What is a Cube?
A cube is a three dimensional shape with six square faces, eight vertices and twelve edges. All twelve edges of a cube are of the same length.
Properties of a Cube
- A cube has six faces, eight vertices and twelve edges of equal length
- Opposite edges of a cube are parallel to each other
- It has square shaped faces
- All faces have the same area
- Each face of the cube is joined to four other faces
- Each vertex of the cube meets three edges and three faces
- Angle between the edges of a cube is a right angle.
What is a Cuboid?
A cuboid is also a three dimensional shape with six faces, eight vertices and twelve edges but when looking at cuboids, we can notice that the edges may be unequal in length. The edges of a cuboid are named as length, width and height.
Properties of a Cuboid
- A cuboid has six faces, eight vertices and twelve edges
- Edges of the cuboid are named as length, width and height
- The faces of the cuboid are rectangular in shape
- The areas of opposite faces are equal
- Angle between the edges is a right angle
- The opposite edges of a cuboid are congruent and parallel
Formulas Applied to Cubes and Cuboids
Solved Examples
Example 1: The length, width and height of a cuboidal gift box are 10 inches, 8 inches and 6 inches. What is the least area of paper needed to cover the box? Express your answer in square inches.
Solution:
The paper covers the surface of the cuboidal box so the least area of paper to cover the cube will be equal to the total surface area of the cuboid and this is expressed in square inches.
Total surface area, S = l \(\times\) w + w \(\times\) h + h \(\times\) l Write the formula
S = 10 \(\times\) 8 + 8 \(\times\) 6 + 6 \(\times\) 10 Substitute
S = 80 + 48 + 60 Multiply
S = 188 Add
So, the least area of paper needed to cover the box is 188 square inches.
Example 2: Find the volume of the cuboid with length, width and height as 20 cm, 15 cm and 12 cm respectively.
Solution:
V = l \(\times\) w \(\times\) h Write the formula
V = 20 \(\times\) 15 \(\times\) 12 Substitute
V = 3600 Multiply
So, the volume of the cuboid is 3600 cubic centimeters.
Example 3: Find the amount of water that can be stored in a cubical container with an edge length of 5 feet.
Solution:
The amount of water that can be stored in the container will be equal to the volume of the container. The container is cubical in shape so let us use the volume of the cube formula.
V = \(side^3\) Write the formula
V = \(5^3\) Substitute
V = 125 Cube of 5
Therefore, 125 cubic feet of water can be stored in the container.
Example 4: The volume of a Rubik’s cube is 27 cubic inches. Find the edge length of the cube?
Solution:
V = \(side^3\) Write the formula
27 = \(side^3\) Substitute
\(\sqrt[3]{27}\) = \(\sqrt[3]{side^3}\) Cube root on both sides
3 = side
So, the edge length of Rubik’s cube is 3 inches.
There are eight vertices in a cuboid.
There are twelve edges in a cube.
Square units like square centimeters, square meters, square feet, or square yards, can be used to measure the surface area of a cuboid.
Cubic units like cubic centimeters, cubic meters, cubic feet or cubic yards, can be used to measure the volume of a cube.