The area of a given triangle can be calculated in various ways depending on what is known about the triangle.
*** Area = half base multiplied by vertical height ***
Three cases must be considered: (a) the right triangle, (b) an acute triangle, (c) a scalene triangle. In all three cases, the same formula emerges
Area △ ABC = ½ base x vertical height = ½ b.h
Video (Youtube) - Deriving the formula for the area of a triangle - video by Maths Whenever
*** Area when two sides and the included angle are known ***
The area of the triangle is given by
Area △ ABC = ½ bc. sin Θ
Letter b is the triangle base and the other sides have lengths a and c.
*** Heron's Formula ***
If the three side lengths are known then the area of the triangle can be calculated using Heron's formula
where A is the area. The side lengths are a, b and c. Letter s is the semi-perimeter that is
½(a + b + c)
An online calculator is available at Wolfram Mathworld
*** Proof of Heron's formula ***
The following links offer proofs of Heron's formula
Proof using trigonometry - (pdf) - uses trigonometry to prove the formula
A proof without using trigonometry is at Jamie York Press - HERE (pdf)
Jim Wilson - University of Georgia offers a further proof using algebra
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