Class 10 Maths Formula Sheet (CBSE 2026-27)
NCERT Class 10 · All 14 Chapters · Free PDF
Every formula from all 14 NCERT Class 10 Maths chapters in one place, checked against
the NCERT textbook (Reprint 2026-27) and the official CBSE 2026-27 syllabus. Covers Maths Standard and Maths Basic (same syllabus).
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1Real Numbers
Fundamental Theorem of Arithmetic
- Every composite number can be written as a product of primes, and this factorisation is unique apart from the order of the factors
- HCF = product of the smallest power of each common prime factor
- LCM = product of the greatest power of each prime factor involved
- For two positive integers: \(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\)
The product rule works for two numbers only, not for three or more.
Irrational numbers
- If a prime \(p\) divides \(a^2\), then \(p\) divides \(a\) (\(a\) a positive integer)
- \(\sqrt2, \sqrt3, \sqrt5\) (and \(\sqrt p\) for any prime \(p\)) are irrational
- Proof method (contradiction): assume \(\sqrt p = \dfrac{a}{b}\) with \(a, b\) coprime, then show \(p\) divides both \(a\) and \(b\)
- Rational \(\pm\) irrational is irrational; non-zero rational \(\times\) irrational is irrational (e.g. \(3 + 2\sqrt5\), \(7\sqrt5\))
Not in 2026-27 syllabus: decimal expansions
- A rational \(\dfrac{p}{q}\) (in lowest terms) has a terminating decimal expansion \(\iff\) \(q\) is of the form \(2^n 5^m\); otherwise it is non-terminating repeating. Irrational numbers are non-terminating non-repeating.
2Polynomials
Basics
- Degree 1, 2, 3: linear, quadratic, cubic
- Quadratic: \(ax^2 + bx + c,\ a \ne 0\)
- \(k\) is a zero of \(p(x)\) if \(p(k) = 0\)
- Zeroes of \(p(x)\) = x-coordinates of the points where \(y = p(x)\) meets the x-axis
- A polynomial of degree \(n\) has at most \(n\) zeroes
- Graph of \(ax^2 + bx + c\) is a parabola: opens upward if \(a > 0\), downward if \(a < 0\)
Zeroes and coefficients (quadratic)
- If \(\alpha, \beta\) are the zeroes of \(ax^2 + bx + c\):
- \(\alpha + \beta = -\dfrac{b}{a} = -\dfrac{\text{coefficient of } x}{\text{coefficient of } x^2}\)
- \(\alpha\beta = \dfrac{c}{a} = \dfrac{\text{constant term}}{\text{coefficient of } x^2}\)
- Quadratic with given zeroes: \(k\left[x^2 - (\alpha + \beta)x + \alpha\beta\right]\), \(k \ne 0\)
- Useful: \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\); \(\ \dfrac{1}{\alpha} + \dfrac{1}{\beta} = \dfrac{\alpha + \beta}{\alpha\beta}\)
Not in 2026-27 syllabus: cubic polynomials (still in the NCERT book)
- If \(\alpha, \beta, \gamma\) are the zeroes of \(ax^3 + bx^2 + cx + d\): \(\ \alpha + \beta + \gamma = -\dfrac{b}{a}\), \(\ \alpha\beta + \beta\gamma + \gamma\alpha = \dfrac{c}{a}\), \(\ \alpha\beta\gamma = -\dfrac{d}{a}\)
3Pair of Linear Equations in Two Variables
Consistency for \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\)
| Condition | Graph | Solutions | Type |
|---|
| \(\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}\) | intersecting lines | exactly one (unique) | consistent |
| \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\) | coincident lines | infinitely many | dependent (consistent) |
| \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}\) | parallel lines | none | inconsistent |
Algebraic methods
- Substitution: express one variable in terms of the other from one equation, substitute into the other
