Prism

Prism

A solid with parallel congruent bases which are both polygons. The bases must be oriented identically. The lateral faces of a prism are all parallelograms or rectangles.

Oblique and right prisms with labels: h=height, B=base area, P=perimeter. Formulas: Volume=Bh; Lateral Surface=hP; Total...

Key Formula

V = B \cdot h \qquad \text{and} \qquad SA = 2B + Ph

Where:

$V$ = Volume of the prism
$SA$ = Total surface area of the prism
$B$ = Area of one base
$h$ = Height (perpendicular distance between the two bases, also called the altitude)
$P$ = Perimeter of one base

Worked Example

Problem: Find the volume and surface area of a right triangular prism whose triangular base has sides 3 cm, 4 cm, and 5 cm (a right triangle), and whose height (length between the two bases) is 10 cm.

Step 1: Find the area of the triangular base. Since the base is a right triangle with legs 3 cm and 4 cm:

B = \frac{1}{2} \times 3 \times 4 = 6 \text{ cm}^2

Step 2: Calculate the volume using the prism volume formula:

V = B \cdot h = 6 \times 10 = 60 \text{ cm}^3

Step 3: Find the perimeter of the triangular base:

P = 3 + 4 + 5 = 12 \text{ cm}

Step 4: Calculate the total surface area. This includes both bases and the lateral area:

SA = 2B + Ph = 2(6) + 12 \times 10 = 12 + 120 = 132 \text{ cm}^2

Answer: The volume is 60 cm³ and the total surface area is 132 cm².

Another Example

This example uses a non-triangular, non-rectangular base (a regular pentagon) to show that the same formulas work for any polygon base. It also demonstrates using the apothem to find the area of a regular polygon.

Problem: A pentagonal prism has a regular pentagon base with side length 6 cm and an apothem of approximately 4.13 cm. The prism has a height of 15 cm. Find the volume and surface area.

Step 1: Find the area of the regular pentagon base. For a regular polygon, Area = ½ × perimeter × apothem. First, find the perimeter:

P = 5 \times 6 = 30 \text{ cm}

Step 2: Now compute the base area:

B = \frac{1}{2} \times 30 \times 4.13 = 61.95 \text{ cm}^2

Step 3: Calculate the volume:

V = B \cdot h = 61.95 \times 15 = 929.25 \text{ cm}^3

Step 4: Calculate the total surface area:

SA = 2B + Ph = 2(61.95) + 30 \times 15 = 123.9 + 450 = 573.9 \text{ cm}^2

Answer: The volume is approximately 929.25 cm³ and the surface area is approximately 573.9 cm².

Frequently Asked Questions

What is the difference between a right prism and an oblique prism?

In a right prism, the lateral edges are perpendicular to the bases, so the lateral faces are rectangles. In an oblique prism, the lateral edges are tilted at an angle, making the lateral faces parallelograms instead of rectangles. Both types have the same volume formula V = Bh, but for an oblique prism, h is the perpendicular distance between the bases, not the length of the lateral edge.

Is a cube a prism?

Yes. A cube is a special case of a right rectangular prism (also called a right square prism) where all six faces are congruent squares. It has two parallel square bases and four square lateral faces, which satisfies the definition of a prism.

How is a prism different from a cylinder?

A prism has polygon bases — flat shapes with straight edges — while a cylinder has circular bases. You can think of a cylinder as the limiting case of a prism as the number of sides on the base polygon increases toward infinity. Both share the same volume formula structure: V = Bh.

Prism vs. Pyramid

	Prism	Pyramid
Number of bases	Two parallel, congruent bases	One base
Top	A second base identical to the first	A single point (apex)
Lateral faces	Parallelograms or rectangles	Triangles
Volume formula	V = Bh	V = ⅓Bh
Cross-section parallel to base	Same size and shape as the base at every height	Gets smaller as you move toward the apex

Why It Matters

Prisms appear throughout geometry courses when you study three-dimensional shapes, surface area, and volume. Many real-world objects — boxes, rooms, steel beams, and even some crystal structures — are prisms. Understanding the prism volume formula V = Bh is also foundational because the pyramid and cylinder volume formulas are built directly from it.

Common Mistakes

Mistake: Confusing the height of the prism with the slant length of a lateral edge, especially in oblique prisms.

Correction: The height h in the volume formula is always the perpendicular distance between the two bases. In a right prism, this equals the lateral edge length, but in an oblique prism it does not. Always measure height at a right angle to the bases.

Mistake: Using the surface area formula SA = 2B + Ph for an oblique prism.

Correction: The formula SA = 2B + Ph applies only to right prisms, where each lateral face is a rectangle of width equal to a base side and height h. For an oblique prism, you must calculate each parallelogram lateral face individually, since the slant height differs from the perpendicular height.

Related Terms

Volume — Quantity measured by the prism volume formula
Lateral Surface — The non-base faces of the prism
Right Prism — Prism with lateral edges perpendicular to bases
Oblique Prism — Prism with tilted lateral edges
Regular Prism — Right prism with a regular polygon base
Altitude of a Prism — Perpendicular distance between the two bases
Polygon — Shape of both bases of a prism
Solid — General category that includes prisms