Class 10 Maths All Formulas: The Complete CBSE Formula Sheet (2026)

Use this when you need to find the HCF of two numbers using repeated division. Start with the larger number, divide by the smaller, then keep dividing the divisor by the remainder until the remainder is zero.

HCF and LCM Relationship:

This is incredibly useful in problems where you know one of HCF or LCM and need to find the other.

Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes, and this factorisation is unique (apart from the order of factors).

Memory tip: Think of Euclid's lemma as "divide, replace, repeat." The divisor becomes the new dividend, the remainder becomes the new divisor.

For a quadratic polynomial $ax^2 + bx + c$ with zeroes $\alpha$ and $\beta$:

Use these when a question asks you to find the sum or product of zeroes without actually computing the zeroes.

For a cubic polynomial $ax^3 + bx^2 + cx + d$ with zeroes $\alpha$, $\beta$, $\gamma$:

Division Algorithm for Polynomials:

where degree of $r(x)$ < degree of $g(x)$. This is the polynomial version of Euclid's lemma.

Memory tip: For quadratics, the signs alternate. Sum of zeroes has a negative sign ($-b/a$), product does not ($c/a$).

For the pair of equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:

Unique solution (consistent, intersecting lines):

Infinitely many solutions (dependent, coincident lines):

No solution (inconsistent, parallel lines):

Use these conditions to quickly check if a system has a solution, especially in MCQs where you do not need to actually solve.

Cross Multiplication Method:

This is the fastest algebraic method for solving linear pairs. Very handy for board exams when time is short.

Memory tip: If the ratios of $a$ and $b$ coefficients are equal but the ratio of $c$ is different, the lines are parallel (no solution). Equal ratios everywhere means the lines overlap completely.

For $ax^2 + bx + c = 0$:

Quadratic Formula:

This works for every quadratic equation, even when factorisation is messy.

Discriminant:

The discriminant tells you the nature of roots before you solve:

• $D > 0$: Two distinct real roots
  - $D = 0$: Two equal real roots (the parabola just touches the x axis)
  - $D < 0$: No real roots (the parabola does not cross the x axis)

Sum and Product of Roots:

Same as the polynomial chapter but specifically for quadratic equations.

Forming a Quadratic Equation from Roots:

Use this when a question gives you the roots and asks you to form the equation.

Memory tip: For the discriminant, just remember "$b$ squared minus $4ac$." If it is positive, two roots. If zero, one root (repeated). If negative, no real roots.

nth Term of an AP:

where $a$ is the first term and $d$ is the common difference. Use this to find any specific term in the sequence.

Sum of First n Terms:

Alternatively, if you know the last term $l$:

This second form is simpler when you already know the first and last terms.

Common Difference:

The difference between any two consecutive terms is constant in an AP.

Useful connection: The nth term can also be written as $a_n = S_n - S_{n-1}$ for $n \ge 2$. This comes up in tricky board exam questions.

Memory tip: Think of $S_n$ as "number of terms divided by 2, times the sum of first and last." The formula $\frac{n}{2}(a + l)$ is basically averaging the first and last terms, then multiplying by how many terms there are.

Basic Proportionality Theorem (BPT / Thales' Theorem):
If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides in the same ratio:

Use this whenever you see a line parallel to one side of a triangle.

Converse of BPT: If a line divides two sides of a triangle in the same ratio, then that line is parallel to the third side.

Criteria for Similarity of Triangles:
- AAA (or AA): If two angles of one triangle equal two angles of another, they are similar.
- SSS: If all three pairs of corresponding sides are in the same ratio, they are similar.
- SAS: If one angle is equal and the sides including that angle are in the same ratio, they are similar.

Area Ratio Theorem:

The ratio of areas of similar triangles equals the square of the ratio of their corresponding sides.

Pythagoras Theorem:

And its converse: if this relation holds, the triangle is right angled.

Memory tip: For similarity criteria, remember "AA is enough." If two angles match, the third automatically matches too. For BPT, think "parallel line, proportional sides."

Distance Formula:

Finds the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$. Also used to check if a triangle is equilateral, isosceles, or right angled.

Section Formula (Internal Division):
A point dividing the line joining $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m:n$ internally:

Use this when a point divides a segment in a given ratio.

Midpoint Formula:
Special case of the section formula when $m:n = 1:1$:

Use this to find the centre of a line segment or to verify that diagonals of a parallelogram bisect each other.

Area of a Triangle:
With vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$:

If the area is zero, the three points are collinear (lie on the same line).

Memory tip: For the distance formula, think Pythagoras. The horizontal and vertical differences are the two legs of a right triangle. The distance is the hypotenuse.

For a right triangle with angle $\theta$, opposite side, adjacent side, and hypotenuse:

Also remember: $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and $\cot\theta = \frac{\cos\theta}{\sin\theta}$

Standard Angle Values:

Memory tip: For sin values at 0, 30, 45, 60, 90 degrees, think $\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$. That gives you $0, \frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}, 1$. Cos is the same sequence in reverse.

These are the three identities you absolutely must know:

From the first identity, you can also derive:
- $\sin^2\theta = 1 - \cos^2\theta$
- $\cos^2\theta = 1 - \sin^2\theta$

From the second: $\sec^2\theta - \tan^2\theta = 1$, which factors as $(\sec\theta + \tan\theta)(\sec\theta - \tan\theta) = 1$.

From the third: $\csc^2\theta - \cot^2\theta = 1$.

These rearranged forms appear frequently in "prove that" type questions.

Complementary Angles:

Use complementary angle formulas when a question involves angles that add up to 90 degrees, like $\sin 50° = \cos 40°$.

