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Surface Area of Cube, Cuboid and Cylinder

Solid shapes :-

In this chapter we shall study about some solid figures like cuboid, cube, cylinder and sphere. These figures are three dimensional figures. We shall also learn to find the surface area and volumes of these figures.
 


Faces of different solids :

Some solid shapes have two or more identical faces. For example, consider the solid cylinder which has two circular faces which are identical and one curved surface area.


 

The square pyramid has four identical triangles and a square base.
 

Surface area:

For any three dimensional figure, we can calculate the two types of Surface areas. They are lateral surface area and total surface area.
 

Lateral Surface area:

If we roll any solid on the ground the portion which touches the ground is the lateral surface area of the given solid. Lateral surface area is the area of three dimensional figures excluding the areas of their base and the top. It is expressed in square units. Lateral surface area is also known as curved surface area.
 

Total Surface Area:

If we want to paint the whole solid, we have to paint all the faces including the top and the bottom of the solid. In such cases we find the total surface area of the solid. Total surface area is the sum of all faces including the top and bottom of the solid.

In general, Total surface area = Lateral surface area + 2 × Area of the base.

Here we consider three solids cube, cuboid and cylinder and find their surface areas.

Cuboid:

A solid bounded by six regular plane faces is called cuboid. A match box, a chalk box, a tea packet, a brick, a tile, a book are all examples of a cuboid.

Here are some more examples of cuboid.


A cuboid has 6 rectangular faces. They are ABCD, EFGH, EFBA, HFCD, EHDA and FGCB.

Area of face ABCD = Area of face EFGH =

Area of face AEHD = Area of face BFGC =

Area of face ABFE = Area of face DHGC =



 

Lateral surface area of a cuboid = Sum of the sides excluding the bases

                                            


Total surface area of a cuboid = Sum of the area of all its six faces

                                         


 

Example

Find the total surface area and lateral surface area of a chalk box whose height, length, and breadth are 25cm, 20cm and 15cm respectively.

Solution

The total surface area    =

                                  =

Lateral surface area =

                           

 

Cube: A cuboid whose length, breadth and height are all equal is called a cube.

Examples of cube: ice cubes, sugar cubes, dice.

Some more examples of cube:

 

Since all the six faces of a cube are squares and of the same size. For a cube we have l = b = h.

Therefore

Lateral surface area of the cube

    

Total surface area of a cube
                                       

 

Example : Find the lateral surface area and total surface area of a cube whose edge is 5m.

Lateral surface area of the cube = 4(Edge)2 = 4(5)2 = 4 × 25 = 100m2.

Total surface area of the cube = 6(Edge)2 = 6(5)2 = 6 × 25 = 150m2.

 

Cylinder:

A cylinder has one curved surface and two circular bases. If the line segment joing the centers of the bases is perpendicular to the base of the cylinder, then the cylinder is a right circular cylinder. If the line segment is not perpendicular to the base then the cylinder is a skew cylinder.

Examples of cylinder:

Let us consider a cylinder of height h and radius r. Take a strip of paper of length h and wrap it exactly once around the cylinder. Cut off this strip and unfold it. We will find that the strip is a rectangle and the length of the strip is the circumference of the circle which is and breadth of the strip is the height of the cylinder which is h.

 


Area of the lateral surface = area of the paper strip which is in the form of a recangle

                                    

Lateral surface area of the cylinder =

Total surface area = area of curved surface + 2 (base area)

                         


 

Example

Find the area of the curved surface and the total surface area of a cylinder whose height is 18cm and radius is 10.5 cm.

Solution

r = 10.5cm h = 18cm

The area of the curved surface =

                                          

The total surface area

                               



 

Example : A small indoor greenhouse is made entirely of glass plates(including base) held together with tape. It is 30cm long, 25cm wide and 25cm high.

(i) What is the area of the glass.

(ii) How much of tape is needed for all the 12 edges?

Solution:

(i) Length = 30 cm, breadth = 25cm, height = 25cm

Area of the glass = Total surface area of the cuboid of length 30 cm, breadth 25cm and height 25cm

                        

Length of the tape = Length of all 12 edges

                           = 4 (Length + Breadth + Height)

                           = 4 (30 + 25 + 25) cm = 320 cm.



 

Example: In a marriage hall there are 15 cylindrical pillars. The radius of each pillar is 18 cm and height 3m. Find the total cost of painting the curved surface area of all pillars at the rate of Rs 5 per m2.

Radius of cylindrical pillar r = 18 cm = 0.18 m

Height h = 3m

Curved surface and

                          

Curved surface area of 15 such pillar = 3.39 × 15

                                                   = 50.91 m2.

Cost of painting 1m2 = Rs. 5

Cost of painting 50.91 m2 = 50.91 × 5 = Rs. 254.57



 




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