For the first problem: Triangles are fascinating mathematical topics. Many other mathematical topics typically encounter or utilize elemements discovered within triangles.
Let us explore some aspects of triangles in an interactive fashion. This site will take you through some historical explorations, some interactive activities, as well as some intriguiging connections in mathematics.
A triangle in typical plane geometry is descibed as a three-sided figure or polygon whose interior angle sum is equal to 180 degrees.
The perimeter of a triangle is the sum of all three of the sides
1. Experiment with this file.
The area of a triangle is quite interesting. Often it will be represented by
Area=(1/2) x Base x Height. Where the height is an altitude drawn from the base to the opposite angle. This formula makes for a relatively easy calculation of the area of a triangle but it is rather difficult to naturally find a triangle that is given in terms of at least one side (the base) and a height. We typically can determine or are given the sides of a triangle when a triangle is present. A formula does exist that can calculate the area of a triangle when all three sides are known.
This formula is attributed to Heron of Alexandria but can be traced back to Archimedes.