Regular Prism — Definition, Formula & Examples

Regular Prism

A prism with bases that are regular polygons. The bases are not necessarily aligned one directly above the other. Note: For some mathematicians regular prism means the same as right regular prism.

3D regular hexagonal prism with height h labeled on right side. Formulas: B = area of base, Volume = Bh.

See also

Hexagon, height of a prism, volume

Key Formula

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

Where:

$V$ = Volume of the prism
$B$ = Area of one regular-polygon base
$h$ = Height (altitude) of the prism — the perpendicular distance between the two bases
$SA$ = Total surface area of the prism
$P$ = Perimeter of the regular-polygon base
$l$ = Lateral edge length (slant length of a lateral face); equals h when the prism is a right prism

Worked Example

Problem: Find the volume and total surface area of a right regular hexagonal prism whose base edge is 4 cm and whose height is 10 cm.

Step 1: Find the area of the regular hexagonal base. A regular hexagon with side length s has area:

B = \frac{3\sqrt{3}}{2}\,s^2 = \frac{3\sqrt{3}}{2}(4)^2 = \frac{3\sqrt{3}}{2}\cdot 16 = 24\sqrt{3} \approx 41.57\;\text{cm}^2

Step 2: Compute the volume using V = B · h.

V = 24\sqrt{3}\cdot 10 = 240\sqrt{3} \approx 415.69\;\text{cm}^3

Step 3: Find the perimeter of the hexagonal base.

P = 6s = 6(4) = 24\;\text{cm}

Step 4: For a right prism the lateral edge length l equals the height h, so compute the lateral area.

\text{Lateral Area} = P \cdot l = 24 \cdot 10 = 240\;\text{cm}^2

Step 5: Add the two bases to get the total surface area.

SA = 2B + 240 = 2(24\sqrt{3}) + 240 = 48\sqrt{3} + 240 \approx 323.14\;\text{cm}^2

Answer: Volume ≈ 415.69 cm³ and total surface area ≈ 323.14 cm².

Another Example

This example uses the simplest regular polygon (equilateral triangle) as the base, contrasting with the hexagonal base in the first example. It shows how the same formulas apply regardless of how many sides the base has.

Problem: A right regular triangular prism (equilateral-triangle bases) has a base edge of 6 cm and a height of 8 cm. Find its volume and total surface area.

Step 1: Find the area of one equilateral-triangle base with side s = 6 cm.

B = \frac{\sqrt{3}}{4}\,s^2 = \frac{\sqrt{3}}{4}(6)^2 = \frac{\sqrt{3}}{4}\cdot 36 = 9\sqrt{3} \approx 15.59\;\text{cm}^2

Step 2: Compute the volume.

V = B \cdot h = 9\sqrt{3}\cdot 8 = 72\sqrt{3} \approx 124.71\;\text{cm}^3

Step 3: Find the perimeter and lateral area.

P = 3(6) = 18\;\text{cm}, \qquad \text{Lateral Area} = 18 \cdot 8 = 144\;\text{cm}^2

Step 4: Compute the total surface area.

SA = 2(9\sqrt{3}) + 144 = 18\sqrt{3} + 144 \approx 175.18\;\text{cm}^2

Answer: Volume ≈ 124.71 cm³ and total surface area ≈ 175.18 cm².

Frequently Asked Questions

What is the difference between a regular prism and a right prism?

A regular prism requires the bases to be regular polygons (all sides and angles equal), but the lateral edges do not have to be perpendicular to the bases. A right prism requires the lateral edges to be perpendicular to the bases, but the bases can be any polygon — they need not be regular. A right regular prism satisfies both conditions.

Is a cube a regular prism?

Yes. A cube has square bases (a square is a regular polygon) and its lateral edges are perpendicular to the bases, so it is a right regular prism. Every edge of a cube has the same length, making it the most symmetric example of a regular prism.

How do you find the volume of a regular prism?

Use V = B · h, where B is the area of the regular polygon base and h is the perpendicular distance between the two bases. First calculate B using the appropriate regular polygon area formula, then multiply by h.

Regular Prism vs. Right Regular Prism

	Regular Prism	Right Regular Prism
Base requirement	Bases must be regular polygons	Bases must be regular polygons
Lateral edges	May be oblique (tilted) relative to the bases	Must be perpendicular to the bases
Lateral faces	Parallelograms (not necessarily rectangles)	Rectangles
Volume formula	V = B · h (h = perpendicular height)	V = B · h (h = lateral edge length, since edge ⊥ base)
Surface area	Lateral area uses slant length of the parallelogram faces	SA = 2B + P · h (simpler because faces are rectangles)

Why It Matters

Regular prisms appear throughout geometry courses when you study surface area and volume of 3-D shapes. Packaging, structural columns, and honeycomb cross-sections are real-world examples built on regular-polygon cross sections. Understanding regular prisms also prepares you for working with antiprisms, regular polyhedra, and more complex solid geometry problems.

Common Mistakes

Mistake: Confusing the height (altitude) with the lateral edge length in an oblique regular prism.

Correction: The height h is always the perpendicular distance between the two base planes. In an oblique prism the lateral edge is longer than h. Always use the perpendicular height in the volume formula V = B · h.

Mistake: Assuming a right prism is automatically a regular prism.

Correction: A right prism only guarantees that lateral edges are perpendicular to the bases. If the bases are irregular polygons (e.g., a non-equilateral triangle), the prism is right but not regular. Both conditions — right angles and regular-polygon bases — must hold for a right regular prism.

Related Terms

Prism — General category that includes regular prisms
Regular Polygon — Shape of the bases in a regular prism
Right Regular Prism — Regular prism with lateral edges perpendicular to bases
Base — The two congruent parallel faces of the prism
Altitude of a Prism — Perpendicular distance between the two bases
Volume — Key measurement calculated using V = B · h
Hexagon — Common regular polygon used as a prism base