Determining Volumes by Slicing

What is the Meaning of Determining Volumes by Slicing?

You have probably found before formulas for finding the volumes of some geometric bodies, for example if you are given a cone, you can find its volume using the formula

\[ V_{\text{cone}} = \frac{1}{3}\pi r^2 h,\]

or maybe you are given a cylinder, whose volume can be found with

\[ V_{\text{cylinder}} = \pi r^2 h.\]

There is also the volume of a pyramid, which regardless of the shape of its base, you can find its volume with

\[ V_{\text{pyramid}} = \frac{1}{3}A h.\]

But where do all these formulas come from? Why are there \( \frac{1}{3} \) involved?

The answer to this question becomes easier if you slice any of these solids, obtaining a cross-section of the body.

A cross-section of a body is a view that shows how the body would look if you were to make a cut across it.

Consider the case of a cone. Here you can see a cross-section obtained by slicing the cone halfway through its height. The slice is done so it is parallel to the base of the cone.

Figure 1. The cross-section parallel to the base of a cone is a circle

This is particularly useful because the formula for finding the area of a circle is

\[ A_{\text{circle}} = \pi r^2, \]

so you can add together all the cross-sections of the cone by means of integration to obtain the volume of a cone.

Finding Volumes by Slicing

To find volumes by slicing, consider a simpler example: the volume of a rectangular prism.

Figure 2. A rectangular prism

Next, slice the prism to obtain a cross-section that is parallel to any of its faces, for example a cross-section parallel to its base.

Figure 3. The cross-section parallel to the base of a rectangular prism is a rectangle

You can find the area of a rectangle by using the formula

\[ A_{\text{rectangle}} = \ell w.\]

To find the volume of the rectangular prism, imagine you stack all the rectangular slices of the prism. Since all slices are equal, you just multiply the area of the rectangular prism (let's just call it \( A \) for simplicity) by the height of the prism, that is

\[ \begin{align} V_{\text{prism}} &= A h \\ &= \ell w h. \end{align} \]

You can imagine this step as if dragging the area of a cross-section of the prism from one side to another

Figure 4. Volume of the rectangular prism obtained by dragging the area of a cross-section from one side to another

A similar idea applies to other geometric bodies, like cylinders.

Figure 5. Volume of a cylinder obtained by dragging the area of a cross-section from a base to another

The volume of a cylinder can be found by multiplying the area of the base by its height, so

\[ \begin{align} V_{\text{cylinder}} &= A h \\ &= \pi r^2 h. \end{align}\]

But what happens if the slices are not equal? You need to integrate!

Determining Formulas for Volumes by Slicing

So far you have seen that if all the areas of a cross-section of a solid are equal, you can just multiply by the length of the corresponding transversal side to obtain the volume of a solid, like in the case of the rectangular prism or with the cylinder.

Now consider the case of a cone. Suppose that the base of this cone is equal to \(R,\) and its height is \( h.\) If you take cross-sections parallel to its base, you will find out that these do not have the same area.

Figure 6. The cross-sections of a cone do not have the same area

For illustrative purposes, place the cone along the \(x-\)axis with its tip in the origin. This way you can find the radius of each cross-section as a function of \(x.\)

Figure 7. The radius of each cross-section is a function of \(x.\)

In fact, the cross-sections are all circles with different radii, which can be found by using similar triangles, that is

\[ \frac{R}{h} =\frac{r}{x},\]

so

\[ r = \frac{R}{h}x.\]

Figure 8. You can find the radius of a cross-section with the help of similar triangles

The function

\[ r(x) = \frac{R}{h}x\]

gives you the radius of each cross-section in terms of its position relative to the tip of the cone, so the area of a cross-section is

\[ \begin{align} A(x) &= \pi \left( r(x) \right) ^2 \\ &= \pi \left( \frac{R}{h}x \right)^2 \\ &= \frac{\pi R^2}{h^2}x^2. \end{align} \]

Now that you know a function for the area of each cross-section, you can add them together by means of integration. The integration limits are \(a=0\) since you are starting from the tip of the cone, and \( b=h\) because the cross-sections align all the way up to the base, so

\[ \begin{align} V &= \int_0^h A(x) \,\mathrm{d}x \\ &= \int_0^h \frac{\pi R^2}{h^2}x^2\,\mathrm{d}x \\ &= \frac{\pi R^2}{h^2}\int_0^h x^2\,\mathrm{d}x. \end{align}\]

You can solve the resulting definite integral using the Power Rule and the Fundamental Theorem of Calculus, that is

\[ \begin{align} \int_0^h x^2 \, \mathrm{d}x &= \left( \frac{1}{3}(h)^3 \right) - \left( \frac{1}{3}(0)^3\right) \\ &= \frac{1}{3}h^3. \end{align} \]

By plugging back the above definite integral into the formula for the volume, you obtain

\[ \begin{align} V &= \left( \frac{\pi R^2}{h^2} \right) \left(\frac{1}{3}h^3 \right) \\ &= \frac{\pi R^2h^3}{3h^2} \\ &= \frac{1}{3}\pi R^2 h,\end{align}\]

which is the formula for the volume of a cone,

\[V_{\text{cone}} = \frac{1}{3}\pi R^2h.\]

So that's where the \( \frac{1}{3} \) comes from! You can find formulas for more solids by following steps similar to above.

Determining Volumes by Slicing Solids of Revolution

Another way of finding volumes of solids is by rotation of two-dimensional figures around a revolution axis. The solids found this way are known as Solids of Revolution, and their volume are found using different methods depending on the solid.

Figure 9. A solid of revolution obtained by rotation of a parabola along the \(x-\)axis

Check out our articles about the Disk Method and the Washer Method for more information on this subject!

Examples of Volumes Determined by Slicing

There are more formulas that can be determined by slicing!

Ever wondered why the formula for the volume of a sphere has \( \frac{4}{3} \) in it?