Solid Geometry

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.

 

Surface Area of Solids in geometry

 

Cube

                    Surface area = 6a2 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 + πr2 square units

Sphere

                    Surface area of a sphere = 4πr2 square units

 

Volume of solids in geometry

 

Cube Volume = a3 cubic units

Cylinder Volume = πr2h cubic units

Cone Volume = 1/3 πr2h cubic units

Sphere Volume = 4/3 πr3 cubic units

 

Examples of Solid Geometry

 

                    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 = πr2h cubic units

  = 3.14 * 52.52 * 3

  = 25963.875 cubic meters

Volume of the conical section = 1/3 πr2h cubic units

                                                         = 1/3 * 3.14 * 52.52 * 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.