Lesson 3 — Prisms

Big idea: A prism has two congruent parallel bases and parallelogram lateral faces; a right prism has rectangular lateral faces.

Words to know

Word	Meaning
Prism	The solid formed by all segments parallel to a line l that connect a polygonal region R in one plane to a second parallel plane.
Bases	The two congruent polygonal faces of a prism; they lie in parallel planes.
Lateral faces	The faces of a prism other than its bases; they are parallelograms.
Lateral edges	The edges where the lateral faces intersect one another; they are parallel.
Right prism	A prism whose lateral edges are perpendicular to the planes of its bases.
Oblique prism	A prism whose lateral edges are oblique to the planes of its bases.
Net	A pattern of all the faces of a solid that can be cut out and folded to form it.
Lateral area	The sum of the areas of the lateral faces.
Total area	The lateral area plus the areas of the two bases.

The bases of a prism are congruent and lie in parallel planes.
The lateral faces of any prism are parallelograms.
In a right prism, the lateral edges are perpendicular to the base planes, so the lateral faces are rectangles.
In an oblique prism, the lateral edges are not perpendicular to the base planes.
A net shows all faces flat; its yellow region is the lateral area and its two base regions are the bases.
If a base is an n-gon, the prism has n lateral faces and n + 2 faces in all.
A rectangular solid is a special prism; a cube is a special right prism.

Lateral area: L.A. = sum of the areas of the lateral faces.
Total area: T.A. = L.A. + 2B, where B is the area of one base.
For a right prism, each lateral face has area (base edge) × h, so L.A. = Ph, where P is the perimeter of the base.
No volume formula in this lesson; volume comes in Lesson 4.

If the homework asks…	Do this…
Name the bases and lateral faces	Bases are the two congruent parallel polygons; all other faces are lateral faces.
Tell right from oblique	Check whether the lateral edges are perpendicular to the base planes.
Find the lateral area	Add the areas of the lateral faces, or use L.A. = Ph for a right prism.
Find the total area	Add the lateral area and the areas of both bases.
Count faces, edges, vertices	An n-gon base gives n lateral faces, n + 2 faces, 3n edges, and 2n vertices.
Draw a net	Draw the bases and unroll the lateral faces into one row.

The bases are not always the “bottom and top” — they are the two congruent parallel faces.
Lateral faces are parallelograms in general, rectangles only in a right prism.
Total area includes both bases, not just one.
Height h is the perpendicular distance between the base planes, not a slanted lateral edge.
Do not count a lateral face as a base.