Surface Area of Cuboid// cylinder//Geometry

 Surface Area of Cuboid

The volume of an object is the amount of space it occupies. The object that occupies more space is said to have a greater volume.

The total surface area of an object is equal to the area of all the faces of the net. In particular, we have:
       * Total surface area of a cube = 6p
    * Total surface area of a cuboid = 2(Ib + Ih + bh)

Cube  =  l³
Cuboid  = lbh
Volume of a cuboid = Length X Breadth  X  Heigth

Ex. Find the volume of a cube with each side 10 cm.

      Volume of a cube = side  X side  X side

                                    = 10 cm  X  10 Cm  X 10 cm

                                    = 1000 cm³

Ex.  What is the volume of a cuboid with length = 1.2 m, breadth = 80 cm and height = 50 cm?

          Length = 1.2 m

                       = 120 cm

          breadth = 80 cm

           Height = 50 cm

    Volume =  Length  X Breadth X  Heigth

                  = 120  cm  X  80 cm  X  50 cm

                 = 480000 cm³  Ans. 

Q.1. A cuboid is 6 cm long, 8 cm wide and 10 cm high. Calculate its total surface area. 

Solution: Surface area of the cuboid = 2 ( lw + lh + wh ) 
                                                               = 2 ( 6 x 8 + 6 x 10 + 8 x 10 )
                                                               = 2 ( 48 + 60 + 80 )
                                                                 = 376

  Q.2. A cuboid is 6 cm long, 4 cm wide and 3 cm high. Calculate 

 i) its volume

ii) its total surface area.

  Solution: 

i) Volume of the cuboid = 6 X  4  X 3

                                       = 72 cm³

ii) Surface area of the cuboid = 2 ( lb + lh + bh ) 
                                                 = 2 ( 6 x 4 + 6 x 3 + 4 x 3)
                                                  = 2 ( 24 + 18 + 12 )
                                                   = 2 x 54
                                                     = 108  cm²

  Q.3. A cuboid is 120 mm long, 10 mm wide and 96 mm high. Calculate its total surface area.

 Solution: 
Surface area of the cuboid = 2 ( lb + lh + bh )
                                            = 2 ( 120 x 10 + 120 x 96 + 10 x 96 ) 
                                             = 2 ( 1200 + 11,520 + 960 ) 
                                               = 2 x 13,680 
                                                = 27,360 

Q.4. The interior of a godown is 4 m long and 3 m wide. The ceiling of the godown is 4 m high. If this godown was to be filled completely with shoe boxes measuring 40 cm in length, 30 cm in width, and 10 cm in height, how many shoe boxes could be placed in the godown? 

Solution: 
Volume of the godown = 4 m X  3 m X  4 m 
                                      = 48 Volume of the shoe box 
                                      = 40 cm x 30 cm x 10 cm
                                       = 0.4 m x 0.3 m x 0.1 m 
Maximum number of shoe boxes possible in the godown
                                     = Volume of the godown/volume of one shoe box
                                       = 48/0.012 
                                       = 4000 Ans

Q.5. How many bricks, each of dimensions 20 cm   X    12 cm  X   9 cm,  will be required to build a wall 12 m long, 72 cm thick and 4 m high?

Q. 6. The edges of a cubical container measure 25 cm. What will be the volume of water needed to fill this container completely?

Q. 7. The total surface area of a cube is 433.5 cm² . Find its volume. 614.125 

Q.8. A metal cube has a volume of 64 cm³. It is to be painted on all its faces. Find the total area of the faces that will be coated with paint.  96 cm²

Q.6. Calculate the volume of wood used in making an open rectangular box 2cm thick, given that its internal dimensions are 54 cm by 46 cm by 18 cm.

Solution:

External volume = (54 + 2 + 2) x ( 46 + 2 + 2) x (18 x 2)
                          = 58 x 50 x 20
                         = 58,000 cm³

Internal volume = 54 x 46 x 18
                           = 44712 cm³

Volume of wood used = 58,000 cm³ - 44712 cm³

                                   = 13288 cm³

Q.7. the internal dimensions of an open, concrete rectangular tank are 180 cm by 80 cm by 120 cm. If the concrete has a thickness of 30 cm, find the volume of concrete used.

Solution:
External volume = ( 180 + 30 + 30) x ( 80 + 30 + 30) x (120 + 30)
                             = 240 x 140 x 150
                            = 5,040,000 cm³
Internal volume = ( 180 x 80 x 120) cm³
                           = 1,728,000 cm³
Volume of concrete used = ( 5,040,000 - 1,728,000)
                                        = 3,312,000 cm³

   Q.8. A cuboid, with dimensions 9 cm by 7 cm by h cm, has a volume of 378 cm³. 

i) Calculate the height h, of the cuboid.

ii) The cuboid is melted to form smaller cuboids with dimensions 2 cm by 3 cm by 3 cm. How many smaller cuboids can be obtained?