- Elimination: multiply the equations to make the coefficients of one variable equal, then add or subtract to eliminate it
- A false statement (e.g. \(0 = 5\)) means no solution; a true statement (e.g. \(0 = 0\)) means infinitely many
Reducible equations and word problems
- Equations like \(\dfrac{2}{x} + \dfrac{3}{y} = 13\): put \(u = \dfrac{1}{x}\), \(v = \dfrac{1}{y}\) to make them linear
- Speed \(= \dfrac{\text{distance}}{\text{time}}\); downstream speed \(= u + v\), upstream \(= u - v\)
- Two-digit number with digits \(x\) (tens) and \(y\) (units) \(= 10x + y\)
Not in 2026-27 syllabus: cross-multiplication
- \(\dfrac{x}{b_1c_2 - b_2c_1} = \dfrac{y}{c_1a_2 - c_2a_1} = \dfrac{1}{a_1b_2 - a_2b_1}\), provided \(a_1b_2 - a_2b_1 \ne 0\)
4Quadratic Equations
Standard form and roots
- \(ax^2 + bx + c = 0,\ a \ne 0\) (\(a, b, c\) real)
- \(\alpha\) is a root if \(a\alpha^2 + b\alpha + c = 0\)
- Roots of \(ax^2 + bx + c = 0\) = zeroes of the polynomial \(ax^2 + bx + c\)
- Factorisation: split the middle term (find two numbers with sum \(b\) and product \(ac\)), then set each factor to 0
Quadratic formula
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\quad b^2 - 4ac \ge 0$$
- Sum of roots \(= -\dfrac{b}{a}\), product of roots \(= \dfrac{c}{a}\)
Nature of roots (discriminant \(D = b^2 - 4ac\))
| \(D > 0\) | \(D = 0\) | \(D < 0\) |
|---|
| two distinct real roots | two equal (coincident) real roots, each \(= -\dfrac{b}{2a}\) | no real roots |
Word problems: area of rectangle = length × breadth; consecutive integers \(x, x + 1\); consecutive odd/even integers \(x, x + 2\); time = distance ÷ speed. Reject answers that make no sense (e.g. negative lengths).
5Arithmetic Progressions
Terms
- General form: \(a,\ a + d,\ a + 2d,\ a + 3d, \dots\); \(d = a_{k+1} - a_k\) (same for all \(k\))
- \(n\)th term: \(a_n = a + (n - 1)d\)
- \(n\)th term from the end: \(l - (n - 1)d\), \(l\) = last term
- \(a, b, c\) are in AP \(\iff 2b = a + c\)
- Three terms in AP: \(a - d,\ a,\ a + d\)
Sum of first \(n\) terms
- \(S_n = \dfrac{n}{2}\left[2a + (n - 1)d\right]\)
- \(S_n = \dfrac{n}{2}(a + l)\), \(l\) = last (\(n\)th) term
- \(a_n = S_n - S_{n-1}\)
- Sum of first \(n\) natural numbers: \(1 + 2 + \dots + n = \dfrac{n(n + 1)}{2}\)
6Triangles
Similar figures
- Same shape, not necessarily the same size
- All congruent figures are similar; the converse is not true
- Two polygons with the same number of sides are similar if corresponding angles are equal and corresponding sides are in the same ratio
- All circles, all squares, all equilateral triangles are similar
Basic Proportionality Theorem (Thales)
- If \(DE \parallel BC\) in \(\triangle ABC\) (D on AB, E on AC), then \(\dfrac{AD}{DB} = \dfrac{AE}{EC}\)
- Also \(\dfrac{AD}{AB} = \dfrac{AE}{AC}\) and \(\dfrac{DE}{BC} = \dfrac{AD}{AB}\)
- Converse: if \(\dfrac{AD}{DB} = \dfrac{AE}{EC}\), then \(DE \parallel BC\)
Similarity criteria for \(\triangle ABC \sim \triangle DEF\)
| Criterion | Condition |
|---|
| AAA | \(\angle A = \angle D,\ \angle B = \angle E,\ \angle C = \angle F\) |
| AA | two pairs of corresponding angles equal (the third pair is then equal too) |
| SSS | \(\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{CA}{FD}\) |
| SAS | \(\angle A = \angle D\) and \(\dfrac{AB}{DE} = \dfrac{AC}{DF}\) |
If two triangles are similar, all corresponding sides, perimeters, altitudes and medians are in the same ratio. Write vertices in matching order.
