# 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.