Solution:
          i) Volume of the cuboid = 9 x 7 x h
                                                   = 378 cm³
                                                h = 378/9 x 7  
                                                   = 6  
      ii) Volume of each small cuboid = 2 x 3 x 3 = 18 cm³
Number of small cuboids that can be obtained =  378/18 = 21

Q.9. A cuboid, with dimensions l cm by 18 cm by 38 cm, has a volume of 35 568 cm³ .
i) Find the length l, of the cuboid.
ii) The cuboid is melted to form cubes of length 2 cm. How many cubes can be obtained?

Q.10. An open rectangular tank, with dimensions 55 cm by 35 cm by 36 cm, is initially half-filled with water. Find the depth of water in tank after 7700 cm³ of water is added to it.

Solution: 

Q.11. A fish tank measuring 80 cm by 40 cm contains water to a height of 35 cm. Find
i) the volume of water in the tank, giving your answer in litres,
ii) the surface area of the tank that is in contact with the water, giving your answer in m².

12. A metal cube has a volume of 64 cm³ . It is to be painted on all its faces. Find the total area of the faces that will be coated with paint.

13. The total surface area of a cube is 433.5 cm² . Find its volume.

14. A trough, in the form of an open rectangular box, is 1.85 m long, 45 cm wide and 28 cm deep externally. If the trough is made of wood 2.5 cm thick, find the volume of wood used to make this trough, giving your answer in m³.

15. The cross section of a drain is a rectangle 30 cm wide. If water 3.5 cm deep flows through the drain at a rate of 22 cm/s, how many litres of water will flow through in one minute?

16. A cuboid of length 12 cm and breadth 9 cm has a total surface area of 426 cm².

i) Find the height of the cuboid.

ii) Hence, find its volume.

 Volume of a Cylinder:

1. The diameter of the base of a cylinder is 14 cm and its height is half of its base radius. Calculate the volume of the cylinder.

                               

Solution:

                   Base radius  = 14 ➗ 2  
                                              = 7 cm
    Height of the cylinder = ½  x 7

                                                  = 3.5 cm
Volume of the cylinder = πr²h
                                                = π(7)² (3.5)
                                                = 539 cm³

Q.2. The diameter of the base of a cylinder is 18 cm and its height is 2.5 times its base radius. Find the volume of the cylinder.

Solution:

Given here,

               Base of  a cylinder = 18 cm

                Height  = 2.5  x 18

                                 = 45

We know,

      Volume of the cylinder  = πr²h

                                                          = 3.14  x 45  x 9

                                                             = 1,271.7 

Q.3. A pipe of radius 2.8 cm discharges water at a rate of 3 m/s, Calculate the volume of water discharged per minutes, giving your answer in litres.

Solution:

Since water is discharged through the pipe at a rate of 3 m/s, i.e. 300 cm/s, in 1 second, the volume of water discharged is the volume of water that fills the pipe to a length of 300 cm as shown.

In 1 second, volume of water discharged

= volume of pipe of length 300 cm

= πr²h

= π(2.8)²(300)

= 2352 π cm³

In 1 minute, volume of water discharged = 2352 π x 60

                                                                                     = 443 000 cm³

                                                                                     = 443 l

Practice:

1. A pipe of radius 0.6 cm discharges petrol at a rate of 2.45 m/s. Find the volume of petrol discharged in 3 minutes, giving your answer in litres.

2. A pipe of diameter 0.036 m discharges water at a rate of 1.6 m/s into a cylindrical tank wiht a base radius of 3.4 m and a height of 1.4 m. Find the time required to till the tank, giving your answer correct to the nearest minute.

Q.1. A closed metal cylindrical container has a base radius of 5 cm and a height of 12 cm. 

i) Calculate the total surface area of the container.

The lid of the container is now removed. The exterior of the container, including the base, is painted green.

ii) Express the area of the container that is painted as a percentage of the total surface area found in (i).

Solution:

i) Total surface area of the container = 2πr²  + 2πr²h

                            = 2π(5)²  + 2π(5)(12)

             = 50π + 120π

= 170π

     =534 cm²

ii) Area of the container that is painted = πr² + 2πrh

                                                                = π(5)²  + 2π(5) (12)

                                                   = 25 π  + 120π

                                     = 145 π

                                           = 456 cm²

Required percentage = 145 π/170π   x 100%

                       = 85 5/17%