Not in 2026-27 syllabus: areas and Pythagoras
- Ratio of areas of similar triangles = square of the ratio of corresponding sides: \(\dfrac{\text{ar}(ABC)}{\text{ar}(DEF)} = \left(\dfrac{AB}{DE}\right)^2\)
- Pythagoras: in a right triangle, \(\text{hypotenuse}^2 = \text{base}^2 + \text{perpendicular}^2\) (still used throughout Class 10)
7Coordinate Geometry
Distance formula
- \(PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
- Distance from origin: \(OP = \sqrt{x^2 + y^2}\)
- Uses: check collinearity (\(AB + BC = AC\)), identify a square, rhombus, rectangle or the type of triangle
Section formula (internal, ratio \(m_1 : m_2\))
$$P = \left(\frac{m_1x_2 + m_2x_1}{m_1 + m_2},\ \frac{m_1y_2 + m_2y_1}{m_1 + m_2}\right)$$
- Mid-point: \(\left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)\)
- Ratio \(k : 1\) form: \(\left(\dfrac{kx_2 + x_1}{k + 1},\ \dfrac{ky_2 + y_1}{k + 1}\right)\)
- Points of trisection divide in ratios \(1 : 2\) and \(2 : 1\)
Not in 2026-27 syllabus: area and centroid
- Area of triangle \(= \dfrac{1}{2}\left|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right|\); the points are collinear if this is 0
- Centroid \(= \left(\dfrac{x_1 + x_2 + x_3}{3},\ \dfrac{y_1 + y_2 + y_3}{3}\right)\)
8Introduction to Trigonometry
Ratios (right angle at B)
- \(\sin A = \dfrac{\text{opposite}}{\text{hypotenuse}}\)
- \(\cos A = \dfrac{\text{adjacent}}{\text{hypotenuse}}\)
- \(\tan A = \dfrac{\text{opposite}}{\text{adjacent}}\)
- \(\operatorname{cosec} A = \dfrac{1}{\sin A}\), \(\ \sec A = \dfrac{1}{\cos A}\), \(\ \cot A = \dfrac{1}{\tan A}\)
- \(\tan A = \dfrac{\sin A}{\cos A}\), \(\ \cot A = \dfrac{\cos A}{\sin A}\)
- \(\sin A, \cos A \le 1\); \(\ \sec A, \operatorname{cosec} A \ge 1\) (for acute \(A\))
Knowing one ratio lets you find the rest (draw the triangle and use Pythagoras).
Identities
- \(\sin^2 A + \cos^2 A = 1\)
- \(\sec^2 A - \tan^2 A = 1\) \((0^\circ \le A < 90^\circ)\)
- \(\operatorname{cosec}^2 A - \cot^2 A = 1\) \((0^\circ < A \le 90^\circ)\)
- Rearranged: \(\sin^2 A = 1 - \cos^2 A\), \(\ \sec^2 A = 1 + \tan^2 A\), \(\ \operatorname{cosec}^2 A = 1 + \cot^2 A\), \(\ (\sec A - \tan A)(\sec A + \tan A) = 1\)
Values of standard angles
| \(\angle A\) | \(0^\circ\) | \(30^\circ\) | \(45^\circ\) | \(60^\circ\) | \(90^\circ\) |
|---|
| \(\sin A\) | 0 | \(\frac{1}{2}\) | \(\frac{1}{\sqrt2}\) | \(\frac{\sqrt3}{2}\) | 1 |
|---|
| \(\cos A\) | 1 | \(\frac{\sqrt3}{2}\) | \(\frac{1}{\sqrt2}\) | \(\frac{1}{2}\) | 0 |
|---|
| \(\tan A\) | 0 | \(\frac{1}{\sqrt3}\) | 1 | \(\sqrt3\) | not defined |
|---|
| \(\operatorname{cosec} A\) | not defined | 2 | \(\sqrt2\) | \(\frac{2}{\sqrt3}\) | 1 |
|---|
| \(\sec A\) | 1 | \(\frac{2}{\sqrt3}\) | \(\sqrt2\) | 2 | not defined |
|---|
| \(\cot A\) | not defined | \(\sqrt3\) | 1 | \(\frac{1}{\sqrt3}\) | 0 |
|---|
As \(A\) increases from \(0^\circ\) to \(90^\circ\), \(\sin A\) increases from 0 to 1 and \(\cos A\) decreases from 1 to 0.
