pq\) terminates iff \(q\) (lowest terms) has only the primes 2, 5
Recurring decimal, e.g.: \(0.\overline{47}=\frac{47}{99},\) \(0.1\overline6=\frac{15}{90}\) \({}=\frac16\)
Surds: \(\sqrt{ab}=\sqrt a\sqrt b,\) \((\sqrt a+\sqrt b)(\sqrt a-\sqrt b)\) \({}=a-b\)
Rationalise: \(\frac1{\sqrt a+\sqrt b}=\frac{\sqrt a-\sqrt b}{a-b}\)
Exponents: \(a^ma^n=a^{m+n},\) \((a^m)^n=a^{mn},\) \(a^{-m}=\frac1{a^m},\) \(a^{\frac pq}=(\sqrt[q]a)^p\)
2 Polynomials
Polynomial: degree = highest power; linear polynomial and its zero: \(p(x)=ax+b\ (a\ne0);\) \(x=-\frac ba\)
Line \(y=ax+b\): slope \(a\), \(y\)-intercept \(b\): \(a=\frac{y_2-y_1}{x_2-x_1}\)
Linear growth (\(a>0\)) / decay (\(a<0\)): equal change per equal step
3 Sequences and Progressions
AP; GP: \(a_n=a+(n-1)d;\) \(a_n=ar^{n-1}\)
Sum of naturals; of odd numbers: \(1+2+\cdots+n=\frac{n(n+1)}2;\) \(1+3+\cdots+(2n-1)=n^2\)
Fibonacci; Tower of Hanoi: \(F_{n+1}=F_n+F_{n-1};\) \(M_n=2M_{n-1}+1=2^n-1\)
4 Algebraic Identities
Squares: \((a\pm b)^2=a^2\pm2ab+b^2,\) \((a+b)(a-b)=a^2-b^2\)
Product: \((x+a)(x+b)=x^2+(a+b)x+ab\)
Three terms: \((a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\)
Cubes: \((a\pm b)^3=a^3\pm b^3\pm3ab(a\pm b)\)
Sum/difference of cubes: \(a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)\)
5 Linear Equations in Two Variables
\(ax+by+c=0\) (\(a,b\) not both 0): infinitely many solutions, all on one straight line
\(x=a\) ∥ \(y\)-axis; \(y=b\) ∥ \(x\)-axis; intercepts: put \(y=0\), then \(x=0\)
A point is on the line exactly when its coordinates satisfy the equation; \(y=mx\) passes through the origin
6 Coordinate Geometry
Quadrants I to IV: \((+,+)\), \((-,+)\), \((-,-)\), \((+,-)\); on the \(x\)-axis \(y=0\), on the \(y\)-axis \(x=0\)
Distance: \(AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2};\) \(OP=\sqrt{x^2+y^2}\)
Mid-point: \(\left(\frac{x_1+x_2}2,\ \frac{y_1+y_2}2\right)\)
Parallelogram: diagonals share a mid-point; equidistant point \(P\): \(PA^2=PB^2\)
7 Euclid's Geometry
Axioms: things equal to the same thing are equal; equals added to (or taken from) equals give equals; the whole is greater than the part
Postulates: a line through any two points; extend a segment; a circle with any centre and radius; all right angles equal
Postulate 5: interior angles on one side \(<180^\circ\) ⇒ lines meet on that side. Playfair: one parallel through a point not on a line
Measurement: \(C\) between \(A,B\) ⇒ \(AC+CB=AB\); a segment has exactly one mid-point
8 Lines and Angles
Linear pair sums to 180°; vertically opposite angles are equal; reflex angle of \(\theta\) is \(360^\circ-\theta\)
Parallel lines: corresponding and alternate angles equal, co-interior angles sum to 180° (and conversely); \(l\parallel m,\ m\parallel n\Rightarrow l\parallel n\)
