Question

Two cubes, each of volume 512 $c{{m}^{3}}$ are joined end to end. Find the lateral and total surface area of the resulting cuboid.

Answer 1

Hint: To solve the problem, we need to know the basics of finding the lateral surface area and the total surface area. Further, we need to know how to find the surface area for a cube, then we will then make use of this result to then find the surface area of two cubes joined end to end. The lateral surface area of the cube is given by 4 ${{a}^{2}}$ and the total surface area is given by $6{{a}^{2}}$ . (where a is the edge length).
Complete step-by-step answer:
Before beginning to solve the problem, we try to understand the meaning of lateral and total surface area of solid shapes. The lateral surface of an object is all of the sides of the object, excluding its base and top (when they exist). The lateral surface area is the area of the lateral surface. The total surface area of the solid shape is the lateral surface plus the area of the base and the top of the shape. For example, in case of a cube, there are 4 square surfaces excluding the top and the base, thus the lateral surface area would be 4 ${{a}^{2}}$ (where a is the edge length of the square). Now, to calculate the total surface area we include the top and the base square faces too. Thus, we have total surface area as $4{{a}^{2}}+2{{a}^{2}}=6{{a}^{2}}$ . Now coming back to the problem in hand, we have,

We now know that the volume of each cube is 512 $c{{m}^{3}}$ . Thus, the formula for volume of cube is given by ${{a}^{3}}$ (where a is the edge length of the cube). Now, we use this formula to find the edge length a. So, ${{a}^{3}}$ = 512 , thus a = 8 cm. Now, when the two cubes are joined end to end, one of the surfaces gets covered from each cube as seen in the above figure. Now, for two cubes, we have,
Total surface area = $(2)(6{{a}^{2}})-2{{a}^{2}}=10{{a}^{2}}$ (two surfaces are covered since one of the surface is covered from each cube)
Lateral surface area = $(2)(4{{a}^{2}})-2{{a}^{2}}=6{{a}^{2}}$ (two surfaces are covered since one of the surface is covered from each cube)
Substituting the value of a = 8, we get,
Total surface area = $10({{8}^{2}})$ = 640
Lateral surface area = $6({{8}^{2}})$ = 384
Thus, the lateral surface area of the two cubes joined end to end is 384 $c{{m}^{2}}$ and the total surface area of the two cubes joined end to end is 640 $c{{m}^{2}}$.

Note: An alternative to solve the problem is to consider the joined cubes as one cuboid formed. We can see from the figure that the length (l) of the cuboid formed is 16 cm, breadth (b) and height (h) are 8 cm each. The lateral surface area of the cuboid is 2h (l+b) and the total surface area is 2 (lb +bh + lh). Thus, substituting the values, we get,
Lateral surface area = 2(8) (16+8) = 384
Total surface area = 2[(16) (8) + (8)(8) + (16) (8)] = 600
Thus, we get the same answer from this approach too.