CBSE Class 10 Maths Revision Notes for Surface Areas and Volumes

Surface Areas and Volumes

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
   
   Diagonal of a cuboid = 
   
   
   
   
   

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
   Diagonal of a cube =  x edge
   
   

   
   
   

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 = 2π rh
   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 (πR² - πr²)
   
   
   
   
   

7. If r, h and l respectively denote the radius, height and slant height of a right circular cone, then:
   Slant height (ℓ) =
   Area of curved surface = πrℓ = πr
   Total surface area = Area of curved surface + Area of base = πrℓ + πr² = πr (ℓ + r)
   Volume =
   
   
   

   
   
   

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

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
   Volume =
   
   

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]
    
    Volume = 

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.