Prisms

Learn what prisms are, their properties, and how to calculate their volume and surface area.

Intermediate25 minLesson
Printable worksheet

Definition

A prism is a three-dimensional shape with two identical, parallel bases connected by rectangular faces.
Key Properties:
  • The two bases are congruent (same shape and size)
  • The bases can be any polygon (triangle, rectangle, pentagon, hexagon, etc.)
  • The lateral faces (sides) are always rectangles
  • The height is the perpendicular distance between the bases
Naming Prisms: Prisms are named by the shape of their bases:
  • Triangular prism → bases are triangles
  • Rectangular prism → bases are rectangles (includes cubes)
  • Pentagonal prism → bases are pentagons
  • Hexagonal prism → bases are hexagons
Volume Formula:
V=B×hV = B \times h
where BB = area of the base, hh = height of the prism
Surface Area Formula:
SA=2B+PhSA = 2B + Ph
where BB = area of base, PP = perimeter of base, hh = height

Try it now

What shape are the lateral faces (sides) of any prism?

Worked Examples

A rectangular prism has a length of 5 cm, width of 4 cm, and height of 8 cm. Find its volume.
1

Identify the formula

V=B×hV = B \times h where BB is the base area → V=B×hV = B \times h
2

Calculate the base area

Base is a rectangle: B=l×w=5×4B = l \times w = 5 \times 4 → B=20B = 20 cm²
3

Multiply by height

V=20×8V = 20 \times 8 → V=160V = 160 cm³
4

Write with units

Volume is measured in cubic units → V=160V = 160 cm³

Common Mistakes

Confusing height of the base with height of the prism

Why it's wrong: In a triangular prism, there are two different heights: the height of the triangular base (used to find base area) and the height/length of the prism (the distance between the bases).

Correct: Label your diagram clearly. Use 'base height' for the triangle's height and 'prism height' for the length of the prism.

Forgetting to multiply the base area by 2 for surface area

Why it's wrong: A prism has TWO bases (top and bottom), not just one.

Correct: The surface area formula is SA=2B+PhSA = 2B + Ph. The '2B' accounts for both bases.

Using the wrong units (cm² for volume, cm³ for area)

Why it's wrong: Volume measures 3D space (cubic units), while area measures 2D surfaces (square units).

Correct: Volume always uses cubic units (cm³, m³). Surface area always uses square units (cm², m²).

Interactive Visual

3D Shape Viewer

Faces

6

Edges

12

Vertices

8

Euler's formula: V − E + F = 8 − 12 + 6 = 2

Volume

V = lwh

V = 6 · 4 · 8 = 192

192 units³

Surface Area

SA = 2(lw + lh + wh)

SA = 2(6 · 4 + 6 · 8 + 4 · 8) = 208

208 units²

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

15 problems
Problem 1 of 15
Easy
What shape are the lateral faces (sides) of any prism?

Why It Matters

Prisms are everywhere in our daily lives:
  • Architecture: Buildings often have prism-shaped sections. Glass atriums, doorways, and roof structures frequently use triangular prisms
  • Packaging: Most boxes are rectangular prisms. Understanding prisms helps companies minimize material while maximizing storage
  • Engineering: Bridges use triangular prism supports because triangles distribute weight efficiently
  • Science: Optical prisms split white light into a rainbow of colors
  • Food: Many products come in prism shapes: Toblerone chocolate is a famous triangular prism!
Understanding how to calculate volume and surface area helps with practical problems like determining how much a container holds or how much material is needed to wrap a package.

Real World Applications

Packaging and Shipping

Companies use prism calculations to design efficient packaging that minimizes material waste while maximizing product protection.

Example:

A shipping box is 30 cm × 20 cm × 15 cm. The volume is 30×20×15=900030 \times 20 \times 15 = 9000 cm³, and the cardboard needed (surface area) is 2(30×20+30×15+20×15)=27002(30 \times 20 + 30 \times 15 + 20 \times 15) = 2700 cm².

1Try It Yourself

You need to ship a gift in a box that is 25 cm long, 15 cm wide, and 10 cm tall.

How much wrapping paper do you need (surface area)?

Step 1: Write the mathematical expression

Calculate: 2×(25×15+25×10+15×10)2 \times (25 \times 15 + 25 \times 10 + 15 \times 10)

Aquarium Design

Aquarium builders use volume calculations to determine water capacity and ensure proper filtration systems.

Example:

A rectangular aquarium is 80 cm × 40 cm × 50 cm. It holds 80×40×50=16000080 \times 40 \times 50 = 160000 cm³ = 160 liters of water.

2Try It Yourself

You want to build a fish tank that is 60 cm long, 30 cm wide, and 40 cm tall.

How many liters of water will it hold? (1 liter = 1000 cm³)

Step 1: Write the mathematical expression

Volume = length × width × height, then convert

Architecture and Construction

Architects use prism calculations for building materials, room dimensions, and structural planning.

Example:

A triangular roof section has a base of 8 m, height 3 m, and length 12 m. The volume of attic space is 12×8×3×12=144\frac{1}{2} \times 8 \times 3 \times 12 = 144 m³.

3Try It Yourself

A tent has a triangular cross-section with base 4 m and height 2.5 m. The tent is 6 m long.

What is the volume of air inside the tent?

Step 1: Write the mathematical expression

Calculate: 12×4×2.5×6\frac{1}{2} \times 4 \times 2.5 \times 6

Key Takeaways

  • 1A prism has two identical, parallel polygon bases connected by rectangular lateral faces
  • 2Prisms are named by their base shape: triangular prism, rectangular prism, pentagonal prism, etc.
  • 3Volume of any prism: V=B×hV = B \times h (base area × prism height)
  • 4Surface area of any prism: SA=2B+PhSA = 2B + Ph (2 bases + lateral faces)
  • 5The height of the prism is the perpendicular distance between the two bases

Frequently Asked Questions

Yes! A cube is a special rectangular prism where all edges are equal. The base is a square, and the height equals the side of the square.
Yes! A cube is a special rectangular prism where all edges are equal. The base is a square, and the height equals the side of the square.
A prism has two identical parallel bases and rectangular sides. A pyramid has one base and triangular sides that meet at a single point (apex).
No, by definition prisms have flat polygonal bases and flat rectangular lateral faces. A cylinder looks similar but has circular bases, it's not technically a prism.

Glossary

Prism
A 3D shape with two identical, parallel polygon bases connected by rectangular faces
Base
The two identical polygon faces at the top and bottom of a prism
Lateral face
The rectangular faces on the sides of a prism connecting the two bases
Height
The perpendicular distance between the two bases of a prism
Congruent
Having the same shape and size

Formula Card

Volume of Any Prism

V=B×hV = B \times h

$B$ = area of base, $h$ = height of prism

Surface Area of Any Prism

SA=2B+PhSA = 2B + Ph

$B$ = area of base, $P$ = perimeter of base, $h$ = height

Rectangular Prism Volume

V=l×w×hV = l \times w \times h

$l$ = length, $w$ = width, $h$ = height

Triangular Prism Volume

V=12bhtri×HV = \frac{1}{2}bh_{\text{tri}} \times H

$b$ = triangle base, $h_{\text{tri}}$ = triangle height, $H$ = prism height

Today's challengeOne problem, three tries

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