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CBSE Class 10 MathematicsAll chapters · one-page formula sheet · 2026–27 rationalised syllabus
CBSE gives no formula booklet: learn every result on this sheet. Organised by NCERT chapter.
1 Real Numbers
Fundamental Theorem of Arithmetic: every composite number is a product of primes, uniquely apart from orderUse: Write composite numbers as a unique product of prime powers
HCF = product of the smallest power of each common prime; LCM = product of the greatest power of each primeUse: HCF and LCM by comparing the powers of each prime
For two positive integers: \(\text{HCF}(a,b)\times\text{LCM}(a,b)\) \({}=a\times b\)Use: A missing HCF or LCM; when two bells next ring together
\(\sqrt2,\sqrt3,\sqrt5\) and \(\sqrt p\) (\(p\) prime) are irrational: prove by contradiction (\(p\mid a^2\Rightarrow p\mid a\))Use: Prove \(\sqrt{2}\), \(\sqrt{3}\) and \(\sqrt{5}\) are irrational by contradiction
Watch for: Show the formula, the substitution and the simplification
2 Polynomials
Zeroes \(\alpha,\beta\) of \(ax^2+bx+c\): \(\alpha+\beta=-\frac ba,\) \(\alpha\beta=\frac ca\)Use: Symmetric expressions such as \(\alpha^{2} + \beta^{2}\) without finding the zeroes
Quadratic with zeroes \(\alpha,\beta\): \(k\big[x^2-(\alpha+\beta)x+\alpha\beta\big]\)Use: Form a quadratic polynomial from the sum and product of its zeroes
Zeroes of \(p(x)\) = \(x\)-coordinates where \(y=p(x)\) meets the \(x\)-axis; degree \(n\) ⇒ at most \(n\) zeroesUse: Read the number of zeroes from a graph
3 Pair of Linear Equations
\(a_1x+b_1y+c_1=0\), \(a_2x+b_2y+c_2=0\)Use: Graph a pair of linear equations; read the solution
Unique solution (lines intersect): \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\)Use: Compare \(\frac{a_{1}}{a_{2}}\), \(\frac{b_{1}}{b_{2}}\), \(\frac{c_{1}}{c_{2}}\) to decide one, no or infinitely many solutions
No solution (parallel, inconsistent): \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\)Use: Decide from the ratios whether a pair is inconsistent
Infinitely many (coincident): \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\) \({}=\frac{c_1}{c_2}\)Use: Find unknown constants for infinitely many solutions
Solve by substitution or elimination; consistent = at least one solutionUse: Word problems about ages, numbers and speeds
4 Quadratic Equations
\(ax^2+bx+c=0,\ a\ne0\): \(x=\frac{-b\pm\sqrt D}{2a},\) \(D=b^2-4ac\)Use: Solve with the formula, checking the discriminant firstWatch for: Bracket negatives when substituting
Nature of roots: \(D>0\text{ two distinct real};\) \(D=0\text{ two equal real }\big(x\) \({}=-\tfrac b{2a}\big);\) \(D<0\text{ no real roots}\)Use: The value of \(k\) for equal or real roots
5 Arithmetic Progressions
\(n\)th term: \(a_n=a+(n-1)d\)Use: Any term of an AP; find \(a\) or \(d\) from given termsWatch for: \(a_{n} = a + (n - 1)d\): check your formula with \(n = 1\)
Sum of first \(n\) terms: \(S_n=\frac n2\big[2a+(n-1)d\big]\) \({}=\frac n2(a+l)\)Use: Unknown \(n\), \(a\) or \(d\), such as how many terms give 300
From \(S_n\): \(a_n=S_n-S_{n-1}\)Use: Find the nth term from a given sum formula
\(n\)th term from the end (last term \(l\)): \(l-(n-1)d\)Use: A term counted from the end
Three terms in AP; middle term: \(a-d,\ a,\ a+d;\) \(b=\frac{a+c}2\)Use: Decide whether a given list forms an AP
Natural numbers: \(1+2+\cdots+n=\frac{n(n+1)}2\)Use: Sums of consecutive natural numbers
