Question

what is the formula of volume of cone​

Answer 1

The formula for the volume of a cone is V=1/3hπr².

Answer 2

$\red\bigstar$ Explanation $\red\bigstar$

$\leadsto$ Solution:-

$\sf{Volume\:of\:cone} = \frac{1}{3} \times \pi r^2h$

$\leadsto$ Derivation:-

We can prove that $\sf{Volume\:of\:cone} = \frac{1}{3} \times \pi r^2h$ using two methods

i) Using practical proof

ii) Using calculus

• Using practical proof:-

Now let's assume that there is a cone placed in a cylinder such that the cone is exactly fitted into the cylinder. Now we have poured water into a cone as well as the cylinder now you will notice that the amount of water present in the cone is one-third of the amount of present in the cylinder.

We know that amount is equal to the volume of that container

We also know that volume of the cylinder is $\pi r^2 h$

Therefore,

$\sf{Volume\:of\:cone} = \frac{1}{3} \times \pi r^2h$

• Using Calculus:-

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If we make a graph of radius and the height, then it would look like this

Here the slant height of the cone equals to the slope of the line.

We also know that $\sf Slope\:of\:line = \frac{y_2 - y_1}{x_2 - x_1}$

Please refer the attachment for the figure with co-ordinates.

We also know that the slant height is a straight line

Therefore,

The equation of slant height is $\sf y = mx + c$

Now let's calculate the slope for the equation of the slant height = $\frac{(r - 0)}{(h-0)}$ = $\frac{r}{h}$

We have got the value of m as $\frac{r}{h}$

We know that,

$\sf m(x_2 - x_1) = y_2 - y_1$

Put the value of m as $\frac{r}{h}$

Here the value of x and y are variable but the slope as the line equation does not change.

Let $\sf x_2 = x$ and $\sf y_2 = y$ and $\sf x_1 = y_1 = 0$ as it starts from the origin

Therefore,

$\sf \frac{r}{h}(x - 0) = y - 0\\y = \frac{rx}{h}$

We know that volume of a figure = $\int\limits^b_a {\pi y^2} \, dx$

We know that $y = \frac{rx}{h}$

Therefore,

$\sf V = \int\limits^h_0 {\pi [\frac{rx}{h}]^2 } \, dx \\\sf Here\:\pi,r,h\:are\:constants\\V = \pi \times \frac{r^2}{h^2} \int\limits^0_h {x^2} \, dx \\\sf Therefore,\\V = \pi \times \frac{r^2}{h^2} \times [\frac{x^3}{3}]^h _ 0\\V = \pi \times \frac{r^2}{h^2} \times \frac{h^3}{3}\\ \\\sf Therefore\\V = \frac{\pi r^2 h}{3}$