Pappus Centroid Theorems

Pappus's Centroid Theorems are results from geometry that relate the surface area and volume of solids of revolution to the centroid of the generating region. They provide an alternative to direct integration. For the disk and rectangle used here, symmetry places both the area centroid and the boundary centroid at (0, R), so the same centroid distance applies to both theorems.

Experiment Steps

1. Set Generating shape to Disk, Half-height or radius r to 1 and Centroid distance R to 2. Observe volume 39.47842 and surface area 78.95684.
2. Read the dashed centroid orbit in the oblique diagram. It is a circle in the yz-plane about the x-axis, shown as an ellipse by projection, and its length is 4π.
3. Switch Generating shape to Rectangle and set Rectangle width w to 3. Keeping r = 1 and R = 2, check area 6, perimeter 10, volume 75.39822 and full surface area 125.6637.
4. Set Centroid distance R to 1. The tangent-axis boundary R = r is rejected in this experiment; no dependent results remain. Restore R to 2.
5. Drag the diagram to pan, then press Reset view to restore its framing. Length controls retain their values.

Link to module: Pappus Centroid Theorems