Revision Notes: Mensuration: Cylinder, Cone and Sphere | Mathematics Class 10 ICSE PDF Download

Right Circular Cylinder

A solid having uniform circular cross-section is called a cylinder.
If r and h respectively denote the radius of the circular cross-section and the height of the cylinder, then
Area of cross-section = πr²
Perimeter of cross-section = 2πr
Area of curved surface or lateral surface area = Perimeter of cross-section x Height = 2πrh
Total surface area = Curved surface area + 2(Area of cross-section)
= 2πrh + 2πr²
= 2πr(h + r)
Volume = Area of cross-section x Height = πr²h

Right Circular Hollow Cylinder

If R and r respectively denote the external and internal radii of a right circular hollow cylinder and h denotes its height, then
Thickness of its wall = R – r
Area of cross-section = πR² – πr² = π(R² – r²)
External curved surface = 2πRh
Internal curved surface = 2πrh
Total surface area = External curved surface area + Internal curved surface area + 2(Area of cross section)
= 2πRh + 2πrh + 2π(R² - r²)
Volume of material = External volume – Internal volume
= πR²h – πr²h)
= π(R² – r²)h

Right circular cone

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
Total surface area = Area of curved surface + Area of base
= πrl + πr²
= πr(l + r)
Volume = (1/3) πr²h

Sphere

If r is the radius of the sphere, then
Surface area = 4πr²
Volume = 4/3 πr³

Spherical shell

If R is the external radius and r is the internal radius of a spherical shell, then
Surface area (outer) = 4πR²
Volume of material =

Hemisphere

If r is the radius of the hemisphere, then
Total surface Area = 1/2 x Surface area of sphere + Area of base 
= (1/2) x 4πr² + πr²
= 3πr²
Volume = (1/2) x Volume of sphere 

Conversion of Solids

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.

Combination of Solids

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