Rectangular Prism Calculator

Calculate the volume, surface area, and space diagonal of a rectangular prism.

Enter the length, width, and height of the prism to get detailed results. This calculator is useful for geometry problems, construction projects, and more.

Examples

Click on an example to load its data into the calculator.

Standard Packing Box

Standard Box

A common example of a standard-sized packing box.

Length: 12

Width: 8

Height: 6

Perfect Cube

Cube

A special case where all sides are equal, forming a perfect cube.

Length: 10

Width: 10

Height: 10

Flat Rectangular Prism

Flat Prism

An example of a flat object, like a book or a chocolate bar.

Length: 25

Width: 15

Height: 2

Prism with Decimal Dimensions

Decimal Values

A prism with dimensions that are not whole numbers.

Length: 7.5

Width: 4.2

Height: 3.8

Other Titles
Understanding the Rectangular Prism: A Comprehensive Guide
Explore the properties, formulas, and applications of rectangular prisms in geometry and the real world.

What is a Rectangular Prism? Core Concepts

  • A three-dimensional solid shape with six rectangular faces.
  • Characterized by three dimensions: length, width, and height.
  • Also known as a cuboid.
A rectangular prism is a fundamental three-dimensional geometric shape. It's a polyhedron with six faces, all of which are rectangles. Opposite faces are equal and parallel. It has 12 edges and 8 vertices (corners). The angles between adjacent faces are all right angles (90 degrees).
The three key dimensions that define a rectangular prism are its length (l), width (w), and height (h). These dimensions determine all other properties of the prism, including its volume, surface area, and the length of its diagonals.
Key Properties:
  • Faces: 6 rectangular faces.
  • Edges: 12 edges.
  • Vertices: 8 vertices.
  • Special Case: A cube is a special type of rectangular prism where length, width, and height are all equal.

Basic Rectangular Prism Examples

  • A shoebox with length 30cm, width 15cm, height 10cm.
  • A standard brick with dimensions 8in x 3.5in x 2.25in.
  • A swimming pool that is 25m long, 10m wide, and 2m deep.

Step-by-Step Guide to Using the Rectangular Prism Calculator

  • Input the dimensions of the prism accurately.
  • Understand the calculated results: volume, surface area, and space diagonal.
  • Utilize the reset and example features for ease of use.
Our calculator simplifies the process of finding the key properties of a rectangular prism. Follow these steps for accurate calculations.
Inputting Dimensions:
1. Length (l): Enter the measurement of the longest side of the prism's base in the 'Length' field.
2. Width (w): Enter the measurement of the shorter side of the prism's base in the 'Width' field.
3. Height (h): Enter the perpendicular distance between the two bases in the 'Height' field.
Ensure all measurements are in the same unit (e.g., all in cm or all in inches) for the results to be meaningful.
Interpreting the Results:
  • Volume: The total space enclosed by the prism. The unit will be cubed (e.g., cm³, in³).
  • Surface Area: The total area of all six faces of the prism. The unit will be squared (e.g., cm², in²).
  • Space Diagonal: The length of the diagonal connecting two opposite vertices through the interior of the prism. The unit will be the same as the input unit (e.g., cm, in).

Practical Usage Examples

  • Input: l=10, w=5, h=2 → Results: Volume=100, Surface Area=160, Space Diagonal=11.358
  • Input: l=7, w=7, h=7 (a cube) → Results: Volume=343, Surface Area=294, Space Diagonal=12.124

Mathematical Formulas and Derivations

  • The formula for the volume of a rectangular prism.
  • The formula for the surface area of a rectangular prism.
  • The formula for the space diagonal of a rectangular prism derived from the Pythagorean theorem.
Understanding the formulas behind the calculations can provide deeper insight into the geometry of a rectangular prism.
Volume (V)
The volume is the product of its three dimensions. It represents the capacity of the prism.
Formula: V = length × width × height = l × w × h
Surface Area (SA)
The surface area is the sum of the areas of all six rectangular faces. Since there are three pairs of identical faces (top/bottom, front/back, left/right), the formula is:
Formula: SA = 2 × (lw + lh + wh)
Space Diagonal (d)
The space diagonal is the longest diagonal of the prism, running from one vertex to the opposite one. It can be found by applying the Pythagorean theorem twice. First, we find the diagonal of the base (d_base = √(l² + w²)). Then, we use this and the height to find the space diagonal.
Formula: d = √(l² + w² + h²)

Formula Application

  • For a prism with l=3, w=4, h=5: Volume = 3*4*5 = 60.
  • Surface Area = 2 * (3*4 + 3*5 + 4*5) = 2 * (12 + 15 + 20) = 2 * 47 = 94.
  • Space Diagonal = √(3² + 4² + 5²) = √(9 + 16 + 25) = √50 ≈ 7.071.

Real-World Applications of Rectangular Prisms

  • Architecture and Construction: Designing rooms, buildings, and structures.
  • Packaging and Shipping: Calculating box sizes and shipping volumes.
  • Everyday Objects: From aquariums to smartphones.
Rectangular prisms are one of the most common shapes we encounter in our daily lives and various industries.
In Construction:
  • Architects and engineers calculate the volume of rooms to determine heating and cooling requirements.
  • The amount of materials like bricks, concrete, or drywall needed for a building is based on surface area and volume calculations.
In Logistics:
  • Shipping companies use volume calculations to determine how many packages can fit into a container or truck.
  • The cost of shipping is often directly related to the volumetric weight of a package.
In Manufacturing and Design:
  • Designers calculate the volume of containers like cereal boxes or milk cartons to ensure they hold the correct amount of product.
  • The dimensions of electronic devices like phones and laptops are examples of rectangular prisms.

Industry Examples

  • Calculating the amount of water needed to fill a rectangular swimming pool.
  • Determining the storage capacity of a warehouse.
  • Designing a custom-fit box for a product.

Common Questions and Clarifications

  • Difference between a cuboid and a cube.
  • How to handle different units of measurement.
  • Why the space diagonal is important.
Is a cube a rectangular prism?
Yes. A cube is a special kind of rectangular prism where all edges (length, width, and height) are equal. All cubes are rectangular prisms, but not all rectangular prisms are cubes.
What if my measurements are in different units?
It is crucial to convert all your measurements to a single, consistent unit before using the calculator. For example, if your length is in feet and your width is in inches, you must convert one of them so they are both in either feet or inches. Mixing units will lead to incorrect results.
What is the practical use of the space diagonal?
The space diagonal determines the longest straight object that can fit inside the rectangular prism. For example, it tells you the maximum length of a rod, pipe, or umbrella that can be packed into a box.

Clarification Examples

  • A box with l=12, w=12, h=12 is a cube and a rectangular prism.
  • A box with l=12, w=10, h=10 is a rectangular prism but not a cube.