how can i find the area of a polygon by using plain geometry?

Question

i have a project in maths and i wanna do something cool and something new and i need an easy formula which is not famous and one which does not use trigonometry

Answer 1

Place your polygon in the system of x-y cordinate axis.
Find (x,y) for every point of your polygon.
Now sort this (x,y) in a way that you begin from one point and continue clockwise or counterclockwise till you are in the first point again.
Example:polygon ABCDEF
(xA,yA)
(xB,yB)
(xC,yC)
(xD,yD)
(xE,yE)
(xF,yF)
(xA,yA)
Now
a=xA*yB+xB*yC+...+xF*yA
and
From bottum to top
b=xA*yF+xF*yE+...+xB*yA
(if you have a bigger polygon you can expand it, no problem)
Area=Absolute(a-b)/2
Sorry for my English.

Answer 2

1. Pythagoras: (3√2)² = side² + side² 2side² = 18 side = √(18/2) side = 3 The area of this square is side*side=3*3=9 cm² 2. sin60 = height / 10 height = 10sin60 = 10√(3)/2 = 5√3 area = ½*base*height = ½*10*5√3 = 25√3 dm² 3. Pythagoras: (√74)² = 7² + side² side² = 74 - 49 side = √25 side = 5 The area of this rectangle is side1*side2=7*5=35 square units 4. It being an isosceles right triangle means a right angle where the legs meet. Pythagoras: base² = 15² + 15² base² = 450 base = √450 = √9*25*2 = 15√2 Pythagoras again to find the height: 15² = height² + (base/2)² 225 - (15√2 / 2)² = height² height = √(225 - 112.5) = 15√2 / 2 area = ½*base*height = ½ * 15√2 * 15√2 / 2 = 225 / 2 = 112.5 cm² 5. area = 88 = ½*base*height = ½*base*14 = 7*base base = 88 / 7 = 12.571... units

Answer 3

I am not sure exactly what you are asking, but you may want to look at Hero's formula (Herron's formula-varies from book to book). The formula is not as well known and you kind find the area of a triangle from it. You can also aply this in plane (not plain) geometry. If you are graphing on a plane and need to find areas, use the distance formula to find the lengths of the sides of the triangle.

Answer 4

dear the easiest way is to find, by integration, the center of mass of the polygon, join all corners to it and find the areas of the triangle.
but if u don't want to do this, u can find the incentre of the polygon, i.e. the meeting point of the angle bisectors and find the areas of the triangles thus formed but to find the sides using angles will use trigonometry, of course.

Answer 5

This is for a pentagon:
A = a²/4 * √[25 + 10 * √(5)]
where "a" is the length of one of the sides.

I already answered this one for you in your previous question.

There is something we use in land surveying to determine the area of any polygon... regular or irregular. It requires determining the coordinate (x,y) values for each point. It is called the "double meridian distance" method. If you search for it, you should be able to find a more detailed explanation.

'Mamad' below explains it very well.... for a foreigner! LOL!

Answer 6

this can be done graphically.
first draw the figure correctly on a graph paper and count the number of units it enclose. that will be its area
or
divide the polygon into series of triangles and find their areas
and finally sum them up to get gross area
by doing this you can derive a formula by yourselves
try it out

Answer 7

Dear friend,
this may done by simply converting into triangles,
ie, by drawing lines (imaginary) from corners to centre (or common point inside the polygon)

this will convert complex polygons to number of simple triangles.

i hope you find area of triangles easily.

Answer 8

Just cut it up into a bunch of triangles and add the areas.

Answer 9

Break in to n-2 triangles now its easy to find anything.

Answer 10

geometry and trigonometry are basically the same thing

http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html

the only different is, trig is more detailed.