Right Regular Prism

Right Regular Prism

A right prism with bases that are regular polygons.

Hexagonal right prism with height h labeled. Formulas: Volume=Bh, Lateral Surface Area=hP, Total Surface Area=hP+2B.

Key Formula

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

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
$\text{SA}$ = Total surface area of the prism
$P$ = Perimeter of one base

Worked Example

Problem: A right regular hexagonal prism has a base edge length of 4 cm and a height of 10 cm. Find its volume and total surface area.

Step 1: Find the area of the regular hexagonal base. The area of a regular hexagon with side length s is:

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. A regular hexagon has 6 equal sides.

P = 6 \times 4 = 24\;\text{cm}

Step 4: Compute the lateral surface area (the area of the six rectangular faces).

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

Step 5: Compute the total surface area by adding the two bases and the lateral area.

\text{SA} = 2B + P \cdot h = 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 works backward from a known surface area to find the height, demonstrating how to rearrange the surface area formula. It also uses a triangular base instead of a hexagonal one.

Problem: A right regular triangular prism has a total surface area of 108 + 12√3 cm² and a base edge length of 4 cm. Find the height of the prism.

Step 1: Find the area of the equilateral-triangle base with side length 4 cm.

B = \frac{\sqrt{3}}{4}\,s^2 = \frac{\sqrt{3}}{4}(4)^2 = 4\sqrt{3}\;\text{cm}^2

Step 2: Find the perimeter of the base.

P = 3 \times 4 = 12\;\text{cm}

Step 3: Set up the surface area equation and substitute known values.

\text{SA} = 2B + P \cdot h \implies 108 + 12\sqrt{3} = 2(4\sqrt{3}) + 12h

Step 4: Solve for h. Simplify the right side and isolate h.

108 + 12\sqrt{3} = 8\sqrt{3} + 12h \implies 12h = 108 + 4\sqrt{3} \implies h = 9 + \frac{\sqrt{3}}{3}

Answer: The height of the prism is 9 + √3/3 ≈ 9.577 cm.

Frequently Asked Questions

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

A right prism has lateral edges perpendicular to its bases, but the bases can be any polygon (even an irregular one). A right regular prism adds the requirement that the bases are regular polygons — polygons with all sides equal and all interior angles equal. So every right regular prism is a right prism, but not every right prism is a right regular prism.

Is a cube a right regular prism?

Yes. A cube can be viewed as a right regular prism whose bases are squares (a regular 4-sided polygon) and whose height equals the side length of the base. It is a special case of a right square prism where all edges are equal.

How do you find the lateral surface area of a right regular prism?

Multiply the perimeter of the base by the height of the prism: Lateral Area = P · h. Because all base edges are equal in a regular polygon with n sides and edge length s, this simplifies to Lateral Area = n · s · h. Each of the n rectangular lateral faces has area s × h.

Right Regular Prism vs. Oblique Prism

	Right Regular Prism	Oblique Prism
Lateral edges	Perpendicular to the bases	Tilted (not perpendicular) to the bases
Lateral faces	Rectangles	Parallelograms (not necessarily rectangles)
Base shape	Regular polygon (all sides and angles equal)	Any polygon
Volume formula	V = B · h (h is the prism's height and equals the lateral edge length)	V = B · h (h is the perpendicular distance between bases, not the lateral edge length)
Lateral surface area	P · h (simple product)	Requires finding the slant height of each parallelogram face

Why It Matters

Right regular prisms appear frequently in geometry courses when students study three-dimensional solids, surface area, and volume. They also serve as the foundation for understanding more complex solids like antiprisms and cylinders (which can be thought of as prisms with infinitely many sides). In real life, many everyday objects — hexagonal pencils, triangular Toblerone boxes, and square columns — are right regular prisms.

Common Mistakes

Mistake: Confusing the height of the prism with the slant height or the apothem of the base.

Correction: In a right regular prism the height h is the perpendicular distance between the two bases (equal to the lateral edge length). The apothem is the distance from the center of the base polygon to the midpoint of a side and is used only when computing the base area.

Mistake: Forgetting to include both bases when calculating total surface area.

Correction: The total surface area formula is SA = 2B + P · h. Students sometimes compute only P · h (the lateral area) and forget to add the two base areas (2B).

Related Terms

Right Prism — Parent category — bases need not be regular
Prism — General solid with two congruent parallel bases
Regular Polygon — Shape of the bases in this prism
Base — The two congruent polygonal faces of the prism
Right Square Prism — Special case with square bases
Lateral Surface Area — Area of the rectangular side faces
Surface Area — Total area including bases and lateral faces
Volume — Space enclosed, found with V = B · h