To calculate the volume of a cone, you can use the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height of the cone. However, we need to find the height \( h \) using the side (slant height) of the cone.
Given:
- The radius \( r = 14 \) mm
- The slant height \( s = 12 \) mm
We can use the Pythagorean theorem to find the height of the cone:
\[ s^2 = r^2 + h^2 \]
Substituting the known values:
\[ 12^2 = 14^2 + h^2 \]
Calculating \( 12^2 \) and \( 14^2 \):
\[ 144 = 196 + h^2 \]
Now, rearranging the equation to solve for \( h^2 \):
\[ h^2 = 144 - 196 \] \[ h^2 = -52 \]
This indicates a negative value, suggesting that there is a misunderstanding since the slant height should be greater than either the radius or height. This is not possible for a cone, as any cone must have the slant height being the longest side.
Re-checking the values showed that the slant height cannot be shorter than the radius. Therefore, it appears that either the side length or height has been incorrectly stated or values need to be deemed impractical as described.
If we re-evaluated either the radius or slant height, ensure these fit the calculations needed as logical outputs.
For the given values to fit a valid cone, we may re-evaluate with known typical cone measurements or hypothetical attributes. A standard cone's side isn't shorter than radius measurements. If you can check these or provide other valid components, a new evaluation could be made.
Given the contradiction, assuming a height based on typical cone outcomes rather than provided side lengths can help solve overall volume calculations.
If you have any clarifications please share.