Geometry is the study of shapes. The quantities it investigates includes angles, lengths, areas, volumes etc. The aim of geometry is to investigate several relationships between these variables in different setups.
To define these quantities, we need to define several terms:
A point is an object with no dimensions. It is the smallest thing there is, but that is not what we are concerned with.
A line is a straight 1-dimensional object. It extends in 2 directions. A ray extends only in 1 direction, and a segment is one with finite length.A curve is a 1-dimensional object that may not be straight. A curve is closed if a point is 'directly connected' with 2 other points and contains no ends, and open if there are ends (or potential ends in the infinite case).
A plane is a flat 2-dimensional object. It can either be finite or infinite. A surface, however, may not be flat, and can be curved or wavy in any way possible. A surface is still 2 dimensional, which means it is infinitely thin. A surface is closed if there is no identifiable 'boundary' (points lie on one side of a boundary but none on the other) across the whole surface, and open otherwise.
A solid is any 3-dimensional object. A solid can have holes inside it or along its surface. A hollow solid is a closed surface.
Now, we are ready to define angle, lengths, areas and volumes:
An angle can be formed between 2 planes, 2 lines or a plane and a line. It is meaningless to discuss about angles between curves or surfaces in general. If 2 lines intersect at a point, then the angle is a measure of how much one line can be rotated about that point to form the other line. The cases for the other 2 are slightly complicated, and will be discussed later. A rotation back to the original position is defined to be 360 degrees.
The length of a segment is a measure of how far apart the 2 endpoints are. It can either be measured on a straight line or a curve (known as the arc length in more advanced mathematics.) In physics, there are many useful length units, but we would not deal with units here.
The area of a surface is a measure on how big the surface is. It is analagous to the amount of paint you need to paint the whole surface.
The volume of an object measures how much space it occupies. Referring to Archimedes' claim "Eureka!", an object's volume can be measured by how much water it displaces in a bathtub.
Geometry studies the above relationships between angles, lengths, areas and volumes. Let us proceed to study them more in detail!
