# Filling a cone part way, find formula for height expressed in V

I have an ice cream cone, the pointy side downwards. It's 6 cm tall and the diameter of the opening is 3 centimeter. I fill it to some height $h$ . Now how do I find a formula for its height as a function of volume.

I understand the volume of a cone is calculated by $V = \frac{\pi}{3}r^2 h$ and I know how to isolate $h$. But I'm totally in the dark as to how to approach this. If I only fill it partly, let's just say I fill it to 5 centimeters, the diameter of the surface of the ice cream is smaller then the diameter of the opening of the cone itself. How do I calculate the volume then? Since I don't know the diameter of the cone at that point...

How does the diameter of the surface of the ice cream depend on the amount of ice cream in the cone? How do I approach this?

• So you need some way to relate h with 6 and 3. What makes a shape be a cone? Your cone could be created (as a mind experiment) by revolving a cardboard right triangle about the side which is 6. Mark the height h on the cardboard and revolve again. The right triangle contains all information you need. – euler1944 Jul 26 '17 at 15:53

You can use the similar triangles/ratio of edge lengths that appear when you look at the cone from the side (at which point it looks like two adjacent right-angled triangles).

If the original "gradient" of the cone's side was $\frac{r_0}{h_0}$, then at a new height $h$ the new radius is $r = h \frac{r_0}{h_0}$. You can then plug $r$ and $h$ into your original formula for the volume.

Draw a right angle triangle representing the half of the vertical cross section of the cone. Draw a smaller triangle inside the previous triangle, this represents the partially filled ice cream. By using ratios of $\text{opening radius}:\text{cone height}=\text{radius if ice cream surface}:\text{height of the ice cream}$, you'll get the radius of the ice cream surface, hence u can get the volume.