Cuboid: Definitions and Examples

Cuboid: Definitions, Formulas, & Examples

GET TUTORING NEAR ME!

(800) 434-2582

By submitting the following form, you agree to Club Z!'s Terms of Use and Privacy Policy

    Home / Educational Resources / Math Resources / Cuboid: Definitions and Examples

    A cuboid is a three-dimensional shape that is often used in mathematics and geometry. It is also known as a rectangular parallelepiped or rectangular prism. A cuboid is made up of six rectangular faces or sides, where each face meets at right angles with the other two adjacent faces. The three dimensions of a cuboid are its length, width, and height.

    The length of a cuboid is the longest side, while the width is the shorter side, and the height is the vertical dimension. The length, width, and height are usually denoted by the letters l, w, and h, respectively. A cuboid can be thought of as a stretched cube, where the three dimensions are different from each other.

    The surface area of a cuboid is the total area of all its six rectangular faces. The formula for the surface area of a cuboid is 2lw + 2lh + 2wh, where l, w, and h are the length, width, and height of the cuboid, respectively. The surface area is measured in square units. For example, if the length of a cuboid is 4 units, the width is 3 units, and the height is 5 units, the surface area of the cuboid would be:

    2(4 x 3) + 2(4 x 5) + 2(3 x 5) = 24 + 40 + 30 = 94 square units.

    The volume of a cuboid is the amount of space enclosed by the six rectangular faces. The formula for the volume of a cuboid is l x w x h, where l, w, and h are the length, width, and height of the cuboid, respectively. The volume is measured in cubic units. Using the same example as above, the volume of the cuboid would be:

    4 x 3 x 5 = 60 cubic units.

    Cuboids are used in many different fields, such as architecture, engineering, and physics. In architecture, cuboids are often used to represent the dimensions of buildings, rooms, and other structures. Engineers use cuboids to design and model objects with rectangular shapes, such as buildings, bridges, and machinery. In physics, cuboids are used to represent objects with rectangular shapes, such as boxes, containers, and other objects with fixed dimensions.

    Cuboids are also used in many everyday objects, such as bookshelves, refrigerators, and televisions. These objects are often designed with rectangular shapes to make them easier to manufacture, transport, and use. Cuboids are also commonly used in packaging and shipping industries, where products are often packaged in boxes with rectangular shapes.

    Cuboids have several properties that make them unique. One of the most important properties of a cuboid is that all its six faces are rectangles. This means that the opposite faces of a cuboid are always equal in size and shape. Another property of a cuboid is that its opposite faces are always parallel to each other. This means that the cuboid can be slid along any of its three dimensions without changing its shape or size.

    Cuboids also have several special cases, depending on their dimensions. For example, if a cuboid has equal dimensions in all three directions, it is called a cube. A cube is a special case of a cuboid where all its sides are equal in length. The formula for the surface area and volume of a cube is the same as that for a cuboid, where the length, width, and height are all equal.

    Definition:

    A cuboid is a three-dimensional shape that has six faces, all of which are rectangles. It is a type of rectangular prism that has a width, height, and length. The width, height, and length of a cuboid are also known as its three dimensions. The three dimensions of a cuboid are perpendicular to each other, and they intersect at right angles.

    Properties:

    Cuboids have several properties that are important in understanding their characteristics and applications. Some of the main properties of cuboids are as follows:

    1. Faces: Cuboids have six faces, and each face is a rectangle. The faces of a cuboid are opposite and parallel to each other.
    2. Edges: Cuboids have 12 edges, and each edge is the intersection of two adjacent faces.
    3. Vertices: Cuboids have eight vertices, and each vertex is the point where three edges intersect.
    4. Dimensions: The dimensions of a cuboid are its length, width, and height. The length is the longest side of the cuboid, the width is the shortest side, and the height is the distance between the two opposite faces.
    5. Volume: The volume of a cuboid is calculated by multiplying its length, width, and height. The formula for the volume of a cuboid is V = l x w x h, where V is the volume, l is the length, w is the width, and h is the height.
    6. Surface area: The surface area of a cuboid is the sum of the areas of its six faces. The formula for the surface area of a cuboid is A = 2lw + 2lh + 2wh, where A is the surface area, l is the length, w is the width, and h is the height.
    7. Diagonal: The diagonal of a cuboid is the longest line segment that can be drawn from one vertex to another vertex that is not on the same face. The formula for the diagonal of a cuboid is d = ?(l^2 + w^2 + h^2), where d is the diagonal, l is the length, w is the width, and h is the height.

    Examples:

    Cuboids have a wide range of applications in engineering, architecture, and other fields. Here are five examples of how cuboids are used in various industries:

    • Shipping containers: Shipping containers are typically cuboids that are used to transport goods over long distances. They are designed to be stackable and to fit on standard-sized cargo ships.
    • Bricks: Bricks are rectangular cuboids that are used in construction to build walls and other structures. They are made from materials such as clay, concrete, and stone.
    • Bookshelves: Bookshelves are often cuboids that are designed to hold books and other items. They are typically made from wood, metal, or plastic and come in various sizes and shapes.
    • Computer screens: Many computer screens are cuboids that are designed to display digital images and text. They are typically made from materials such as glass, plastic, and metal.
    • Pools: Swimming pools are often cuboids that are designed to hold water for recreational purposes. They are typically made from materials such as concrete, fiberglass, or vinyl.

