Volume of a Cone
Volume of a Cone
Using the definition of volume, we can calculate the sum of unit squares for any object with a closed space. In the case of a cone, a cone has a defined radius and height.
In the simulation below, you can input an arbitrary radius and height of your choice. To calculate the volume you can count each individual unit cubes to get the volume of a cone. Although we would get the correct answer, this is not practical as a cone with a large enough radius and height would take a long time to count up all the individual unit cubes each of length, width and height 1m by 1m by 1m.
Since a cone is a three dimensional representation of a right triangle with a base of a circle, and to plot the radius of a circle reuires two dimensions and the height to form a triangular figure a third dimension, we can assume that a cone will be in three dimensions. Since every cone has an radius r and a height h, we can take the integral of the height to find the volume of a cone.
We can first start out by writing the equation of a line:
\[ y = mx + b \]
In the case of a cone, the slope would be r/h as the radius is the rise in the slope, meaning that as the cone gets larger the radius of the circle gets larger and h as the height is the run of the slope, meaning that as the cone also gets larger the height of the cone gets larger as well.
So our m becomes:
\[ m = \frac{r}{h} \]
So our equation becomes:
\[ y = \frac{r}{h} x \]
Taking the integral of the area of a circle, and replacing r with y, we can then calculate the integral and find the volume of a cone:
\[ A = \pi r^2 \]
\[ V = \int_{0}^{h} \pi r^2 dx \]
\[ V = \int_{0}^{h} \pi y^2 dx \]
\[ V = \int_{0}^{h} \pi \left(\frac{r}{h} x\right)^2 dx \]
\[ V = \int_{0}^{h} \pi \left(\frac{r^2}{h^2} x^2\right) dx \]
\[ V = \pi \frac{r^2}{h^2} \int_{0}^{h} x^2 dx \]
\[ V = \pi \frac{r^2}{h^2} \frac{x^3}{3}_{0}^{h} \]
\[ V = \pi \frac{r^2}{h^2} \left(\frac{h^3}{3} - \frac{0^3}{3}\right) \]
\[ V = \pi \frac{r^2}{h^2} \left(\frac{h^3}{3} - 0\right) \]
\[ V = \pi \frac{r^2}{h^2} \left(\frac{h^3}{3}\right) \]
\[ V = \pi \frac{r^2 h}{3} \]
\[ V = \pi r^2 \frac{h}{3} \]
An equation we can create therefore to calculate the volume of a cone is the radius squared times one third of the height times π.
\[ V = \pi r^2 \frac{h}{3} \]
Since a cone is an enclosed space in three dimensions, that would mean that the units would be meters cubed.
\[ m^3 \]
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