SURFACE AREA OF PRISM AND CYLINDER

About "Surface area of prism and cylinder"

Surface area of prism and cylinder :

The surface area of any prism equals the sum of the areas of its faces, which include the floor, roof and walls. Because the floor and the roof of a prism have the same shape, the surface area can always be found as follows:

Cube : 

A cube is a right rectangular prism with square upper and lower bases and a square vertical faces.

Now let us see how to find the surface of square 

Surface area of cube  =  a² +  a² + a² + a² + a² + a² 

  =  6 a²

Lateral surface area  =  4 a²(Area of side 1, 2, front and back)

Total surface area  =  6 a²(Area of side 1, 2, front, back, top and bottom)

Cuboid :

cuboid is a three-dimensional shape with a length, width, and a height. The cuboid shape has six sides called faces

Area of top face  =  (length x breadth) ==> lb

Area of bottom face  =  (length x breadth) ==> lb

Area of front face  =  length x height ==>  lh

Area of back face  =  length x height ==>  lh

Area of side1 face  =  breadth x height ==>  bh

Area of side2 face  =  breadth x height ==>  bh

Surface area of cube  =  lb +  lb + lh + lh + bh + bh

   = 2lb + 2hl + 2bh

   = 2(lb + hl + bh) 

Lateral surface area  =  bh + bh + hl + hl

=  2bh + 2hl  = 2h (l+b)

(Area of side 1, 2, front and back)

Total surface area  =   2(lb + hl + bh) 

(Area of side 1, 2, front, back, top and bottom)

Cylinder :

A cylinder is a solid with congruent parallel circular bases connected by a curved surface.

The vertical surface of the cylinder is curved and hence its area is called the curved surface or lateral surface area of the cylinder.

Curved Surface Area of a cylinder,

CSA = Circumference of the base x Height

= 2πr x h ==> 2πrh

Total Surface Area, TSA = Area of the Curved Surface                                               2 x Base Area

2πrh + 2 πr²

Thus, TSA = 2πr (h + r) sq.units.

Sphere :

If a circular disc is rotated about one of its diameter, the solid thus generated is called sphere. Thus sphere is a 3- dimensional object which has surface area.

The following activity may help us to visualize the surface area of a sphere as four times the area of the circle with the same radius.

  • Take a plastic ball.
  • Fix a pin at the top of the ball.
  • Wind a uniform thread over the ball so as to cover the whole curved surface area.
  • Unwind the thread and measure the length of the thread used.
  • Cut the thread into four equal parts.
  • Place the strings as shown in the figures.
  • Measure the radius of the sphere and the circles formed.

Now, the radius of the sphere = radius of the four equal circles.

 The curved surface area of a sphere = 4πr² square units

Hemi sphere :

A plane passing through the centre of a solid sphere divides the sphere into two equal parts. Each part of the sphere is called a solid hemisphere.

Curved surface area of hemisphere = Cuverd surface area/2

  =  4πr²/2  =  2πr² 

Total surface area of hemisphere = curved surface area + area of base 

=  2πr² + πr²  =  3πr²

Cone :

Let us consider a sector with radius l and central angle ic. Let L denote the length of the arc. Thus,

2rl/L = 360°/θ°

 L = 2 π l x θ°/360°   ----(1)

Now, join the radii of the sector to obtain a right circular cone.

Let r be the radius of the cone. Hence, L = 2πr

From (1) we obtain,

2πr = 2 πl x (θ°/360°)

r = l (θ°/360°)

(r/l) = (θ°/360°)

Let "A" be the area of sector. Then 

πl²/A = 360°/θ°

Then the curved surface area of the cone = Area of the sector

Thus, the area of the curved surface of the cone A

πl² (θ°/360°) =  πl² (r/l)

Hence, the curved surface area of the cone = πrl sq.units.

Triangular prism :

In the above triangular prism, there are five faces. The shape of the base and the two slanting faces is rectangle. The shape of two faces on the left side and right side  is triangle. 

For the given triangular prism, 

Area of the base  =  Lb 

Area of the first slanting face  =  Ls

Area of the other slanting face  =  Ls

Area of the front face  =  (1/2)bh

Area of the back face   =  (1/2)bh

So,

surface area  =  sum of the area of 5 faces

surface area  =  Lb + 2Ls +2x(1/2)bh

Surface area of triangular prism = Lb+2Ls+bh

After having gone through the stuff given above, we hope that the students would have understood "Surface area of prism and cylinder". 

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