# Surface Area of a Cube Cuboid Cylinder

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## Surface Area of a Cube Cuboid Cylinder

### Cuboid

1) Total Surface Area = 2 (h × l + b × h + b × l) = 2(lb + bh + hl ) where h, l and b are the height, length and width of the cuboid respectively.
Suppose the height, length and width of the box shown above are 20 cm, 15 cm and 10 cm respectively.
Then the total surface area = 2 (20 × 15 + 20 × 10 + 10 × 15) = 2 ( 300 + 200 + 150) = 1300 m2.
2) The side walls (the faces excluding the top and bottom) make the lateral surface area of the cuboid. For example, the total area of all the four walls of the cuboidal room in which you are sitting is the lateral surface area of this room.
Hence, the lateral surface area of a cuboid is given by 2(h × l + b × h) or 2h (l + b).

### Cylinder

1) The lateral (or curved) surface area of a cylinder is 2πrh.
2) The total surface area of a cylinder = πr2 + 2πrh + πr2 = 2πr2 + 2πrh or 2πr (r + h)
3) Note that lateral surface area of a cylinder is the circumference of base × height of cylinder.

### Example 1:

An aquarium is in the form of a cuboid whose external measures are 80 cm × 30 cm × 40 cm. The base, side faces and back face are to be covered with a coloured paper. Find the area of the paper needed?

### Solution:

The length of the aquarium = l = 80 cm
Width of the aquarium = b = 30 cm
Height of the aquarium = h = 40 cm
Area of the base = l × b = 80 × 30 = 2400 sq. cm
Area of the side face = b × h = 30 × 40 = 1200 sq. cm
Area of the back face = l × h = 80 × 40 = 3200 sq. cm
Required area = Area of the base + area of the back face + (2 × area of a side face) = 2400 + 3200 + (2 × 1200) = 8000 sq. cm
Hence the area of the coloured paper required is 8000 sq. cm.

### Example 2:

The internal measures of a cuboidal room are 12 m × 8 m × 4 m. Find the total cost of whitewashing all four walls of a room, if the cost of white washing is Rs 5 per sq. m. What will be the cost of white washing if the ceiling of the room is also whitewashed.

### Solution:

Let the length of the room = l = 12 m
Width of the room = b = 8 m
Height of the room = h = 4 m
Area of the four walls of the room = Perimeter of the base × Height of the room = 2 (l + b) × h = 2 (12 + 8) × 4
= 2 × 20 × 4 = 160 sq. m.
Cost of white washing per sq. m = Rs 5
Hence the total cost of white washing four walls of the room = Rs (160 × 5) = Rs 800
Area of ceiling is 12 × 8 = 96 sq. m
Cost of white washing the ceiling = Rs (96 × 5) = Rs 480
So the total cost of white washing = Rs (800 + 480) = Rs 1280

### Example 3:

In a building there are 24 cylindrical pillars. The radius of each pillar is 28 cm and height is 4 m. Find the total cost of painting the curved surface area of all pillars at the rate of Rs 8 per sq. m.

### Solution:

Radius of cylindrical pillar, r = 28 cm = 0.28 m
height, h = 4 m
curved surface area of a cylinder = 2πrh
curved surface area of a pillar = 2π(0.28)(4) = 7.04 sq. m
curved surface area of 24 such pillar = 7.04 × 24 = 168.96 sq. m
cost of painting an area of 1 sq. m = Rs 8
Therefore, cost of painting 1689.6 sq. m = 168.96 × 8 = Rs 1351.68

### Example 4:

Find the height of a cylinder whose radius is 7 cm and the total surface area is 968 sq. cm.

### Solution:

Let height of the cylinder = h, radius = r = 7cm
Total surface area = 2πr (h + r) = 2π × 7 × (7 + h) = 968
h = 15 cm
Hence, the height of the cylinder is 15 cm.