Not in 2026-27 syllabus: complementary angles
- \(\sin(90^\circ - A) = \cos A\), \(\ \cos(90^\circ - A) = \sin A\), \(\ \tan(90^\circ - A) = \cot A\), \(\ \cot(90^\circ - A) = \tan A\), \(\ \sec(90^\circ - A) = \operatorname{cosec} A\), \(\ \operatorname{cosec}(90^\circ - A) = \sec A\)
9Some Applications of Trigonometry
Definitions
- Line of sight: line from the observer's eye to the object
- Angle of elevation: between the line of sight and the horizontal, object above eye level
- Angle of depression: between the line of sight and the horizontal, object below eye level
- The angle of elevation of B from A equals the angle of depression of A from B (alternate angles)
Solving heights and distances
- \(\tan\theta = \dfrac{\text{height}}{\text{horizontal distance}}\) (most used)
- \(\sin\theta = \dfrac{\text{height}}{\text{slant length}}\) (e.g. a ladder, kite string)
- Add the observer's height if the eye is above the ground
- Board problems use only \(30^\circ, 45^\circ, 60^\circ\) and at most two right triangles. \(\sqrt3 \approx 1.732\), \(\sqrt2 \approx 1.414\).
10Circles
Tangents
- A tangent meets the circle at exactly one point (the point of contact); a secant meets it at two
- Point inside the circle: no tangent; on the circle: exactly one; outside: exactly two
- Theorem 1: the tangent at any point is perpendicular to the radius through the point of contact
- Theorem 2: the two tangents from an external point are equal in length
Useful results
- Tangent length from \(P\), distance \(d\) from centre: \(\ \sqrt{d^2 - r^2}\)
- Tangents from an external point subtend equal angles at the centre and are equally inclined to the line joining the point to the centre
- Angle between two tangents from \(P\) + angle between the radii to the contact points \(= 180^\circ\)
- Tangents at the ends of a diameter are parallel
11Areas Related to Circles
Circle, sector and arc (angle \(\theta\) in degrees)
- Circumference \(= 2\pi r\); area \(= \pi r^2\)
- Arc length: \(\ l = \dfrac{\theta}{360^\circ} \times 2\pi r\)
- Area of sector: \(\ \dfrac{\theta}{360^\circ} \times \pi r^2 = \dfrac{1}{2}\,l\,r\)
- Perimeter of sector \(= 2r + l\)
- Major sector area \(= \pi r^2 -\) minor sector area (angle \(360^\circ - \theta\))
- Semicircle area \(= \dfrac{\pi r^2}{2}\); quadrant \(= \dfrac{\pi r^2}{4}\)
Segment
- Minor segment = area of sector \(-\) area of triangle \(OAB\)
- Major segment = \(\pi r^2 -\) minor segment
| \(\theta\) | Area of \(\triangle OAB\) |
|---|
| \(60^\circ\) | \(\dfrac{\sqrt3}{4}r^2\) (equilateral) |
| \(90^\circ\) | \(\dfrac{1}{2}r^2\) |
| \(120^\circ\) | \(\dfrac{\sqrt3}{4}r^2\) |
Board problems on segments use only \(60^\circ, 90^\circ, 120^\circ\). Clock: the minute hand turns \(6^\circ\) per minute. Use \(\pi = \frac{22}{7}\) or 3.14 as instructed.
12Surface Areas and Volumes
Basic solids
| Solid | Curved / lateral surface area | Total surface area | Volume |
|---|
| Cuboid \(l \times b \times h\) | \(2h(l + b)\) | \(2(lb + bh + hl)\) | \(lbh\) |
| Cube, side \(a\) | \(4a^2\) | \(6a^2\) | \(a^3\) |
| Cylinder | \(2\pi rh\) | \(2\pi r(r + h)\) | \(\pi r^2h\) |
| Cone, \(l = \sqrt{r^2 + h^2}\) | \(\pi rl\) | \(\pi r(l + r)\) | \(\dfrac{1}{3}\pi r^2h\) |
| Sphere | \(4\pi r^2\) | \(4\pi r^2\) | \(\dfrac{4}{3}\pi r^3\) |
| Hemisphere | \(2\pi r^2\) | \(3\pi r^2\) | \(\dfrac{2}{3}\pi r^3\) |
Diagonal of a cuboid \(= \sqrt{l^2 + b^2 + h^2}\); of a cube \(= \sqrt3\,a\).
