Rectangular prisms and cylinders

From 2D to 3D

In plane geometry, we work with two-dimensional shapes (length, width). In solid geometry, we add a third dimension: height. This results in solids, also known as polyhedra (with flat faces) or solids of revolution (with curved faces).

The right prism

A right prism is a solid consisting of two parallel, superimposable faces called bases, connected by lateral faces which are rectangles. The height of the prism is the distance between the two bases.

Example: a right prism whose base is a triangle with sides of 3 cm, 4 cm and 5 cm, and a height of 10 cm, is called a right prism with a triangular base.

General formula for volume: V = Base_area × height

The cylinder of revolution

A cylinder of revolution is formed by rotating a rectangle about one of its sides. Its two bases are identical and parallel discs of radius r.

• Volume: V = π × r² × h
• Lateral area: A = 2 × π × r × h

Numerical example

A cylinder with radius r = 3 cm and height h = 8 cm:

V = π × 3² × 8 = π × 9 × 8 = 72 × π ≈ 226.19 cm³

Solid	Base	Volume formula
Right prism	Polygon	Base area × h
Cylinder	Disc	π × r² × h

Common pitfall

Do not confuse the radius with the diameter: if the question gives a diameter of 6 cm, you must use r = 3 cm in the formulae, not 6 cm. Another common mistake is to forget to include the unit cubed (cm³) for volume, whilst area is expressed in square units (cm²).