What Is a Prism Formula Properties Surface Area and Volume Examples
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 in Geometry Definition Types and Formulas
1. What is a prism in maths?
A prism is a three-dimensional solid that has two parallel, congruent polygonal bases and rectangular or parallelogram-shaped side faces. In geometry, a prism is classified by the shape of its base.
- The two bases are identical and parallel.
- The side faces are called lateral faces.
- The height is the perpendicular distance between the two bases.
2. What is the formula for the volume of a prism?
The volume of a prism is calculated using the formula V = B × h, where B is the area of the base and h is the height. To find the volume:
- Step 1: Calculate the area of the base (B).
- Step 2: Multiply it by the perpendicular height (h).
3. How do you find the surface area of a prism?
The surface area of a prism is the sum of the areas of all its faces. The formula is Surface Area = 2B + Ph, where B is base area, P is the perimeter of the base, and h is height.
- 2B represents the two bases.
- Ph gives the lateral surface area.
4. What is the difference between a prism and a pyramid?
The main difference between a prism and a pyramid is that a prism has two parallel bases, while a pyramid has one base and a single vertex (apex).
- A prism has rectangular or parallelogram lateral faces.
- A pyramid has triangular lateral faces that meet at one point.
- Volume of prism: V = B × h.
- Volume of pyramid: V = (1/3)B × h.
5. What are the types of prisms in geometry?
Prisms are classified based on the shape of their base polygons. Common types of prisms include:
- Triangular prism
- Rectangular prism
- Square prism
- Pentagonal prism
- Hexagonal prism
6. What is a right prism?
A right prism is a prism in which the lateral edges are perpendicular to the base. In a right prism:
- The side faces are rectangles.
- The height equals the length of a lateral edge.
- Volume is still calculated as V = B × h.
7. How many faces, edges, and vertices does a prism have?
The number of faces, edges, and vertices in a prism depends on the number of sides (n) of its base polygon. For an n-sided prism:
- Faces = n + 2
- Edges = 3n
- Vertices = 2n
8. How do you find the lateral surface area of a prism?
The lateral surface area of a prism is calculated using LSA = P × h, where P is the perimeter of the base and h is the height. To calculate:
- Step 1: Find the perimeter of the base.
- Step 2: Multiply it by the height.
9. Can you give an example of finding the volume of a triangular prism?
The volume of a triangular prism is found using V = B × h, where B is the area of the triangular base. Example:
- Base triangle: base = 4 cm, height = 3 cm.
- Area of triangle B = (1/2) × 4 × 3 = 6 cm².
- Prism height = 10 cm.
10. What are some real-life examples of prisms?
Many everyday objects are examples of prisms because they have uniform cross-sections and parallel bases. Common real-life examples include:
- A rectangular prism: brick, book, cereal box.
- A triangular prism: camping tent, Toblerone chocolate bar.
- A hexagonal prism: pencil.
