Prisms
Learn what prisms are, their properties, and how to calculate their volume and surface area.
Definition
- 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
- Triangular prism → bases are triangles
- Rectangular prism → bases are rectangles (includes cubes)
- Pentagonal prism → bases are pentagons
- Hexagonal prism → bases are hexagons
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Worked Examples
A rectangular prism has a length of 5 cm, width of 4 cm, and height of 8 cm. Find its volume.
Identify the formula
where is the base area →
Calculate the base area
Base is a rectangle: → cm²
Multiply by height
→ cm³
Write with units
Volume is measured in cubic units → cm³
Answer: The volume is 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 . 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
5
Edges
9
Vertices
6
Volume
V = Bh
34.64 units³
Surface Area
SA = 2B + Ph
73.86 units²
Interactive Sandbox
Expression Calculator
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History
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Practice Problems
15 problemsWhat shape are the lateral faces (sides) of any prism?
Why It Matters
- 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!
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 cm³, and the cardboard needed (surface area) is cm².
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:
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 cm³ = 160 liters of water.
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 m³.
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:
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: (base area × prism height)
- 4Surface area of any prism: (2 bases + lateral faces)
- 5The height of the prism is the perpendicular distance between the two bases
Frequently Asked Questions
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
$B$ = area of base, $h$ = height of prism
Surface Area of Any Prism
$B$ = area of base, $P$ = perimeter of base, $h$ = height
Rectangular Prism Volume
$l$ = length, $w$ = width, $h$ = height
Triangular Prism Volume
$b$ = triangle base, $h_{\text{tri}}$ = triangle height, $H$ = prism height