Memory tip: The three identities all come from the Pythagoras theorem. Divide the Pythagoras equation by hypotenuse squared to get the first identity, by adjacent squared to get the second, and by opposite squared to get the third.

There are no new formulas here. You use the same trigonometric ratios from Chapter 8, applied to practical situations.

Angle of Elevation: The angle formed between the horizontal line of sight and the line of sight looking upward at an object. If you are looking up at the top of a building, the angle your line of sight makes with the horizontal is the angle of elevation.

Angle of Depression: The angle formed between the horizontal line of sight and the line of sight looking downward. If you are standing on top of a building looking down at something, the angle your line of sight makes with the horizontal is the angle of depression.

Key relationships used in problems:

Important: The angle of depression from point A to point B equals the angle of elevation from point B to point A (alternate interior angles with a horizontal line).

Memory tip: Always draw the diagram first. Label the right angle, the known angle, the known side, and what you need to find. Then pick the right trig ratio. "SOH CAH TOA" tells you which ratio involves which sides.

Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Use this whenever a question involves a tangent and a radius meeting at a point.

Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.

This is used in a huge number of board exam problems. It also means that the angle between the two tangents is bisected by the line joining the external point to the centre.

Additional results:
- The angle between a tangent and a chord drawn from the point of contact equals the inscribed angle on the opposite side.
- If two tangents are drawn from an external point, the centre lies on the bisector of the angle between them.
- $OP$ bisects $\angle APB$ where $PA$ and $PB$ are tangents from $P$ to the circle with centre $O$.

Memory tip: "External point, equal tangents." If you see a point outside a circle with two tangent lines, immediately write $PA = PB$. This single fact solves most circle problems.

Construction 1: Dividing a line segment in a given ratio
To divide a line segment $AB$ in the ratio $m:n$, draw a ray $AX$ making an acute angle with $AB$. Mark $m + n$ equal arcs on $AX$. Join the last point to $B$. Draw a line through the $m$th point parallel to this join. It intersects $AB$ at the required point.

Construction 2: Constructing a tangent to a circle from an external point
Join the centre $O$ to the external point $P$. Find the midpoint $M$ of $OP$. Draw a circle with centre $M$ and radius $MO$. This circle intersects the original circle at the points of tangency.

Construction 3: Constructing a triangle similar to a given triangle
Use the ratio of similarity to scale the sides. The construction involves drawing a ray, marking arcs, and using parallel lines to get the proportional sides.

Memory tip: For tangent construction, the key insight is that the angle in a semicircle is 90 degrees. By drawing a circle on $OP$ as diameter, you ensure the tangent meets the radius at 90 degrees.

Circumference of a Circle:

Area of a Circle:

Area of a Semicircle:

Area of a Sector (with angle $\theta$ in degrees):

Use this when you have a "pizza slice" shaped region.

Length of an Arc:

This gives the curved length of the sector boundary.

Area of a Segment (minor segment):

A segment is the region between a chord and the arc. Area of segment equals area of sector minus area of the triangle formed by the two radii and the chord.

Area of a Ring (annulus, region between two concentric circles):

where $R$ is the outer radius and $r$ is the inner radius.

Memory tip: Notice the pattern. Both sector area and arc length use the fraction $\frac{\theta}{360}$. You are basically taking that fraction of the full circle's area or circumference. Think of it as "what fraction of the full circle is my sector?"

Direct Method:

where $f_i$ is the frequency and $x_i$ is the class mark (midpoint) of each class interval.

Assumed Mean Method:

where $a$ is the assumed mean and $d_i = x_i - a$. Use this when the class marks are large numbers to simplify calculations.

Step Deviation Method:

where $h$ is the class width and $u_i = \frac{x_i - a}{h}$. This is the most efficient method when class widths are uniform.

Memory tip: Direct method is straightforward but slow with big numbers. Assumed mean saves effort. Step deviation is the fastest when all class intervals have the same width.

Median Formula:

where:
- $l$ = lower limit of the median class
- $n$ = total frequency ($\sum f_i$)
- $cf$ = cumulative frequency of the class before the median class
- $f$ = frequency of the median class
- $h$ = class width

The median class is the class whose cumulative frequency is just greater than or equal to $\frac{n}{2}$.

Mode Formula:

where:
- $l$ = lower limit of the modal class
- $f_1$ = frequency of the modal class (the class with the highest frequency)
- $f_0$ = frequency of the class before the modal class
- $f_2$ = frequency of the class after the modal class
- $h$ = class width

Empirical Relationship:

This is useful as a quick check or when one of the three values is missing.

Memory tip: For median, think "$l$ plus adjustment." The adjustment is basically how far into the median class you need to go. For mode, the modal class is simply the class with the highest frequency.

Basic Probability Formula:

This is the foundation of every probability problem in Class 10.

Complementary Events:

where $\bar{E}$ means "the event E does NOT happen." Use this when it is easier to calculate the probability of something NOT happening.

Key facts:
- $0 \le P(E) \le 1$ for any event $E$
- $P(\text{sure event}) = 1$
- $P(\text{impossible event}) = 0$
- The sum of probabilities of all elementary events is 1

Common scenarios and their total outcomes:
- One coin toss: 2 outcomes
- Two coin tosses: 4 outcomes
- Three coin tosses: 8 outcomes ($2^n$ for $n$ coins)
- One die roll: 6 outcomes
- Two dice rolls: 36 outcomes
- A deck of cards: 52 cards (26 red, 26 black, 4 suits of 13 each)

Memory tip: Always start by listing or counting the total outcomes. Then count the favourable ones. The formula is just a fraction. For complementary events, think "the probability of rain plus the probability of no rain always equals 1."