Watch for: Check \(a_{n} = a + (n - 1)d\) with \(n = 1\)
6 Triangles
Basic Proportionality Theorem: if \(DE\parallel BC\) in \(\triangle ABC\): \(\frac{AD}{DB}=\frac{AE}{EC}\)Use: Unknown lengths when a line is parallel to one side
Converse: if a line divides two sides in the same ratio, it is parallel to the third sideUse: Show a line is parallel to a side from given ratios
Similar triangles (AA, SSS, SAS): corresponding angles equal, corresponding sides in the same ratioUse: Shadows, heights and ladders with similar triangles
Pythagoras (right angle at \(B\)): \(AC^2=AB^2+BC^2\)Use: Any right-angled triangle, in 2D or 3D
7 Coordinate Geometry
Distance: \(PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2};\) \(OP=\sqrt{x^2+y^2}\)Use: Show points are collinear or form a given triangle
Section formula, ratio \(m:n\) (internal): \(\left(\frac{mx_2+nx_1}{m+n},\ \frac{my_2+ny_1}{m+n}\right)\)Use: A point dividing a segment internally in a given ratio
Midpoint: \(\left(\frac{x_1+x_2}2,\ \frac{y_1+y_2}2\right)\)Use: Missing vertices of parallelograms: diagonals bisect each other
Ratio in which a line or axis divides \(AB\): take \(k:1\) and solve for \(k\)Use: The ratio in which an axis divides a segment
8 Introduction to Trigonometry
Ratios (right \(\triangle\)): \(\sin A=\frac PH,\) \(\cos A=\frac BH,\) \(\tan A=\frac PB=\frac{\sin A}{\cos A}\)Use: All the ratios when one is given, such as \(\sin A = \frac{3}{5}\)
Reciprocals: \(\cosec A=\frac1{\sin A},\) \(\sec A=\frac1{\cos A},\) \(\cot A=\frac1{\tan A}\)Use: Write each ratio from a right-angled triangle
Identities: \(\sin^2A+\cos^2A=1,\) \(1+\tan^2A=\sec^2A,\) \(1+\cot^2A=\cosec^2A\)Use: Prove identities from one side to the otherWatch for: Name each identity used and end with "\(=\) RHS"
\(0^\circ,30^\circ,45^\circ,60^\circ,90^\circ\): \(\sin:\ 0,\tfrac12,\tfrac1{\sqrt2},\tfrac{\sqrt3}2,1;\) \(\cos:\ 1,\tfrac{\sqrt3}2,\tfrac1{\sqrt2},\tfrac12,0\)Use: Evaluate expressions such as \(2 \sin 30^\circ \cos 60^\circ + \tan^{2} 45^\circ\)
\(\tan\) at the same angles: \(0,\ \tfrac1{\sqrt3},\ 1,\ \sqrt3,\ \text{not defined}\)Use: Exact answers from the ratios of standard angles
Watch for: Work from one side to the other and end with "\(=\) RHS"
9 Applications of Trigonometry
Height and distance: \(\tan\theta=\frac{\text{height}}{\text{horizontal distance}}\)Use: Height of a tower from an angle of elevation
Angle of elevation (looking up) = angle of depression (looking down) between the same two points; use 30°, 45°, 60°Use: How far a boat is from the foot of a lighthouse
10 Circles
Tangent ⟂ radius at the point of contactUse: Use the tangent ⟂ radius result to find angles
Tangents from an external point are equal: \(PA=PB\)Use: Quadrilaterals and triangles that circumscribe a circle
Length of tangent from \(P\), distance \(d\) from centre: \(\sqrt{d^2-r^2}\)Use: Length of a tangent from the radius and the distance from the centre
Tangents from a point: inside 0, on the circle 1, outside 2Use: How many tangents from points inside, on and outside
11 Areas Related to Circles
Circle: \(C=2\pi r,\) \(A=\pi r^2\)Use: Circles, sectors, cylinders and compound shapesWatch for: Square \(r\), use the \(\pi\) given, finish with the unit
Arc length; sector area (angle \(\theta\)): \(l=\frac\theta{360^\circ}\times2\pi r,\) \(A=\frac\theta{360^\circ}\times\pi r^2\)Use: Distance moved by a clock hand; areas swept
