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Surface Area of a Cube

(a is the length of the side of each edge of the cube) .

If we want to calculate the area of a cube then it’s always better to see each surface as a square and a cube is made up of six equal squares. Considering each side of the square as “a”, then the area of each square will be a
2. Therefore, the surface area of the cube will be 6a2.

Surface area is the measure of how much exposed of area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra (objects with flat polygonal faces) the surface area is the sum of the areas of its faces. A cube is called a hexahedron because it is a polyhedron that has 6 (hexa- means 6) faces. The sum of the areas of all external surfaces of a three-dimensional object is called its total surface area (TSA).

The longest diagonal of the cube (i.e., the line joining one vertex on the top face to the diagonally opposite vertex on the bottom face) is called the diagonal of the cube. The length of the diagonal of the cube is 3√(a) and The lateral surface area of a cube = 4 a
2.

Example 1: Find the surface area if the length of one side is 1/2 cm.
Solution:
SA = 6 × a2
SA = 6 × (1/2)2
SA = 6 × 1/2 × 1/2
SA = 6 × 1/4
SA = 1.5 cm2

Example 2: Find the Surface area of a cube of side length 8 cm.
Solution:
SA = 6 × a2
SA = 6 × (8)2
SA = 6 × (64)
SA = 384 cm2

Example 3: Find the total surface area of a box whose edges are all 4.5 cm long.
Solution:

SA = 6 × l2
SA = 6 × 4.52
SA = 121.5

Example 4: A 5 cubic centimeter cube is painted on all its side. If it is sliced into smaller cubes, that each has a volume of 1 cubic centimeter, how many smaller cubes will have exactly one of their sides painted?
Solution:
When a 5 cc (cubic centimeter) cube is sliced into 1 cc cubes, we will get 5*5*5 = 125 1 cc cubes. In each side of the larger cube, the smaller cubes on the edges will have more than one of their sides painted. Therefore, the cubes which are not on the edge of the larger cube and that lie on the facing sides of the larger cube will have exactly one side painted.

In each face of the larger cube, there will be 5*5 = 25 cubes. Of these, there will be 16 cubes on the edge and 3*3 = 9 cubes which are not on the edge. Therefore, there will be 9 × 1 cc cubes per face that will have exactly one of their sides painted.

In total, there will be 9*6 = 54 such cubes.

Example 5: A cube of length 4 cm is cut into smaller cubes with 1 cm in length. What is the percentage increase in the surface area after such cutting?
Solution:
The volume of the big cube:

V = a3
V = 43
V = 64 cc

When it is cut into 1 cm cube, the volume of each of the cubes = 1cc. Hence, there will be 64 such cubes.

The surface area of the smaller cubes
= 6 (12) = 6 cm2.

Therefore, the surface area of 64 such cubes
= 64 * 6 = 384 cm2.
The surface area of the big (original) cube
= 6(42) = 6*16 = 96 cm2.

% increase = (384 – 96) / 96 × 100 = 300%
Example 6: A cube whose sides are 10.7 cm in length. Find the surface area of the cube.
Solution:
Given that:
Side length (a) = 10.7 cm
Surface area of the cube:

SA = 6 a2
SA = 6 × (10.7)2
SA = 686.94 cm2

Example 7: Find the surface area of a cube whose side is 1/6 cm.
Solution:
Given that:
Length of side is 6 cm or
a = 1/6

Surface area of the cube:
SA = 6 a2
SA = 6 × (1/6)2
SA = 1/6 cm2

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