Volume of Prisms and Cylinders

A prism: the same shape all the way along

A prism is a solid with the same cross-section all along its length — slice it anywhere across and you always get the same flat shape. Because every slice is identical, the volume is simply that one cross-section's area, repeated all the way along:

V = (\text{area of cross-section}) \times \text{length}

A cuboid (a box) is a prism whose cross-section is a rectangle. Its cross-section has area l \times w, so stretching it along a height h gives

V = l \times w \times h

A cylinder is a prism too

A cylinder is a prism whose cross-section is a circle. The circle of radius r has area \pi r^2, so carrying it along a height h gives

V = \pi r^2 \times h = \pi r^2 h

For any solid with a constant cross-section:

Prism: V = (\text{cross-section area}) \times \text{length}.
Cuboid (rectangular cross-section): V = l \times w \times h.
Cylinder (circular cross-section): V = \pi r^2 h.

Two prisms to picture

Step through the sketch: first a cuboid (rectangular cross-section), then a cylinder (circular cross-section). Each is a flat drawing of a 3D solid.