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 Area of the given triangle:

In general a triangle is defined by its side length and the angle between the two sides. First we need to collect this information for the given triangle in order to obtain the Area.

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drawing%28+160%2C+160%2C+0.5%2C+4.5%2C+0.5%2C+4.5%2C+line%28+1%2C+1%2C+4%2C+1+%29%2C+line%28+1%2C+1%2C+3%2C+4+%29%2Cline%28+4%2C+1%2C+3%2C+4%29%2C+locate%28+1%2C+1%2C+A+%29%2Clocate%28+3.2%2C+4%2C+B+%29%2Clocate%28+4%2C+1%2C+C+%29%2Ccircle%28+3%2C+1%2C+0.1+%29%2Clocate%28+3%2C+1%2C+red%28+D+%29+%29%2Cred%28+line%28+3%2C+1%2C+3%2C+4+%29+%29%29

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%2BBC%2BCA

.

Semi-perimeter:
It is denoted by s and it is defined as the half of of the perimeter
s=+%28AB%2BBC%2BCA%29%2F2

.

Calculation of Area
Very basic formula to calculate the area of the triangle is Area=+b%2Ah%2F2

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%2Ah%2F2

By simple trigonometry we can replace h by h+=+a%2A+Sin%28C%29

Which can be written in term of the any two sides and the angle between them. i.e.
Area++=++a%2Ab%2ASin%28C%29%2F2%0D%0A+++++++++=++b%2Ac%2ASin%28A%29%2F2%0D%0A+++++++++=++c%2Aa%2ASin%28B%29%2F2

Area++=++a%2Ab%2ASin%28C%29%2F2%0D%0A+++++++++=++b%2Ac%2ASin%28A%29%2F2%0D%0A+++++++++=++c%2Aa%2ASin%28B%29%2F2

example:
Lets say two sides of the triangle are b=4 and c=3 and the common angle A(90)
drawing%28+160%2C+160%2C+-0.5%2C+4.5%2C+-0.5%2C+4.5%2C+line%28+0%2C+0%2C+3%2C+0+%29%2C+line%28+0%2C+0%2C+3%2C+4+%29%2Cline%28+3%2C+0%2C+3%2C+4%29%2C+locate%28+0%2C+0%2C+B+%29%2Clocate%28+3.2%2C+4%2C+C+%29%2Clocate%28+3.2%2C+0%2C+A+%29%2Clocate%28+1.5%2C+0%2C+3+%29%2Clocate%28+3.2%2C+2%2C+4+%29%29

drawing%28+160%2C+160%2C+-0.5%2C+4.5%2C+-0.5%2C+4.5%2C+line%28+0%2C+0%2C+3%2C+0+%29%2C+line%28+0%2C+0%2C+3%2C+4+%29%2Cline%28+3%2C+0%2C+3%2C+4%29%2C+locate%28+0%2C+0%2C+B+%29%2Clocate%28+3.2%2C+4%2C+C+%29%2Clocate%28+3.2%2C+0%2C+A+%29%2Clocate%28+1.5%2C+0%2C+3+%29%2Clocate%28+3.2%2C+2%2C+4+%29%29

Hence the Area of the triangle after plugging all the values in the formula
Area++=++4%2A3%2ASin%2890%29%2F2+=+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%28s%2A%28s-a%29%2A%28s-b%29%2A%28s-c%29%29

Area+=+sqrt%28s%2A%28s-a%29%2A%28s-b%29%2A%28s-c%29%29

,
where s is the semi-perimeter (defined above).

This formula is called Heron's formula.
example:
Lets say side lengths are a=5, b=4 and c=3
s+=+%283%2B4%2B5%29%2F2+=+6

Hence the Area of the triangle after plugging all the values in the formula
Area+=+sqrt%286%2A%286-5%29%2A%286-4%29%2A%286-3%29%29+=+sqrt%2836%29+=+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=+%28a%5E2%2ASin%28B%29%2ASin%28C%29%29%2F%282%2ASin%28A%29%29+=+%28b%5E2%2ASin%28C%29%2ASin%28A%29%29%2F%282%2ASin%28B%29%29+=+%28c%5E2%2ASin%28A%29%2ASin%28B%29%29%2F%282%2ASin%28C%29%29

Area=+%28a%5E2%2ASin%28B%29%2ASin%28C%29%29%2F%282%2ASin%28A%29%29+=+%28b%5E2%2ASin%28C%29%2ASin%28A%29%29%2F%282%2ASin%28B%29%29+=+%28c%5E2%2ASin%28A%29%2ASin%28B%29%29%2F%282%2ASin%28C%29%29

Where the third angle can be calculated by A=+180+-%28B+%2B+C%29

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-+%2860%2B30%29=+90

Hence the Area of the triangle after plugging all the values in the formula
Area=+%285%5E2%2ASin%2830%29%2ASin%2860%29%29%2F%282%2ASin%2890%29%29+=+5.41

Method 4
If the coordinates of the vertices are given as
%28x1+y1%29

then the Area is calculated as
+Area+=+%281%2F2%29%2Aabs%28det%28matrix%28+3%2C+3%2C+x1%2C+y1%2C+1%2C+x2%2C+y2%2C+1%2C+x3%2C+y3%2C+1+%29%29%29

,
where det denotes the determinant, i.e.
+Area+=+%281%2F2%29%2Aabs%28%28x1%2Ay2%29-%28x2%2Ay1%29%2B%28x2%2Ay3%29-%28x3%2Ay2%29%2B%28x3%2Ay1%29-%28x1%2Ay3%29%29

example:
Lets say the coordinates of the given triangle are as follows:
A(1,1), B(3,4) and C(4,1)
drawing%28+160%2C+160%2C+0.5%2C+5.5%2C+0.5%2C+4.5%2C+line%28+1%2C+1%2C+4%2C+1+%29%2C+line%28+1%2C+1%2C+3%2C+4+%29%2Cline%28+4%2C+1%2C+3%2C+4%29%2C+locate%28+1%2C+1%2C+A%281%2C1%29%29%2Clocate%28+3.2%2C+4%2C+B%283%2C4%29%29%2Clocate%28+4%2C+1%2C+C%284%2C1%29%29%29

drawing%28+160%2C+160%2C+0.5%2C+5.5%2C+0.5%2C+4.5%2C+line%28+1%2C+1%2C+4%2C+1+%29%2C+line%28+1%2C+1%2C+3%2C+4+%29%2Cline%28+4%2C+1%2C+3%2C+4%29%2C+locate%28+1%2C+1%2C+A%281%2C1%29%29%2Clocate%28+3.2%2C+4%2C+B%283%2C4%29%29%2Clocate%28+4%2C+1%2C+C%284%2C1%29%29%29

Hence the Area of the triangle after plugging all the values in the formula
Area=+%281%2F2%29%2Aabs%281%2A1+-+4%2A1+%2B+4%2A4+-+3%2A1+%2B+3%2A1+-+1%2A4%29+=+9%2F2

See this article about Computing Area of triangle

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