What is Prism in Maths? Definition, Types and Volume Formula
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 × Height2. 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 × 102. 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!
FAQs on Prism – Meaning, Types, Formulas and Examples
1. What is a prism in Maths?
A prism is a three-dimensional geometric shape with two identical and parallel polygonal bases connected by rectangular lateral faces. The shape of the base determines the prism's name (e.g., a triangular prism has triangular bases, a rectangular prism has rectangular bases).
2. What are the different types of prisms?
Prisms are classified by the shape of their bases. Common types include:
- Triangular prisms
- Rectangular prisms (also called cuboids)
- Square prisms
- Pentagonal prisms
- Hexagonal prisms
3. How do you calculate the volume of a prism?
The volume of a prism is calculated using the formula: Volume = Base Area × Height. First, find the area of the base polygon. Then, multiply this area by the prism's height (the perpendicular distance between the two bases).
4. What is the formula for the surface area of a prism?
The surface area of a prism is the sum of the areas of all its faces. There isn't one single formula; it depends on the type of prism. However, a general approach is to find the area of each face (two bases and the lateral faces) and add them together.
5. How is a prism different from a pyramid?
A key difference is the number of bases: prisms have two parallel and congruent bases, while pyramids have only one base. Prisms have rectangular lateral faces, whereas pyramids have triangular lateral faces.
6. What are some real-world examples of prisms?
Prisms are found in many everyday objects. Examples include:
- Boxes (rectangular prisms)
- Building blocks (various prisms)
- Crystals (various prism shapes)
- Parts of bridges (triangular prisms are used for strength)
7. How do you find the surface area of a rectangular prism?
For a rectangular prism with length (l), width (w), and height (h), the surface area is calculated as: Surface Area = 2(lw + lh + wh)
8. What is a right prism?
A right prism is a prism where the lateral faces are perpendicular to the bases. In other words, the lateral edges are perpendicular to the base. If the lateral edges are not perpendicular, it's called an oblique prism.
9. Explain the concept of cross-sectional area in a prism.
The cross-sectional area of a prism is the area of a plane section perpendicular to the height and parallel to the bases. In a right prism, this is identical to the area of the base. It's crucial because, for volume calculations, the base area (or cross-sectional area) is multiplied by the height.
10. What is the difference between a regular and irregular prism?
A regular prism has regular polygons as its bases (e.g., a square prism). An irregular prism has irregular polygons as its bases (e.g., a prism with a parallelogram base).
11. Can you explain how to draw a net of a triangular prism?
A net is a two-dimensional representation of a three-dimensional shape that can be folded to form the 3D shape. To draw a triangular prism net, start with two congruent triangles (the bases). Then, attach three rectangles to the sides of the triangles. The rectangles represent the lateral faces of the prism.
12. How are prisms used in architecture?
Prismatic shapes are widely used in architecture for various reasons. Triangular prisms are very strong and appear in bridge supports. Other prism shapes create visually interesting structures and allow for efficient use of space.
