Class 10 Maths formulas you need most: algebra, trigonometry and mensuration

Before an exam, a lot of time goes into hunting for the formula you remember seeing somewhere. This post collects a selection of the most important Class 10 Maths formulas in chapter order, so you can revise them quickly.

1. Real Numbers

$a = bq + r,\quad 0 \le r < b$

$\mathrm{HCF}(a,b)\times \mathrm{LCM}(a,b) = a \times b$

When you work from prime factorisations, HCF takes the smallest power of each common prime, and LCM takes the greatest power of every prime.

$\tfrac{p}{q}\ (\text{in lowest terms})\ \text{terminates} \iff q = 2^n 5^m$

Example: if $\mathrm{HCF}(a,b)=12$ and $a\times b = 1800$, then $12\times \mathrm{LCM} = 1800$, so $\mathrm{LCM}=150$.

2. Polynomials

A quadratic polynomial $ax^2+bx+c$ has at most 2 zeroes and a cubic has at most 3. Zeroes are the $x$-values where the graph meets the $x$-axis.

$\alpha+\beta = -\frac{b}{a},\qquad \alpha\beta = \frac{c}{a}$

$k\left[x^2 - (\alpha+\beta)x + \alpha\beta\right],\quad k\neq 0$

3. Quadratic Equations

$ax^2+bx+c=0\ (a\neq 0),\qquad D = b^2-4ac$

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

4. Arithmetic Progressions (AP)

$a_n = a + (n-1)d$

$S_n = \frac{n}{2}\left[2a+(n-1)d\right] = \frac{n}{2}(a+l)$

Useful relations: $a_n = S_n - S_{n-1}$ and $d = a_{k+1}-a_k$. To take three terms in AP, use $a-d,\ a,\ a+d$.

5. Trigonometry

$\sin\theta=\frac{P}{H},\quad \cos\theta=\frac{B}{H},\quad \tan\theta=\frac{P}{B}$

$\sin^2\theta+\cos^2\theta=1,\quad 1+\tan^2\theta=\sec^2\theta,\quad 1+\cot^2\theta=\csc^2\theta$

$\sin(90^\circ-\theta)=\cos\theta,\qquad \tan(90^\circ-\theta)=\cot\theta$

In heights and distances problems, draw the figure first and mark the right angle. The angle of elevation is measured upward from the horizontal, and the angle of depression is measured downward from the horizontal.

6. Mensuration

$l=\sqrt{r^2+h^2}$

When one solid is melted and recast into another, the volume stays the same.

7. Statistics

$\bar{x}=\frac{\sum f_ix_i}{\sum f_i}$

$\text{Mode}=l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h$

$\text{Median}=l+\frac{\frac n2-cf}{f}\times h$

$\text{Mode}=3\,\text{Median}-2\,\text{Mean}$

What to do next

Knowing the formulas is only the first step. Real preparation starts when you apply them to questions. This post covers a selection of Maths formulas only. The DepLearn Class 10 revision kit has formulas for both Maths and Science, plus 12 worked practice questions, and you can download it after you join the waitlist.