Important Formulas

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Important Formulas - AreaPythagorean Theorem (Pythagoras' theorem)

In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
$c^2=a^2+b^2$
where $c$ is the length of the hypotenuse and $a$ and $b$ are the lengths of the other two sides.

Length of the longest rod that can be placed in a box
of length $l$, breadth $b$ and height $h$

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

Pi is a mathematical constant which is the ratio of a circle's circumference to its diameter. It is denoted by $\pi$$\pi \approx 3.14\approx \frac{22}{7}$

Sum of Interior Angles of a polygonSum of the interior angles of a polygon $=180(n-2)$ degrees where $n$ = number of sides

Example 1:
Number of sides of a triangle $=3.$
Hence, sum of the interior angles of a triangle
$=180(3-2)=180\times 1=180°$

Example 2:
Number of sides of a quadrilateral $=4.$
Hence, sum of the interior angles of any quadrilateral
$=180(4-2)=180\times 2=360°$

If each of side of a rectangle or any two dimensional shape is increased by $x\%$, its area is increased by $(\frac{x^2}{100}+2x)\%$

If radius of a circle is increased by $x\%$, its area is increased by $(\frac{x^2}{100}+2x)\%$

Example $1$
If each side of a triangle is doubled, what is the percentage increase in its area?

Here, each side is increased by $100\%$ (because each side is doubled)

percentage increase in its area
=$(\frac{{100}^2}{100}+2\times 100)\%=300\%$

Example $2$
If circumference of a circle is increased by $50\%$, what is the percentage increase in its area?

Circumference of the circle is increased by $50\%$. Since circumference of the circle is $2\pi r$, radius, $r$ is increased by $50\%$.

percentage increase in its area
$=(\frac{{50}^2}{100}+2\times 50)\%=125\%$