Lesson Various methods to calculate the Area of Triangle

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The aim of this lesson is to explain the various methods to calculate the <A HREF=Area.wikipedia>Area</A> of the given triangle:

In general a <A HREF=Triangle.wikipedia>triangle</A> is defined by its side length and the <A HREF=Angle.wikipedia>angle</A> between the two sides. First we need to collect this information for the given triangle in order to obtain the Area.

{{{drawing( 160, 160, 0.5, 4.5, 0.5, 4.5, line( 1, 1, 4, 1 ), line( 1, 1, 3, 4 ),line( 4, 1, 3, 4), locate( 1, 1, A ),locate( 3.2, 4, B ),locate( 4, 1, C ),circle( 3, 1, 0.1 ),locate( 3, 1, red( D ) ),red( line( 3, 1, 3, 4 ) ))}}}

Before we get into the real business, one needs to know the notations and basic parameters of a triangle.

As in the above diagram:

Sides:
BC, CA and AB are the three sides of the triangle and a,b,c are the corresponding side lengths.

Angle:
Angle between two sides is denoted by the name of Common point.i.e. Angle between AB and BC is denoted as angle B.

Perimeter: 
It is denoted by p and it is defined as the sum of all side lengths, which is equal to {{{p= AB+BC+CA}}}.

Semi-perimeter: 
It is denoted by s and it is defined as the half of of the perimeter
{{{s= (AB+BC+CA)/2}}}.

Calculation of Area
Very basic formula to calculate the area of the triangle is {{{Area= b*h/2}}}
where 
b is the base of the triangle.
h is the height of the triangle shown in the diagram.

Method 1
If you have two sides (say a and b) and the angle between them(say C), then the area can be calculated from the formula as {{{Area= b*h/2}}}

By simple trigonometry we can replace h by {{{h = a* Sin(C)}}}

Which can be written in term of the any two sides and the angle between them. i.e.

{{{Area  =  a*b*Sin(C)/2
         =  b*c*Sin(A)/2
         =  c*a*Sin(B)/2}}}

example: 
Lets say two sides of the triangle are b=4 and c=3 and the common angle A(90)

{{{drawing( 160, 160, -0.5, 4.5, -0.5, 4.5, line( 0, 0, 3, 0 ), line( 0, 0, 3, 4 ),line( 3, 0, 3, 4), locate( 0, 0, B ),locate( 3.2, 4, C ),locate( 3.2, 0, A ),locate( 1.5, 0, 3 ),locate( 3.2, 2, 4 ))}}}

Hence the Area of the triangle after plugging all the values in the formula
{{{Area  =  4*3*Sin(90)/2 = 6}}}

Method 2

If you have all the side lengths only. No information about the angles of the triangle. 
Then the area of the triangle is given by
{{{Area = sqrt(s*(s-a)*(s-b)*(s-c))}}},
where s is the semi-perimeter (defined above).

This formula is called <A HREF=Triangle.wikipedia>Heron's formula</A>.

example: 
Lets say side lengths are a=5, b=4 and c=3
{{{s = (3+4+5)/2 = 6}}}
Hence the Area of the triangle after plugging all the values in the formula
{{{Area = sqrt(6*(6-5)*(6-4)*(6-3)) = sqrt(36) = 6}}}

Method 3

If you have two angles (say B and C) and the side common to them(say a), then the area can be calculated as 
{{{Area= (a^2*Sin(B)*Sin(C))/(2*Sin(A)) = (b^2*Sin(C)*Sin(A))/(2*Sin(B)) = (c^2*Sin(A)*Sin(B))/(2*Sin(C))}}}

Where the third angle can be calculated by {{{A= 180 -(B + C)}}}

example: 
Lets say two angles of the triangle are B=30 and C=60 and the common side length a= 5
So to apply the above formula we need third Angle of the triangle.
Angle {{{A= 180- (60+30)= 90}}}

Hence the Area of the triangle after plugging all the values in the formula 
{{{Area= (5^2*Sin(30)*Sin(60))/(2*Sin(90)) = 5.41}}}

Method 4

If the coordinates of the vertices are given as 
{{{(x1 y1)}}} , {{{(x2 y2)}}} , {{{(x3 y3)}}}

then the Area is calculated as
{{{ Area = (1/2)*abs(det(matrix( 3, 3, x1, y1, 1, x2, y2, 1, x3, y3, 1 )))}}},
where det denotes the determinant, i.e. 
{{{ Area = (1/2)*abs((x1*y2)-(x2*y1)+(x2*y3)-(x3*y2)+(x3*y1)-(x1*y3))}}}

example: 
Lets say the coordinates of the given triangle are as follows:
A(1,1), B(3,4) and C(4,1)

{{{drawing( 160, 160, 0.5, 5.5, 0.5, 4.5, line( 1, 1, 4, 1 ), line( 1, 1, 3, 4 ),line( 4, 1, 3, 4), locate( 1, 1, A(1,1)),locate( 3.2, 4, B(3,4)),locate( 4, 1, C(4,1)))}}}
Hence the Area of the triangle after plugging all the values in the formula
{{{Area= (1/2)*abs(1*1 - 4*1 + 4*4 - 3*1 + 3*1 - 1*4) = 9/2}}}

See this <A HREF=Area_of_triangle.wikipedia> article about Computing Area of triangle </A>