Prisms are three-dimensional closed surfaces.
At this level, the focus will be on rectangular prisms and triangular prisms.


A prism is a three-dimensional solid figure with two parallel faces, called bases, that are congruent polygons, and lateral flat faces which are rectangles in a right prism, and are parallelograms in an oblique prism.

Right Triangular Prism
Regarding right prisms:
Prisms are called polyhedra since their faces are polygons.
The bases are parallel and congruent.
The lateral faces are rectangles.
The lateral edges are parallel and congruent.
Prisms are named for the shape of the bases.
Prisms do NOT always sit on their bases

The following diagrams show various prisms with their bases shaded.

Right Pentagonal Prism
Right Triangular Prism
Right Rectangular Prism
Oblique Octagonal Prism

All prisms discussed will be right prisms.
In a right prism, the congruent bases will appear directly above
one another when the prism is sitting on its base.

The segments joining the opposite bases are parallel to one another,
and are perpendicular to the bases.

FYI: There are prisms that are "slanted" as to the location of their bases.
These are called oblique prisms. These will be studied in high school geometry.



congruent Cross Sections
(parallel to bases)

Cross Sections:

All cross sections of a prism parallel to the bases will be congruent to the bases (the same size and shape as the bases).

"height" of a prism is the distance between the two bases. In a right prism, the height is a lateral edge.


Volume of a Prism:
The volume of a prism is its base area times its height.        Vprism = Bh
V = volume in cubic units;   B = area of the base in square units;   h = height in units

Find the volume of this right triangular prism.

• Find the area of the base.
Since the base is a triangle, the A = ½ b• h.
            A = ½ • 9 • 6 = 27 sq. units.
• The height = 12 units.
• The volume formula is V = Bh.
            V = 27 sq. units • 12 units =
324 cubic units.


Surface Area of a Prism:
The surface area of a prism is the sum of the areas of the bases plus the areas of the lateral faces. (The sum of the areas of all the faces.)

Find the surface area, SA, of the right triangular prism shown at the right.

• Find the area of the base.
Since the base is a triangle, the A = ½ b• h.
            B = ½ • 8 • 4 = 16 sq. units.
• Find the area of the rectangular lateral faces.             84, 112, 70
• Add the areas of all of the faces. Remember that there are two bases.
            SA = 16 + 16 + 84 + 112 + 70

            = 298 square units

Notice how a net made the computation of the surface area easy to organize.
A net of this prism shows the "surfaces" whose areas, when added, comprise the surface area.
The red values represent areas of the sections.
In this example, B = area of the prism's base
= ½ (triangle's base) • (triangle's height).
Adding all of these sections together will yield the surface area of the solid.

The work that was done in the example above can be combined to create a "formula" for the surface area of a right prism.
SA = 16 + 16 + 6•14 + 8•14 + 5• 14
= 2(16) + (6 + 8 + 5)•14
= 2(B) + (base perimeter)• height

The surface area, SA, of a right prism can be found using the formula:
SA = 2B + ph
B = area of base, p = perimeter of base, h = height.


In the Real World:
Reflective Prisms

In the study of optics, prisms are used to reflect light, such as occurs in binoculars.  Prisms are also used to disperse light, or break light into its spectral colors of the rainbow.  The most commonly used optic prism is a triangular prism, which has a triangular base and rectangular sides.


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