Request a call back

# Volume and Surface Area of Solids

## Volume and Surface Area of Solids Synopsis

Synopsis

1. Surface area of a solid is the sum of the areas of all its faces.

2. The space occupied by a solid object is the volume of that object.

3. If l, b, h denote respectively the length, breadth and height of a cuboid, then:
Lateral surface area or Area of four walls = 2(ℓ + b) h
Total surface area = 2(ℓb + bh + hℓ)
Volume = ℓ x b x h

4. If the length of each edge of a cube is 'a' units, then:
Lateral surface area = 4 x (edge)2
Total surface area = 6 x (edge)2
Volume = (edge)3

5. If r and h respectively denote the radius of the base and the height of a right circular cylinder, then:
Area of each end or Base area = πr2
Area of curved surface or lateral surface area = perimeter of the base x height = 2πrh
Total surface area (including both ends) = 2πrh + 2πr2 = 2πr (h + r)
Volume = Area of the base x height = πr2h
6. If R and r respectively denote the external and internal radii of a right circular hollow cylinder and h denotes its height, then:
Area of each end = πR2 - πr2
Area of curved surface = 2π(R + r)h
Total surface area = (Area of curved surface) + 2(Area of each end)
= 2π(R + r)h + 2 (πR- πr2)
7. If r, h and l respectively denote the radius, height and slant height of a right circular cone, then:

Total surface area = Area of curved surface + Area of base = πℓ + πr2 = πr (ℓ + r)

8. If r is the radius of a sphere, then:
Surface area = 4πr²

9. If r is the radius of a hemisphere, then:
Area of curved surface = 2πr2
Total surface Area = Area of curved surface + Area of base
= 2πr2 + πr2
= 3πr2

10. The total surface area of the solid formed by the combination of solids is the sum of the curved surface area of each of the individual solids.

11. The volume of the solid formed by the combination of basic solids is the sum of the volumes of each of the basic solids.

12. If a right circular cone is cut off by a plane parallel to its base, then the portion of the cone between the plane and the base of the cone is called a frustum of the cone.
13. If h is the height, l is the slant height, R and r are the radii of the upper and lower ends of a frustum of a cone, then:
Curved surface area = π (R + r) ℓ
Total surface area = π (R + r) ℓ + π[R2 + r2]

14. When a solid is melted and converted to another, volume of both the solids remains the same, assuming there is no wastage in the conversions. However, the surface area of the two solids may or may not be the same.

15. The solids having the same curved surface do not necessarily occupy the same volume and vice versa.