Math Visualizations

What is Volume of Revolution?

A volume of revolution is created by rotating a 2D curve around an axis. This technique computes the volume of 3D solids like vases, bottles, and bowls.

The Disk Method

If $r(h)$ is the radius of the solid at height $h$, the volume is:

Each infinitesimally thin disk at height $h$ has volume $\pi [r(h)]^2 \, dh$.

The Pottery Wheel

In this visualization, you control a pottery wheel:

• 5 profile points define the vase shape (radius at each height)
• The profile is rotated around the y-axis to create the 3D solid
• Drag to orbit and scroll to zoom the 3D view

Try This

1. Adjust the sliders to change each profile point’s radius
2. Orbit the 3D view by dragging — see the vase from all angles
3. Create a bowl by making the middle points wider
4. Create a bottle by making the neck narrower than the body

Real-Life Applications

• Pottery and ceramics — designing vessels on a wheel
• Bottles and containers — optimizing volume for manufacturing
• Architecture — designing domes and rotunda structures
• Engineering — calculating volumes of rotationally symmetric parts