Cuboid Calculator

Calculator and formulas for calculating the volume and surface of a cuboid

The Rectangular Box - Six Faces of Perfection!

Cuboid Calculator

The Cuboid

A cuboid is a polyhedron with six rectangular faces. Opposite faces are parallel and equal.

Enter the dimensions
Length of the cuboid
Width of the cuboid
Height of the cuboid
Cuboid Calculation Results
Volume V:
Surface S:
Face diagonal d:
Space diagonal e:
Cuboid Properties

The rectangular box: Six rectangular faces with opposite faces parallel and equal

6 Rectangular faces 12 Edges 8 Vertices 3 Dimensions

Cuboid Structure

cuboid

The rectangular box with three different dimensions.
Length × Width × Height.

Properties of a cuboid

A cuboid is a fundamental three-dimensional geometric shape:

  • Definition: Six rectangular faces
  • Opposite faces: Parallel and equal in size
  • Dimensions: Length, width, and height
  • Vertices: 8 corners
  • Edges: 12 edges in total
  • Also known as: Rectangular prism

Geometric Properties of the Cuboid

The cuboid displays fundamental geometric characteristics:

Basic Parameters
  • Faces: 6 rectangles
  • Edges: 12 edges (4 of each length)
  • Vertices: 8 corners
  • Euler Formula: V - E + F = 8 - 12 + 6 = 2
Special Properties
  • Orthogonal: All angles are 90 degrees
  • Parallel faces: Opposite faces are parallel
  • Right angles: All edges meet at right angles
  • Convex: All vertices point outward

Mathematical Relationships

The cuboid follows simple but elegant mathematical laws:

Volume Formula
Simple multiplication

Volume equals length times width times height. Straightforward and elegant.

Surface Area Formula
Sum of all faces

Surface area is twice the sum of the three face areas. Each pair of opposite faces contributes equally.

Applications of the Cuboid

Cuboids are found everywhere in our daily lives:

Architecture & Construction
  • Building foundations
  • Room dimensions
  • Concrete blocks
  • Structural elements
Packaging & Storage
  • Shipping boxes
  • Storage containers
  • Product packaging
  • Warehouse planning
Education & Learning
  • Geometry lessons
  • Volume calculations
  • 3D visualization
  • Mathematical modeling
Engineering & Manufacturing
  • Machine components
  • Material calculations
  • CAD modeling
  • Production planning

Formulas for a cuboid

Volume (V)
\[V = a \cdot b \cdot c\]

The volume is calculated from the product of length, width and height

Surface area (S)
\[S = 2 \cdot (ab + ac + bc)\]

The surface area is the sum of the areas of all six sides

Face diagonal (d)
\[d = \sqrt{a^2 + b^2}\]

The diagonal of a face using the Pythagorean theorem

Space diagonal (e)
\[e = \sqrt{a^2 + b^2 + c^2}\]

The spatial diagonal from one corner to the opposite corner

Perimeter of base area
\[U_{base} = 2 \cdot (a + b)\]

The perimeter of the rectangular base area

Example calculation for a cuboid

Given
Length a = 4 cm Width b = 3 cm Height c = 2 cm

Calculate: All properties of the cuboid

1. Volume calculation
\[V = 4 \times 3 \times 2 = 24 \text{ cm}^3\]

The volume is 24 cubic centimeters

2. Surface area calculation
\[S = 2 \times (4 \times 3 + 4 \times 2 + 3 \times 2)\] \[S = 2 \times (12 + 8 + 6) = 52 \text{ cm}^2\]

The surface area is 52 square centimeters

3. Space diagonal calculation
\[e = \sqrt{4^2 + 3^2 + 2^2}\] \[e = \sqrt{16 + 9 + 4} = \sqrt{29} \approx 5.39 \text{ cm}\]

The space diagonal is approximately 5.39 cm

4. Perimeter of base
\[U_{base} = 2 \times (4 + 3) = 14 \text{ cm}\]

The perimeter of the base area is 14 cm

Summary
Volume: 24 cm³ Surface: 52 cm²
Space diagonal: 5.39 cm Base perimeter: 14 cm

A complete analysis of the 4×3×2 cuboid

The Cuboid: Foundation of 3D Geometry

The cuboid is one of the most fundamental and important shapes in three-dimensional geometry. Also known as a rectangular prism or rectangular box, it forms the basis for understanding many geometric concepts. Its simple yet elegant structure, with six rectangular faces meeting at right angles, makes it an ideal starting point for exploring 3D mathematics and real-world applications.

The Beauty of Rectangular Perfection

The cuboid demonstrates the elegance of orthogonal geometry:

  • Right angles: All edges meet at perfect 90-degree angles
  • Parallel faces: Opposite faces are perfectly parallel and congruent
  • Three dimensions: Length, width, and height define the entire shape
  • Rectangular faces: All six faces are rectangles
  • Symmetry: Multiple planes of symmetry
  • Simplicity: Easy-to-understand formulas
  • Versatility: Foundation for complex geometric constructions

Mathematical Elegance

Formula Simplicity

The formulas for cuboid calculations are beautifully straightforward, demonstrating how fundamental geometric principles can be expressed in simple mathematical terms.

Dimensional Relationships

The cuboid perfectly illustrates how changing one dimension affects volume linearly, while surface area changes in a more complex but predictable pattern.

Practical Applications

From architecture to packaging, the cuboid's practical applications are endless, making it one of the most useful geometric shapes in engineering and design.

Educational Value

As a teaching tool, the cuboid helps students understand the transition from 2D to 3D thinking, spatial relationships, and the concept of volume versus surface area.

Conclusion

The cuboid stands as a testament to the power of geometric simplicity. Its straightforward structure belies its fundamental importance in mathematics, science, and everyday life. From the boxes we ship to the rooms we live in, from the foundations of buildings to the pixels on our screens, cuboids are everywhere. Understanding the cuboid provides a solid foundation for exploring more complex three-dimensional shapes and develops spatial reasoning skills essential for mathematics, engineering, and design. As both a practical tool and a mathematical concept, the cuboid continues to be one of the most important shapes in our three-dimensional world.