When a cube is kept in a three-dimensional space, the area occupied by the sides of the cube in the space is called surface area of cube. In real world, we have been surrounded by many solid objects which have their own area as well volume. The area defines the region occupied by the objects and volume defines the space contained inside that object. The basic solid shapes or 3D shapes which we have learned till now in geometry are:
Here, we will be discussing the cube and its surface area. A cube has six faces, eight edges and eight vertices. The area of all the faces of a cube is the same as they are all squares. One of the popular examples of a cube is a Rubik’s cube. You will also get many examples in real life, such as carton boxes, storage trunks, etc. So whenever we think to keep such objects in a space or area, we need to know the surface area to find a suitable place for that object. Although cuboid also looks the same as a cube, the difference between cube and cuboid is cuboid has only its parallel sides equal whereas a cube has al its sides equal.
A cube consists of ‘n’ number of square units. Hence the space covered by these square units on the surface of the cube is the surface area. Basically, the surface area is the sum of all the area of all the shapes that cover the surface of the shape or object. In the case of a cube, there are 6 faces. So the surface area will be sum of all the area of six faces.
Let us derive the formula for surface area for a given cube, to solve problems based on it.
Surface Area of a Cube Formula
As per the definition of the cube, we know, the cube consists of 6 square faces. Let us consider, a cube whose length of the edges is ‘a’.
Now, we know, by the formula of area of a square;
Area = Side2 = a2
Therefore, the total surface area of a cube = 6 × (area of each side)
= 6 × a2 = 6a2 Square Unit
|TSA = 6 a2|
Length of Edge of the Cube
From the formula of the surface area of the cube, we can also find the length of the edge of the cube by rearranging the formula, such as;
A = 6 (side)2
side2 = A/6
side = √(A/6) = Edge length
where A is the area.
Q.1: Calculate the cost required to paint an aquarium which is in cube shape having an edge length of 10m. If the painting cost of an aquarium is INR 3/m2.
Solution: Total surface area of aquarium = 6 (side)2
= 6 (10)2
= 600 sq.m
Total cost of painting the aquarium = 3 × 600 = Rs. 1800
Q.2: If the sidewall of a cubic structure have length 7m, then find the total surface area.
Solution: Given, the length of the sidewall = 7m
As per the formula, we know;
TSA = 6a2
TSA = 6 x 7 x 7 = 294 sq.m
Q.3: Find the length of the edge of the cube, if its area is 2400 sq.cm.
Solution: Given, area = 2400 sq.cm.
Length of edge of cube = √(A/6) = √(2400/6) = √400 = 20 cm.
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