Segment: \(\text{area}=\text{sector}\) \({}-\tfrac12r^2\sin\theta\)Use: Minor or major segment: sector minus triangle
Segment, \(\theta=60^\circ,90^\circ,120^\circ\): subtract \(\tfrac{\sqrt3}4r^2\), \(\tfrac12r^2\), \(\tfrac{\sqrt3}4r^2\)Use: Segments for sectors of \(60^\circ\) and \(90^\circ\)
Perimeter of sector; ring: \(2r+l;\) \(\pi(R^2-r^2)\)Use: Combined figures with circles, sectors and squares
Minute hand sweeps \(6^\circ\) per minute; hour hand \(0.5^\circ\) per minuteUse: Distance moved by the tip of a clock hand
Watch for: Square \(r\), use the \(\pi\) given and finish with the unit
12 Surface Areas and Volumes
Cuboid \(l,b,h\): \(\text{TSA}=2(lb+bh+hl),\) \(\text{LSA}=2h(l+b),\) \(V=lbh\)Use: Surface area and volume of single solids
Cube, edge \(a\): \(\text{TSA}=6a^2,\) \(V=a^3\)Use: Surface area and volume of single solids
Cylinder: \(\text{CSA}=2\pi rh,\) \(\text{TSA}=2\pi r(r+h),\) \(V=\pi r^2h\)Use: Surface area and volume of single solids
Cone, \(l=\sqrt{r^2+h^2}\): \(\text{CSA}=\pi rl,\) \(\text{TSA}=\pi r(l+r),\) \(V=\tfrac13\pi r^2h\)Use: Combined solids, such as a cone on a hemisphere
Sphere: \(\text{SA}=4\pi r^2,\) \(V=\tfrac43\pi r^3\)Use: How many small solids can be made from a larger one
Hemisphere: \(\text{CSA}=2\pi r^2,\) \(\text{TSA}=3\pi r^2,\) \(V=\tfrac23\pi r^3\)Use: Combined solids, such as a cone on a hemisphere
Combined solids: add only the surfaces you can see; volumes simply add (or subtract)Use: Tents, toys and capsules: which surfaces to include
Watch for: Write the formula, square \(r\) and give the right unit
13 Statistics
Class mark: \(x_i=\frac{\text{upper}+\text{lower limit}}2\)Use: Mean of grouped data using class marks
Mean: direct: \(\bar x=\frac{\sum f_ix_i}{\sum f_i}\)Use: Mean of grouped data by the direct method
Assumed mean, \(d_i=x_i-a\): \(\bar x=a+\frac{\sum f_id_i}{\sum f_i}\)Use: Mean of grouped data by the assumed-mean method
Step deviation, \(u_i=\frac{x_i-a}h\): \(\bar x=a+\frac{\sum f_iu_i}{\sum f_i}\times h\)Use: Mean of grouped data by step deviation
Mode: \(l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h\)Use: Mode of grouped data from the modal classWatch for: Modal class: the highest frequency
Median: \(l+\frac{\frac n2-cf}{f}\times h\)Use: Median from the cumulative frequency tableWatch for: Median class: the first \(cf \ge \frac{n}{2}\)
Empirical relation: \(3\,\text{Median}=\text{Mode}\) \({}+2\,\text{Mean}\)Use: One measure of central tendency from the other two
Mode: \(l\), \(f_1\) of the modal class, \(f_0,f_2\) of the classes before/after. Median: \(l\), \(f\) of the median class, \(cf\) = cumulative frequency before it.
Watch for: Median class: first \(cf \ge \frac{n}{2}\); modal class: highest frequency
14 Probability
Classical probability: \(P(E)=\frac{\text{favourable outcomes}}{\text{total outcomes}}\)Use: Probability from equally likely outcomesWatch for: List equally likely outcomes first, then count favourable ones
Complement; range: \(P(E)+P(\bar E)=1;\) \(0\le P(E)\le1\)Use: Complementary, sure and impossible events
Sure event 1, impossible event 0; probabilities of all elementary events add to 1Use: Complementary, sure and impossible events
Cards: 52 = 4 suits × 13; 26 red, 26 black; 12 face cards. Two dice: 36 outcomesUse: Probabilities with a deck of 52 cards or two dice
Watch for: Count equally likely outcomes before favourable ones
