SOLUTION: How do you find the area of a polygon ABCDE with vertices A(0,0), B(0,4), C(8,8), D(14,0), and E(9,-2)

Algebra.Com
Question 1154678: How do you find the area of a polygon ABCDE with vertices A(0,0), B(0,4), C(8,8), D(14,0), and E(9,-2)
Found 5 solutions by Alan3354, MathLover1, MathTherapy, ikleyn, greenestamps:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
How do you find the area of a polygon ABCDE with vertices A(0,0), B(0,4), C(8,8), D(14,0), and E(9,-2)
----------
There's more than one way.
The simplest, IMO, is this:
-----

 A    B    C    D    E    A
 0    0    8   14    9    0
 0    4    8    0   -2    0

----------------
Add the diagonal products starting at the upper left.
0*4 + 0*8 + 8*0 + 14*-2 + 9*0 = 0+0+0-28+0 = -28
---
Add the diagonal products starting at the lower left.
0*0 + 4*8 +8*14 + 0*9 + -2*0 = 0+32+112+0+0 = 144
-----------
The difference is 172
The area is 1/2 that = 86 sq units.
============
The points have to be in order around the figure.
I think it works for all polygons, not just convex.
========================
PS Graphing it is a good idea, to ascertain that the points are in order around the polygon.

Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

first graph it
(,)
(,)
(,)
(,)
(,)


the length of the side
the length of the side :
Solved by pluggable solver: Distance Between 2 points
The distance formula is . Plug in the numbers,

The distance is 9.21954445729289.


=>

the length of the side :
Solved by pluggable solver: Distance Between 2 points
The distance formula is . Plug in the numbers,

The distance is 8.94427190999916.


=>
the length of the side :
Solved by pluggable solver: Distance Between 2 points
The distance formula is . Plug in the numbers,

The distance is 10.


=>
the length of the side :
Solved by pluggable solver: Distance Between 2 points
The distance formula is . Plug in the numbers,

The distance is 5.3851648071345.


=>
The area of any irregular quadrilateral can be calculated by dividing it into triangles.
Heron's Formula for the area of a triangle(Hero's Formula)
A method for calculating the area of a triangle when you know the lengths of all three sides.
Let ,, be the lengths of the sides of a triangle.
The area is given by:
where is half the perimeter, or
divide in triangles:
you have triangles ABC, ACD, and AED
the area of triangles :
sides:


find side
Solved by pluggable solver: Distance Between 2 points
The distance formula is . Plug in the numbers,

The distance is 11.3137084989848.








=>the area of triangle

find the area of triangle








=>the area of triangle

find the area of triangle :








=>the area of triangle
the area of triangles


Answer by MathTherapy(10553)   (Show Source): You can put this solution on YOUR website!
How do you find the area of a polygon ABCDE with vertices A(0,0), B(0,4), C(8,8), D(14,0), and E(9,-2)
I got the same thing Tutor @ALAN got: . I also used the same method!
If you want to TORTURE yourself, then follow the other person's solution, which by the way should be 86 sq units, NOTHING more, NOTHING LESS.
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.

By the way, there is another elementary method calculating the area of a polygon.


Place it into a rectangle with vertical and horizontal sides.


Make this rectangle as small as possible, i.e. get the polygon "inscribed" into the rectangle.


Calculate the area of the rectangle (which is easy).


Then subtract the areas of all excessive triangles, by "cutting" them.


It is very easy, too.


Doing in this way, you will get the answer.


Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Breaking the figure into three triangles and finding the area of each is an astonishingly bad (and tedious!) way to try to get the answer...!

The matrix method for solving the problem, used by a couple of the other tutors, is clearly the easiest way to get the answer. But the method suggested by another tutor of inscribing the given figure in a rectangle is also an easy method.

I will steal another tutor's figure and add to it to find the answer by this method.



The area of rectangle PQRS (dimensions 14x10) is 140.

The area of right triangle BPC (legs 4 and 8) is 16
The area of right triangle CQD (legs 6 and 8) is 24
The area of right triangle DRE (legs 2 and 5) is 5
The area of right triangle ESA (legs 2 and 9) is 9

The area of polygon ABCDE is 140-(16+24+5+9) 140-54 = 86.


RELATED QUESTIONS

Find the area of the pentagon ABCDE with vertices A = (1, 2), B = (0, 0), C = (1, 4), D... (answered by Alan3354)
Find the area of the pentagon ABCDE with vertices A = (1, 2), B = (0, 0), C = (1, 4), D... (answered by Alan3354)
The pentagon ABCDE is graphed on a coordinate plane with vertices A(0, 0), B(3, 5), C(3,... (answered by Alan3354)
A hexagon has the following vertices: A (3, 0), B (7, 0), C (9, 2), D (7, 4), E (3, 4), F (answered by Alan3354)
In the figure below, AXC and BXD are arcs with centers B and A respectively. ABCD is a... (answered by Edwin McCravy)
The vertices of parallelogram ABCD are A(2, 4), B(0, 0), C(6, 2), and D(8, 6). Find the... (answered by ewatrrr)
I've been trying to use the same formula that I use to find the area of a triangle but it (answered by Alan3354,ikleyn)
Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars... (answered by CPhill)
A vertical line divides the triangle with vertices (0, 0), (9, 0) and (8, 4) into two... (answered by ikleyn)