    Quiz

    1. What is a cuboid? A: A cuboid is a three-dimensional shape that has six rectangular faces, with opposite faces parallel and equal in size.
    2. What are the dimensions of a cuboid? A: A cuboid has three dimensions: length, width, and height.
    3. What is the formula for the volume of a cuboid? A: The volume of a cuboid is calculated by multiplying its length, width, and height together: V = l × w × h.
    4. What is the formula for the surface area of a cuboid? A: The surface area of a cuboid is calculated by adding the area of all six faces together: SA = 2lw + 2lh + 2wh.
    5. How many edges does a cuboid have? A: A cuboid has 12 edges.
    6. How many vertices does a cuboid have? A: A cuboid has 8 vertices.
    7. How is a cuboid different from a cube? A: A cube is a special type of cuboid where all six faces are square and all edges are equal in length.
    8. What is the maximum number of right angles that can be found in a cuboid? A: There are always 12 right angles in a cuboid.
    9. What is the longest diagonal of a cuboid? A: The longest diagonal of a cuboid is the space diagonal, which is the distance between two opposite corners of the cuboid. The formula for the space diagonal is ?(l² + w² + h²).
    10. Can a cuboid have all three dimensions of different sizes? A: Yes, a cuboid can have all three dimensions of different sizes, as long as it still has six rectangular faces.

     

    If you’re interested in online or in-person tutoring on this subject, please contact us and we would be happy to assist!

    Cuboid:

    Example plots

    Example plots

    Alternate names

    box | rectangular parallelepiped | rectangular right prism

    Equation

    abs(x)<=a/2 and abs(y)<=b/2 and abs(z)<=c/2

    (for a cuboid with center at the origin and edges along the Cartesian axes of lengths a, b, and c, respectively)

    Solid properties

    (-a/2, -b/2, -c/2) | (-a/2, -b/2, c/2) | (-a/2, b/2, -c/2) | (-a/2, b/2, c/2) | (a/2, -b/2, -c/2) | (a/2, -b/2, c/2) | (a/2, b/2, -c/2) | (a/2, b/2, c/2)

    8

    c

    S = 2 (a b + a c + b c)

    x^_ = (0, 0, 0)

    V = a b c

    I = (1/12 (b^2 + c^2) | 0 | 0 0 | 1/12 (a^2 + c^2) | 0 0 | 0 | 1/12 (a^2 + b^2))

    (for a cuboid with center at the origin and edges along the Cartesian axes of lengths a, b, and c, respectively)

    Distance properties

    sqrt(a^2 + b^2 + c^2)

    χ = 1

    s^_ = (90 a^2 b^2 sqrt(a^2 + b^2 + c^2) c^2 + 105 a b (sinh^(-1)(c/sqrt(a^2 + b^2)) b a^3 + sinh^(-1)(b/sqrt(a^2 + c^2)) c a^3 + sinh^(-1)(c/sqrt(a^2 + b^2)) b^3 a + sinh^(-1)(b/sqrt(a^2 + c^2)) c^3 a + sinh^(-1)(a/sqrt(b^2 + c^2)) b c^3 + sinh^(-1)(a/sqrt(b^2 + c^2)) b^3 c) c - 84 a b (tan^(-1)((b c)/(a sqrt(a^2 + b^2 + c^2))) a^4 + tan^(-1)((a c)/(b sqrt(a^2 + b^2 + c^2))) b^4 + tan^(-1)((a b)/(c sqrt(a^2 + b^2 + c^2))) c^4) c + 21 (sinh^(-1)(b/a) b a^6 + sinh^(-1)(c/a) c a^6 + sinh^(-1)(a/b) b^6 a + sinh^(-1)(a/c) c^6 a + sinh^(-1)(b/c) b c^6 + sinh^(-1)(c/b) b^6 c) - 21 (sinh^(-1)(b/sqrt(a^2 + c^2)) b a^6 + sinh^(-1)(c/sqrt(a^2 + b^2)) c a^6 + sinh^(-1)(a/sqrt(b^2 + c^2)) b^6 a + sinh^(-1)(a/sqrt(b^2 + c^2)) c^6 a + sinh^(-1)(b/sqrt(a^2 + c^2)) b c^6 + sinh^(-1)(c/sqrt(a^2 + b^2)) b^6 c) + 25 (b^2 (sqrt(a^2 + b^2) - sqrt(a^2 + b^2 + c^2)) a^4 + c^2 (sqrt(a^2 + c^2) - sqrt(a^2 + b^2 + c^2)) a^4 + b^4 (sqrt(a^2 + b^2) - sqrt(a^2 + b^2 + c^2)) a^2 + c^4 (sqrt(a^2 + c^2) - sqrt(a^2 + b^2 + c^2)) a^2 + b^2 c^4 (sqrt(b^2 + c^2) - sqrt(a^2 + b^2 + c^2)) + b^4 c^2 (sqrt(b^2 + c^2) - sqrt(a^2 + b^2 + c^2))) + 8 ((a - sqrt(a^2 + b^2) - sqrt(a^2 + c^2) + sqrt(a^2 + b^2 + c^2)) a^6 + b^6 (b - sqrt(a^2 + b^2) - sqrt(b^2 + c^2) + sqrt(a^2 + b^2 + c^2)) + c^6 (c - sqrt(a^2 + c^2) - sqrt(b^2 + c^2) + sqrt(a^2 + b^2 + c^2))))/(630 a^2 b^2 c^2)

    Properties

    convex solids | hexahedra | parallelepipeds | solid polyhedra

    Find the right fit or it’s free.

    We guarantee you’ll find the right tutor, or we’ll cover the first hour of your lesson.