**Introduction **

Solid geometry is a branch of geometry which mainly focuses on the properties like surface area, and volume of solid figures or three dimensional shapes like cube, cone, cylinder, and sphere.

A solid is generated by the revolution of a two dimensional plane. For example a cylinder is generated by the revolution of a rectangle; the revolution of a circle about its diameter generates a sphere.

**Cube**

Surface area = 6a^{2} square units

**Cylinder**

Lateral surface area = 2πrh square units

Total surface area = 2πr (h + r) square units

**Cone**

Lateral surface area = πrs square units

Total surface area = πrs + πr^{2} square units

**Sphere**

Surface area of a sphere = 4πr^{2} square units

**Cube** Volume = a^{3} cubic units

**Cylinder** Volume = πr^{2}h cubic units

**Cone** Volume = 1/3 πr^{2}h cubic units

**Sphere** Volume = 4/3 πr^{3} cubic units

An overhead water tank is in the form of a cylinder with 3 meter height and surmounted by a conical section. The radius and slant height of the conical section is 52.5 m and 53 m respectively. Find the total area of the tank to be painted and the capacity of the tank.

**Solution:**

**Area of the tank to be painted**

Radius of the cylindrical part, r = 52.5m

Height of the cylindrical part, h = 3m

**Lateral surface area of the cylindrical part = 2πrh square
units**

= 2 * π * 52.52 * 3 square meters

= 2 * 3.14 * 52.5 * 52.5 * 3

= **51927.75 square meters**

Radius of the conical part, r = 52.5m

Slant height of the conical part, s = 53m

**Lateral surface area of the conical part = πrs square
units**

= 3.14 * 52.5 * 53

= **8737.05 square meters**

**Area to be painted = lateral surface area of the cylindrical part + lateral surface area of the conical part**

= 51927.75 + 8737.05 square meters

= **60664.8 square meters**

**Capacity of the tank**

**Volume of the cylindrical section = πr ^{2}h cubic units**

= 3.14 * 52.5^{2} * 3

= **25963.875 cubic meters**

**Volume of the conical section = 1/3 πr ^{2}h cubic units**

= 1/3 * 3.14 * 52.5^{2} * 53

= **152898.375 cubic meters**

**Total volume of the tank = volume of the cylindrical part + volume of conical part**

= 25963.875 + 152898.375 cubic meters

**Capacity of the tank =** **178862.25 cubic meters.**