Surface areas &volumes

CHAPTER: Surface area CHAPTER: Surface area and volumeand volume

Cube CUBOIDCylinder Cone

A bundle of many sheets of paper makes up a cuboid.

Area of 6 rectangles = 2(lxb) + 2(bxh) + 2(lxh)

Surface area of a cuboid = 2 (lb + bh + hl)

A cube, whose length, breadth and height are all equal, it is called a Cube. If each edge of the cube is a, then the surface area of this cube would 6a²,

Surface area of a cube =6a²

where a is the edge of the cube.

Area of base (square) = a x b

a

Height of cube = c

Volume of cube = Area of base x height

= (a x b) x c

b

c

b

Volume of Parallelopiped

Volume of Cube

a

a

Area of base (square) = a2

Height of cube = a

Volume of cube = Area of base x height

= a2 x a = a3

a

(unit)3

Take a number of circular sheets of paper and stack them up, if the stack is kept vertically up, we get a Right circular cylinder, since it has been kept at right angles to the base, and the base is circular.

The area of the rectangular sheet gives us the curved surface area of the cylinder. The length of the sheet is equal to the circumference of the circular base which is equal to 2חr.

So, curved surface of the cylinder = area of the rectangular sheet = length x breadth

= perimeter of the base of the cylinder x h

r x hח 2 = Therefore, Curved

surface area of a cylinder = 2חrh.

If the top and the bottom of the cylinder are also to be covered, then we need two circles to do that, each of radius r, and thus having an area of חr² each, giving us the total surface area as

. r²(r+h)ח r² = 2ח rh + 2ח 2So, Total surface area of a cylinder =2 חr (r +h)

Where h is the height of the cylinder and r is it’s radius.

Volume of cylinder

Volume of cylinder = Area of base x vertical height

= π r2 xh

r

h

The area of each triangle = ½ x base of each triangle x l.

So, area of the entire piece of paper = sum of the areas of all the

triangles = ½ b1Ll+ ½b2l+ ½b3L+… = ½l(b1+b2+b3+…) = ½ x l x length of entire

curved boundary of the fig 3

(as b1+b2+b3+… makes up the curved portion of the fig)

But the curved portion of the figure makes up the perimeter of the base of the cone and the circumference of the base of the cone = 2 חr, where r is the base radius of the cone.

So, Curved surface area of a cone = ½ x 1 x 2 חr = חrl. Where r is it’s base radius and l. it’s

slant height. TSA of a cone = חr l. + חr² = חr (l+r)

• A right circular cone has a height h that extends from the tip and is perpendicular to the circular base of the cone

• Given a right circular cone with a height h and radius of its circular base r:

3( V ) = π r2h

r

h h

r

Volume of a Cone

Here the vertical height and radius of cylinder & cone are same.

3( volume of cone) = volume of cylinder

V = 1/3 π r2h

if both cylinder and cone have same height and radius then volume of a cylinder is three times the volume of a cone ,

Volume = 3V Volume =V

• The string which had completely The string which had completely covered the surface area of the covered the surface area of the sphere, has been used to sphere, has been used to completely fill the regions of completely fill the regions of four circles, all of the same four circles, all of the same radius as of the sphere.radius as of the sphere.

• This suggests that the surface This suggests that the surface area of a sphere of radius rarea of a sphere of radius r

= 4 times the area of a = 4 times the area of a circle of radius r =4 x (circle of radius r =4 x ( חחr²)r²)

So, So, Surface area of a sphere =4 Surface area of a sphere =4 חחr².r².Where r is the radius of the Where r is the radius of the

spheresphere.

4( 1/3πr2h ) = 4( 1/3πr3 ) = V

h=rr

Volume of a Sphere

Here the vertical height and radius of cone are same as radius of sphere.

4( volume of cone) = volume of Sphere

V = 4/3 π r3

r