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Revision Notes on Surface Areas and Volumes

Surface Areas and Volumes

Surface Area is the area of the outer part of any 3D figure and Volume is the capacity of the figure i.e. the space inside the solid. To find the surface areas and volumes of the combination of solids, we must know the surface area and volume of the solids separately. Some of the formulas of solids are -

Name Figure Lateral or Curved Surface Area Total Surface Area Volume Length of diagonal and nomenclature
Cube Cube 4l2 6l2 l3

√3

l = edge of the cube

Cuboid Cuboid 2h(l +b) 2(lb + bh + hl) lbh

l^{2}+b^{2}+h^{2}

l = length

b = breadth

h = height

Cylinder Cylinder 2πrh 2πr2πh = 2πr(r + h) πr2h

r = radius

h = height

Hollow cylinder Hollow cylinder 2πh (R + r) 2πh (R + r) + 2πh (R2 - r2) -

R = outer radius

r = inner radius

Cone Cone \pi rl = \pi \sqrt{h^{2}+r^{2} πr2 + πrl = πr(r + l) 1/3 πr2h

r = radius

h = height

l = slant height

Sphere Sphere 4πr2 4πr2 4/3 πr3

r = radius

Hemisphere Hemisphere 2πr2 3πr2 2/3 πr3

r = radius

Spherical shell Spherical shell 4πR2 (Surface area of outer) 4πr2 (Surface area of outer) 4/3 π(R3 – r3

R = outer radius

r = inner radius

Prism Prism Perimeter of base × height Lateteral surface area + 2(Area of the end surface) Area of base × height -
pyramid pyramid 1/2 (Perimeter of base) × slant height  Lateral surface area + Area of the base 1/3 area of base × height -

Surface Area of a Combination of Solids

If a solid is molded by two or more than two solids then we need to divide it in separate solids to calculate its surface area.

Example

Find the total surface area of the given figure.

Cone, Cylinder and hemisphere

Solution

This solid is the combination of three solids i.e.cone, cylinder and hemisphere.

Total surface area of the solid = Curved surface area of cone + Curved surface area of cylinder + Curved surface area of hemisphere

Curved surface area of cone

Given, h = 5cm, r = 3cm (half of the diameter of hemisphere)

Curved surface area of cylinder = 2πrh

Given, h = 8cm (Total height – height of cone – height of hemisphere), r = 3cm

Curved surface area of hemisphere = 2πr2

Given, r = 3 cm

Total surface area of the solid

Total surface

Volume of a combination of solids

Find the volume of the given solid.

Combination of solids

Solution

The given solid is made up of two solids i.e. Pyramid and cuboid.

Total volume of the solid = Volume of pyramid + Volume of cuboid

 Volume of pyramid = 1/3 Area of base x height

Given, height = 6 in. and length of side = 4 in.

Volume of cuboid = lbh

Given, l = 4 in., b = 4 in, h = 5 in.

Total volume of the solid = 1/3 Area of base x height + lbh

= 1/3 x 4 x 4 x 6 + (4) (4) (5)

= 32 + 80

= 112 in3

Conversion of Solid from One Shape to Another

When we convert a solid of any shape into another shape by melting or remoulding then the volume of the solid remains the same even after the conversion of shape.

Example

If we transfer the water from a cuboid-shaped container of 20 m x 22 m into a cylindrical container having a diameter of 2 m and height of 3.5 m. then what will be the height of the water level in the cuboid container if the cylindrical tank gets filled after transferring the water. 

Cuboid and Cylinder

Solution

We know that the volume of the cuboid is equal to the volume of the cylinder.

Volume of cuboid = volume of cylinder

l x b x h = πr2h

20 x 22 x h = 22/7 x 1 x 3.5

440 × h =11

H = 2.5 cm

Frustum of a Cone

If we cut the cone with a plane which is parallel to its base and remove the cone then the remaining piece will be the Frustum of a Cone.

Frustum of a Cone

Volume of the frustum of the cone  Volume of the frustum of the cone
The curved or Lateral surface area of the frustum of the cone Lateral surface area of the frustum of the cone
Total surface area of the frustum of the cone Area of the base + Area of the top + Lateral surface areaTotal surface area of the frustum of the cone
Slant height of the frustum Slant height of the frustum

Example

Find the lateral surface area of the given frustum of a right circular cone. 

Lateral surface area of the frustum

Solution

Given, r =1.8 in.

R = 4 in.

l = 4.5 in.

The lateral surface area of the frustum of the cone = πl (R + r)

= π x 4.5 (4 +1.8)

=3.14 x 4.5 x 5.8

= 81.95 sq. in.

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