Prism - Cross Section, Types, Properties, Formulas ... - Vedantu

The concept of prism in maths plays a key role in geometry and is widely used in maths classrooms, textbooks, project work, and competitive exams. Understanding prisms helps students solve questions on volume, surface area, and visualization of 3D shapes, and also connects deeply with real-life applications.

What Is Prism in Maths?

A prism in maths is a three-dimensional (3D) solid shape with flat sides, having two identical, parallel polygonal bases connected by rectangular (or parallelogram) faces called lateral faces. Prisms do not have any curved surfaces. You’ll find this concept useful in geometry, solid geometry, and mensuration topics. Each prism is named after the shape of its bases, such as triangular prism, rectangular prism, or hexagonal prism.

Types of Prisms

Prisms can be classified in two main ways:

• Based on the shape of the base (triangular, rectangular, pentagonal, hexagonal, etc.)
• Based on orientation: Right prism (side faces are rectangles, axis perpendicular to base) and Oblique prism (side faces are parallelograms, axis not perpendicular to base)

Prism Type	Base Shape	Example
Triangular Prism	Triangle	Tent, Toblerone chocolate
Rectangular Prism (Cuboid)	Rectangle	Bricks, Book
Square Prism	Square	Drawer, Cube
Hexagonal Prism	Hexagon	Nut Bolt, Pencil (old style)

Key Formula for Prism in Maths

Here are the essential prism formulas every student should know:

• Volume of a Prism = Base Area × Height
• Surface Area of Prism = 2 × (Base Area) + (Perimeter of Base × Height)

For example, the volume \( V \) of a rectangular prism with base area \( B \) and height \( h \) is: \( V = B \times h \).

These formulas are crucial for exam problems and real-life applications.

Properties of a Prism

• Two parallel, congruent polygonal bases
• Lateral faces are rectangles (in right prism) or parallelograms (in oblique prism)
• No curved surfaces
• Faces = (Number of base sides + 2), Edges = (3 × number of base sides), Vertices = (2 × number of base sides)
• Uniform cross-section throughout length

Feature	Rectangular Prism	Triangular Prism
Bases	2 rectangles	2 triangles
Lateral Faces	4 rectangles	3 rectangles
Vertices	8	6

Step-by-Step Illustration: Sample Problems

Let’s see how to solve a volume of prism question:

Question: Find the volume of a rectangular prism with base area 30 cm2 and height 12 cm.

1. Identify formula: Volume = Base Area × Height 2. Substitute values: Volume = 30 × 12 3. Calculate: Volume = 360 cm3 4. Final Answer: 360 cm3

Question: A triangular prism has a base area of 50 cm2 and height 10 cm. What is its volume?

1. Volume = 50 × 10 2. Volume = 500 cm3

Frequent Errors and Misunderstandings

• Confusing prism with pyramid (prisms have 2 bases, pyramids only 1)
• Using wrong base area or base shape formula
• Missing the correct unit in answers (cm2 vs cm3)
• Applying prism formula to a cylinder (cylinder base is curved, not polygonal)

Relation to Other Concepts

Understanding prism in maths also helps with advanced topics like cross-sectional area, 3D shapes and their properties, and the difference between prism and pyramid. It anchors your study for solid geometry, nets of solids, and even some physics (optical prisms).

Classroom Tip

A quick way to remember prisms: "If it stands on a base and the cross-section matches from one end to the other, it's a prism!" Vedantu’s teachers often draw prism nets in class to help students visualize how 3D solids unfold.

Try These Yourself

• Draw a net diagram of a triangular prism and color each face differently.
• A hexagonal prism has a base area of 24 cm2 and a height of 8 cm. What is its volume?
• Name two real-life objects shaped like a rectangular prism.
• Find the surface area of a cube (which is a special square prism) with side 5 cm.

Quick Reference Table

Prism Name	Base Shape	# Faces	Volume Formula
Triangular Prism	Triangle	5	(Area of Triangle) × Height
Rectangular Prism	Rectangle	6	Length × Width × Height
Pentagonal Prism	Pentagon	7	(Area of Pentagon) × Height

Related Topic Links

• Rectangular Prism – For more on cuboid formulas and application.
• Cross-Sectional Area – To master volume and surface calculations.
• Properties of 3D Shapes – Compare prisms with spheres, cones, cubes, and others.

We explored prism in maths—from definition, important formulas, types, solved examples, mistakes to avoid, and how prisms connect to other key maths ideas. Keep practicing regularly and join Vedantu’s maths sessions for more confidence with prism questions and all other geometry basics!