Angle sum 180°; exterior angle = sum of the two interior opposite angles; at least two angles of a triangle are acute
9 Triangles – Congruence
Congruence: SSS, SAS, ASA/AAS, RHS (not SSA, not AAA); then CPCT. \(\triangle ABC\cong\triangle PQR\): \(A\leftrightarrow P\), \(B\leftrightarrow Q\), \(C\leftrightarrow R\)
Isosceles: equal sides ⇔ equal opposite angles; median to the base ⟂ base and bisects the vertex angle
"If \(P\) then \(Q\)": converse "if \(Q\) then \(P\)"; one counterexample disproves
10 Quadrilaterals
Parallelogram: opposite sides/angles equal, diagonals bisect; rhombus diagonals ⟂; rectangle diagonals equal
Mid-point theorem: \(DE\parallel BC,\) \(DE=\tfrac12BC\)
Centroid divides each median 2 : 1: \(AG=\tfrac23AD,\) \(G=\left(\tfrac{x_1+x_2+x_3}3,\ \tfrac{y_1+y_2+y_3}3\right)\)
Joining mid-points of any quadrilateral gives a parallelogram, perimeter \(AC+BD\), half the area
11 Circles
Chord \(2l\) at distance \(d\) from the centre: \(r^2=l^2+d^2\)
⟂ from centre bisects a chord; equal chords ⇔ equidistant from centre ⇔ equal angles at centre
Angle at centre = 2 × angle on the remaining arc; angles in the same segment are equal; angle in a semicircle = 90°
Cyclic quadrilateral: \(\angle A+\angle C=\angle B+\angle D\) \({}=180^\circ\)
12 Area and Perimeter
Arc; sector (angle \(\theta^\circ\)): \(l=\frac\theta{360}\cdot2\pi r,\) \(A=\frac\theta{360}\cdot\pi r^2\)
Heron, \(s=\frac{a+b+c}2\): \(A=\sqrt{s(s-a)(s-b)(s-c)}\)
Inradius; equilateral triangle: \(A=rs;\) \(A=\frac{\sqrt3}4a^2\)
Brahmagupta (cyclic quad., \(s=\frac{a+b+c+d}2\)): \(A=\sqrt{(s-a)(s-b)(s-c)(s-d)}\)
Trapezium; rhombus: \(\tfrac12(a+b)h;\) \(\tfrac12d_1d_2\)
13 Surface Area and Volume
Cuboid; cube: \(2(lb+bh+hl),\ lbh;\) \(6a^2,\ a^3\)
Cylinder: \(\text{CSA}=2\pi rh,\) \(\text{TSA}=2\pi r(r+h),\) \(V=\pi r^2h\)
Cone, \(l=\sqrt{r^2+h^2}\): \(\text{CSA}=\pi rl,\) \(\text{TSA}=\pi r(l+r),\) \(V=\tfrac13\pi r^2h\)
Sphere; hemisphere: \(4\pi r^2,\ \tfrac43\pi r^3;\) \(2\pi r^2\ (3\pi r^2\text{ total}),\ \tfrac23\pi r^3\)
\(1\text{ m}^3=1000\) litres; scale factor \(k\): areas × \(k^2\), volumes × \(k^3\)
14 Statistics
Mean; weighted mean: \(\bar x=\frac{\sum f_ix_i}{\sum f_i};\) \(\bar x_w=\frac{\sum w_ix_i}{\sum w_i}\)
Combined mean of two groups: \(\frac{n_1\bar x_1+n_2\bar x_2}{n_1+n_2}\)
Mixture concentration: \(\frac{q_1c_1+q_2c_2}{q_1+q_2}\)
Average speed = total distance ÷ total time; 100% stacked bars compare proportions, not totals
Median: middle value in order (\(n\) even: mean of the two middle values); mode: most frequent value
15 Probability
Experimental: \(P(E)=\frac{\text{trials with }E}{\text{total trials}}\)
Theoretical (equally likely): \(P(E)=\frac{\text{favourable}}{\text{total}};\) \(P(\text{not }E)=1-P(E)\)
\(0\le P\le1\); tree diagram: one branch per stage; two coins 4, coin + die 12, two dice 36 outcomes