What is a Cube?
A three-dimensional solid shape bounded by square shaped faces is known as a cube. It has 6 faces, 12 edges and 8 vertices.
In our daily life we come across a number of objects that are shaped like a cube such as dice, Rubik’s cube, sugar cubes and so on.
Surface Area of a Cube
The surface of an object is its outer or exposed part or faces. As mentioned above, a cube has 6 faces that are squares in shape so all 8 edges of a cube will be equal in length. This implies that the total surface area of a cube will be the sum of the areas of all the square faces.
How do we the calculate Surface Area of a Cube?
If we look at the net of a cube, then we can say that:
Total surface area of a cube = Sum of areas of six square faces
= 6 × Area of one square face
So,
Total surface area of a cube, A = 6 × \(side^2 \)
Total surface area of a cube, A = 6 × \(a^2 \)
Total surface area of a cube, A = 6 \(a^2 \), where a is the side length of one of the square faces.
The lateral surface area is the total area of the lateral faces. Since a cube has 4 lateral faces its lateral surface area is given by:
Lateral surface area of a cube = \(4a^2 \)
[Note:
- Like the measure of any other type of area of geometric shapes, surface area is also measured in square units like square meters, square inches, square centimeters and so on.
- The side length of the square faces of a cube can also be referred to as the side or edge length of the cube.]
Solved Examples
Example 1: Find the surface area of a Rubik’s cube with a side length of 8 cm.
Solution:
As stated:
Side length of the rubik’s cube, a = 8 centimeters.
Apply the formula for the total surface area of a cube:
A = \(6a^2 \)
A = 6 × 8 × 8 [Substitute the given value]
A = 384 \(cm^2 \) [Multiply]
Therefore, the total surface area of the Rubik’s cube is 384 square centimeters.
Example 2: If the total surface area of a cube is 1600 square inches, find the measure of its side length.
Solution:
As stated:
Total Surface area of the cube, A = 20 square inches.
A = \(6a^2 \) [Formula for the surface area of a cube]
1600 = \(6a^2 \) [Substitute the given value]
\(\frac{1600}{6}=a^2 \) [Divide both sides by 6]
\(\sqrt{\frac{1600}{6}}=a \) [Positive square root on both sides]
\(\sqrt{266.66}=a \) [Simplify]
\(16.329=a \) [Square root]
\(a=16.329~in\)
Therefore, the side length of the cube is 16.329 inches.
Example 3: John wants to wrap a gift for his friend’s birthday in a cube shaped box of side length 5 centimeters. What is the minimum amount of wrapping paper he will need to wrap the box?
Solution:
As stated: Side length of gift box = 5 cm
We can find the minimum amount of wrapping paper required by calculating the total surface area of the gift box.
Since the box is a cube, use the formula for the total surface area of a cube.
A = \(6a^2\)
A = 6 × 5 × 5 [Substitute the value of side length]
A = 150 \(cm^2\) [Multiply]
So the surface area of the gift box is 150 square centimeters.
Therefore, John needs a minimum of 150 square centimeters of wrapping paper to wrap the gift box.
The total area occupied by the side faces of a cube, also known as its lateral faces, is the lateral surface area of the cube.
The surface area of a cube is expressed in square units, such as square centimeters, square feet, square meters and square inches.
The area of the side faces of a cube, excluding the top and bottom faces, is referred to as lateral surface area (LSA). The total surface area (TSA) measures the total area of all surfaces, including the bases of a cube. concepts. By looking at what your child is doing correctly and which concepts they understand, you can determine ways to practice areas that they are still developing.
The area that is covered by the six faces of a cube is known as its total surface area.
Area is a two-dimensional measurement of the size of a flat surface, whereas surface area is a measurement of the exposed surface of a solid shape(three-dimensional). This is the main distinction between area and surface area.