Combinations of two solids
- Surface area of the combined solid = sum of the surface areas that are visible (outside); hidden joining faces are not counted
- Volume of the combined solid = sum of the volumes of the parts
- Hollowed out (e.g. a cone scooped from a cylinder): volume = difference
Common examples
- Capsule (cylinder + 2 hemispheres): SA \(= 2\pi rh + 4\pi r^2\)
- Cone on a hemisphere (toy): SA \(= \pi rl + 2\pi r^2\)
- Cylinder with a hemisphere scooped out of one end: SA \(= 2\pi rh + \pi r^2 + 2\pi r^2\)
Not in 2026-27 syllabus: frustum and conversion
- Frustum (radii \(r_1, r_2\), height \(h\)): \(V = \dfrac{1}{3}\pi h(r_1^2 + r_2^2 + r_1r_2)\); slant height \(l = \sqrt{h^2 + (r_1 - r_2)^2}\); CSA \(= \pi l(r_1 + r_2)\); TSA \(= \pi l(r_1 + r_2) + \pi r_1^2 + \pi r_2^2\)
- Melting and recasting: the volume stays the same
13Statistics
Mean of grouped data
| Method | Formula | When to use |
|---|
| Direct | \(\bar x = \dfrac{\sum f_i x_i}{\sum f_i}\) | small numbers |
| Assumed mean | \(\bar x = a + \dfrac{\sum f_i d_i}{\sum f_i}\), \(\ d_i = x_i - a\) | large \(x_i\) |
| Step deviation | \(\bar x = a + \left(\dfrac{\sum f_i u_i}{\sum f_i}\right) \times h\), \(\ u_i = \dfrac{x_i - a}{h}\) | equal class sizes, large \(x_i\) |
\(x_i\) = class mark \(= \dfrac{\text{upper limit} + \text{lower limit}}{2}\); \(a\) = assumed mean (usually a central \(x_i\)); \(h\) = class size. All three methods give the same answer.
Mode of grouped data
$$\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$
- Modal class = class with the highest frequency
- \(l\) = lower limit of the modal class, \(h\) = class size
- \(f_1\) = frequency of the modal class; \(f_0\) = of the class before it; \(f_2\) = of the class after it
Median of grouped data
$$\text{Median} = l + \left(\frac{\frac{n}{2} - cf}{f}\right) \times h$$
- \(n = \sum f_i\); median class = class whose cumulative frequency first exceeds \(\dfrac{n}{2}\)
- \(l\) = lower limit of the median class, \(cf\) = cumulative frequency of the class before it, \(f\) = frequency of the median class, \(h\) = class size
Empirical relationship
- \(3\,\text{Median} = \text{Mode} + 2\,\text{Mean}\)
- Make classes continuous before using the mode or median formula (e.g. 10–19, 20–29 becomes 9.5–19.5, 19.5–29.5).
14Probability
Classical probability
$$P(E) = \frac{\text{number of outcomes favourable to } E}{\text{number of all possible outcomes}}$$
- Assumes the outcomes are equally likely
- \(0 \le P(E) \le 1\)
- Sure (certain) event: \(P = 1\); impossible event: \(P = 0\)
- \(P(E) + P(\bar E) = 1\), where \(\bar E\) = "not \(E\)" (complementary events)
- Elementary event: only one outcome; the probabilities of all elementary events add up to 1
Common experiments
- One coin: 2 outcomes; two coins: 4; three coins: 8
- One die: 6 outcomes; two dice: \(6 \times 6 = 36\)
- Deck of 52 cards: 4 suits (spades ♠, clubs ♣ black; hearts ♥, diamonds ♦ red) of 13 cards each; 26 red, 26 black
- Face cards: king, queen, jack, so 12 in total (3 per suit); 4 aces
- Prime numbers on a die: 2, 3, 5
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