How to Find the Volume and Surface Area of a Rectangular Prism?
The concept of rectangular prism plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding rectangular prisms helps students solve geometry and mensuration questions more easily and relates mathematical concepts to everyday objects like boxes, rooms, and books.
What Is Rectangular Prism?
A rectangular prism is a three-dimensional solid shape that has six faces, all of which are rectangles. It has twelve edges and eight vertices (corners). You’ll find this concept applied in areas such as geometry, mensuration, and real-life object modeling. A rectangular prism is sometimes called a cuboid. In a right rectangular prism, all angles between adjoining faces are right angles. All opposite faces are identical rectangles.
Key Formula for Rectangular Prism
Here’s the standard formula for the volume and surface area of a rectangular prism:
- Volume: V = length × width × height, or V = l × w × h
- Total Surface Area: A = 2(lw + lh + wh)
- Lateral Surface Area: LSA = 2h(l + w)
Cross-Disciplinary Usage
Rectangular prism is not only useful in Maths but also plays an important role in Physics, Computer Science, and Engineering. For example, in Physics, students use the concept when studying volume, density, and capacity. In Computer Science, 3D modeling often uses rectangular prisms for designing objects. For JEE, NEET, and Olympiad exams, understanding rectangular prism formulas can help solve advanced mensuration and geometry problems.
Step-by-Step Illustration
Let’s go step by step to find the surface area and volume of a rectangular prism with length = 8 cm, width = 5 cm, and height = 3 cm.
2. Calculate the volume:
3. Calculate the surface area:
4. Final answer:
Rectangular Prism Net
A rectangular prism net is a two-dimensional pattern that can be folded to form a three-dimensional rectangular prism. The net helps you see all six rectangular faces laid flat. This is useful for visualizing which faces to measure when calculating surface area. If you cut open a box and spread it out, you’ll get a net made up of six rectangles.
Rectangular Prism Faces, Edges, and Vertices
| Parts | Count |
|---|---|
| Faces | 6 (all rectangles) |
| Edges | 12 |
| Vertices (corners) | 8 |
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for surface area: Add together the products of each pair of dimensions, then double the total. For example, if l = 4, w = 2, h = 3: (4×2) + (2×3) + (3×4) = 8 + 6 + 12 = 26. Double it: 2×26=52. Your answer: 52 units². This trick saves you time in exams. Vedantu teachers give more such shortcuts during live tutorials.
Rectangular Prism in Real Life (Examples)
You’ll find rectangular prisms in many everyday objects. Common examples include:
- Books and notebooks
- Bricks and building blocks
- Lunch boxes
- Shipping cartons and containers
- Laptops and mobile phones
Recognizing these items as rectangular prisms helps you connect math to real-world scenarios, making learning fun!
Practice Problems to Try Yourself
- Find the volume of a rectangular prism with length 12 cm, width 7 cm, and height 5 cm.
- A box measures 10 cm × 4 cm × 6 cm. What is its total surface area?
- How many faces, edges, and vertices does a rectangular prism have?
- Give three real-life objects that are shaped like a rectangular prism.
Frequent Errors and Misunderstandings
- Mixing up length, width, and height measurements.
- Forgetting to multiply by 2 when calculating total surface area.
- Assuming all rectangular prisms are cubes (remember, only if all sides are equal!).
Key Differences: Rectangular Prism, Cuboid, and Cube
| Shape | Faces | Special Property |
|---|---|---|
| Cube | 6 (all squares) | All sides equal |
| Cuboid/Rectangular Prism | 6 (all rectangles) | Opposite faces equal; sides may differ |
Relation to Other Concepts
The concept of rectangular prism is closely connected with related topics such as volume of cuboid, surface area of cuboid, and geometry. Mastery of rectangular prisms helps you solve surface area and volume problems for cubes, cuboids, and other types of prisms. Explore more differences at Cuboid and Cube.
Classroom Tip
A simple way to remember the rectangular prism formula is to write “2(lw + lh + wh)” on a sticky note and place it on your study desk. Vedantu’s teachers recommend using color-coded diagrams to visualize length, width, and height for error-free calculations.
We explored rectangular prism — from its definition, formulas, practical examples, and common mistakes to its connection with cubes and cuboids. Keep practicing with Vedantu’s resources to boost your speed and accuracy in solving 3D geometry problems efficiently!
Useful Links for More Practice:
• Surface Area of Cuboid
• Volume of Cuboid
• Prism
• Cube
• Faces, Edges, and Vertices
FAQs on Rectangular Prism: Meaning, Formula & Examples
1. What is a rectangular prism?
A rectangular prism, also known as a cuboid, is a three-dimensional shape with six rectangular faces. All pairs of opposite faces are congruent (identical in shape and size). It has 8 vertices (corners), 12 edges, and 6 faces.
2. What are the formulas for the volume and surface area of a rectangular prism?
Let l represent length, w represent width, and h represent height. Then:
- Volume (V) = l × w × h
- Total Surface Area (TSA) = 2(lw + wh + lh)
- Lateral Surface Area (LSA) = 2h(l + w)
3. How many faces, edges, and vertices does a rectangular prism have?
A rectangular prism has:
- 6 faces (all rectangular)
- 12 edges
- 8 vertices (corners)
4. What is the difference between a rectangular prism, a cube, and a cuboid?
While the terms are often used interchangeably, there are subtle distinctions:
- Rectangular Prism: A three-dimensional shape with six rectangular faces. Opposite faces are congruent.
- Cuboid: A specific type of rectangular prism where all angles are right angles.
- Cube: A special cuboid where all six faces are squares (equal length, width, and height).
Essentially, a cube is a specific type of cuboid, which in turn is a type of rectangular prism.
5. What are some real-life examples of rectangular prisms?
Many everyday objects are rectangular prisms, including:
- Books
- Boxes
- Bricks
- Buildings (often approximated as)
- Suitcases
- Room
6. How do I calculate the volume of a rectangular prism?
Multiply the length, width, and height: Volume = Length × Width × Height
7. How do I calculate the surface area of a rectangular prism?
Use the formula: Surface Area = 2(lw + wh + lh), where l = length, w = width, and h = height.
8. What is a net of a rectangular prism?
A net is a two-dimensional representation of a three-dimensional shape. For a rectangular prism, the net shows how the six rectangular faces fit together to form the solid. It's useful for visualizing surface area calculations.
9. What is the relationship between a rectangular prism and Euler's formula?
Euler's formula (V - E + F = 2) relates the number of vertices (V), edges (E), and faces (F) of a polyhedron. For a rectangular prism: V = 8, E = 12, F = 6. Therefore, 8 - 12 + 6 = 2, satisfying Euler's formula.
10. Can a rectangular prism have square faces?
Yes! If the length and width of a rectangular prism are equal, then those faces are squares. A cube is a special case where all faces are squares.
11. What are the lateral faces of a rectangular prism?
The lateral faces are the four faces that are not the top or bottom bases. They connect the two bases.
12. How is the volume of a rectangular prism related to its base area?
The volume is the base area multiplied by the height. Since the base is a rectangle, the base area is length × width. Therefore, Volume = (Length × Width) × Height
