Surface Areas and Volumes

Cuboid and Cube

Shapes like the cuboid have always fascinated us. Those that can easily be drawn on notebooks and papers ie. the 2D figures, give us the basic idea of shapes. What about solid shapes like the cuboid? When we talk of solid shape we refer to the 3D outline of any basic shape like Cubes, Cuboids, etc. From rectangles to squares the 3D counterparts like Cuboid and Cube of both give us a different figure of calculation. Let’s see how!

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Surface Area of Cuboid

Before delving into the concept of surface areas and volumes of cube and cuboid, it is essential that we learn to make them. A cuboid is a three-dimensional representation of a rectangle while a cube is the three-dimensional representation of a square. Cuboid is an assemblage of rectangular pieces, similarly, a cube is an assemblage of square shaped pieces as length, breadth, and height.

Cuboid

A cuboid when opened gives the following view:

Cuboid

The figure above shows that a cuboid is made of six rectangles (where, l,b, and h are the three edges of a cuboid). The area of the six rectangles in consecutive order from 1 to 6 are:

1) area = (l × h)
2) area = (l × b)
3) area = (l × h)
4) area = (l × b)
5) area = (b × h)
6) area = (b × h)

When we sum up all the areas we get the surface area of the cuboid = 2(l*b) +2(b*h) + 2(l*h) =2(lb+bh+hl). The surface area of a cuboid hence is:

Surface Area = 2 (lb + bh + hl)

Cuboid

(source: ruwix)

Surface Area of Cube

A cube is a 3d representation of a square and has all equal sides. The length, breadth, and height in a cube are the same and are termed as sides (s). If s is each edge of the cube then, the surface area of a cube:

2 [(s × s) + (s × s) + (s × s)] = 2 ( 3s2) = 6s2

Lateral Surface Area of Cuboid

The lateral surface area of a cuboid is the area of only 4 rectangles. Since we know that a cuboid is teamed up by 6 rectangles, finding the area of only 4 gives us the lateral surface area of that cuboid. So the lateral surface area of a cuboid is:

Area of rectangle 1 + 2 + 3 + 4 i.e. (l × h) + (b × h) + (l × h) + (b × h) = 2lh + 2bh = 2 (l + b) h. The lateral surface area of a cuboid is: 2(l+b)h

Lateral Surface Area of Cube

Similar is the case with a cube. The area of 4 constituting squares gives us the lateral surface area of the cube. So the lateral surface area of the cube shall be s2+s2+s+s= 4s2. The lateral surface area of a cube = 4s2

Volumes

Every object occupies space. The measurement of this occupied space is the volume of that object. The object may be solid or hollow. In case it is solid, then the space occupied by this solid object is its volume. In case of hollow objects, we know that such object is filled with air or a liquid.

So if we are to find the volume of a hollow object, we intend to calculate the space occupied by the air or a liquid in such object.  The volume that fills this hollow object is said to be the capacity of the object or container. Therefore, the volume is the space occupied by the object while capacity is the volume of any substance that the object can accommodate inside it.

The Volume of a Cuboid

For understanding the volume of a cuboid we need to first understand the area of a rectangle since cuboids are a combination of rectangular planes stacked over each other. The height up to which these rectangular planes are stacked helps find the volume of the whole structure.

The area of a rectangle is l×b. Now since a few rectangular planes are stacked upwards to form a 3d rectangular structure, the volume of the object is Base area × height. So the volume = space occupied by the cuboid  = Base area × height.  Volume = A × h. A = l × b. Hence,

Volume of Cuboid (V) = l × b × h

The Volume of a Cube

As already said that cube is a combination of squares. All the sides in a cube are equal. So the volume of the cube is Base area of a square × side. Area = s × s. Volume = Area × s = s × s × s = s3. We use the cubic unit as units to symbolize the volume of an object.

Volume of Cube (V) = s3

Solved Examples For You

Question: A solid cube of side 16cm is cut into eight cubes of equal volume. What will be the side of the new cube? Find the ratio between their surface areas.

Solution: The volume of the cube= volume of the eight cubes. The volume of the cube =  S3.
Side (S) = 16cm. Volume (V)=  16= 4096 cm3. Now, the volume of 8 cubes is 4096 cm3

Volume of 1 cube = 4096/8 = 512 cm3
So if volume of a cube =512 cm3, s3  = 512 cm3
s = 3√ 512 = 8 cm. The side of new cube is 8cm.

For finding the surface area of cubes we use the formula  4s2
The surface area of a cube = 4s2. The side (S) of the larger cube is 16cm. The side (s) of the smaller cube is 8cm
Surface area (SA1) of larger cube= 4 × 16= 4 × 256 = 1024cm.
Surface area (SA2) of smaller cube= 4 × 8= 4 × 64 = 256cm2
SA: SA2 = SA/ SA2
= 1024 / 256 = 4 / 1 ⇒ 4 : 1. The ratio between the surface areas is 4 : 1.

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