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 |  | 4l2 | 6l2 | l3 | √3 l = edge of the cube |
| Cuboid |  | 2h(l +b) | 2(lb + bh + hl) | lbh | l = length b = breadth h = height |
| Cylinder |  | 2πrh | 2πr2 + 2πh = 2πr(r + h) | πr2h | r = radius h = height |
| Hollow cylinder |  | 2πh (R + r) | 2πh (R + r) + 2πh (R2 - r2) | - | R = outer radius r = inner radius |
| Cone |  |  | πr2 + πrl = πr(r + l) | 1/3 πr2h | r = radius h = height l = slant height |
| Sphere |  | 4πr2 | 4πr2 | 4/3 πr3 | r = radius |
| Hemisphere |  | 2πr2 | 3πr2 | 2/3 πr3 | r = radius |
| Spherical shell |  | 4πR2 (Surface area of outer) | 4πr2 (Surface area of outer) | 4/3 π(R3 – r3) | R = outer radius r = inner radius |
| Prism |  | Perimeter of base × height | Lateteral surface area + 2(Area of the end surface) | Area of base × height | - |
| 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.
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
Volume of a combination of solids
Find the volume of the given solid.
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.
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.
| Volume of the frustum of the cone |  |
| The curved or 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 area |
| Slant height of the frustum |  |
Example
Find the lateral surface area of the given frustum of a right circular cone.
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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