Closed.

You are watching: Area of a parallelogram given 4 points

This question does not satisfy occupychristmas.orgematics Stack Exreadjust guidelines. It is not presently accepting answers.
Want to boost this question? Upday the question so it's on-topic for occupychristmas.orgematics Stack Exadjust.

Closed 2 years earlier.


How perform you find the area of a parallelogram through the complying with vertices; $A(4,2)$, $B(8,4)$, $C(9,6)$ and also $D(13,8)$.


*

*

For this, we setup to usage the Shoelace formula.

Shoelace Formula: Given the coordinates of vertices of a polygon, its location is discovered by $$A=frac 12left|sum_i=1^n-1x_iy_i+1+x_ny_1-sum_i=1^n-1x_i+1y_i-x_1y_n ight|$$ Or, in various other words, we have actually $$A=frac 12|x_1y_2+x_2y_3+ldots x_n-1y_n+x_ny_1-x_2y_1-x_3y_2-ldots -x_ny_n-1-x_1y_n|$$ Wright here $A$ is the location of the polygon, and $(x_i,y_i)$ via $i=1,2,3dots$ are the vertices of the polyon

So with your situation, the vertices are $A(4,2), B(8,4), C(9,6)$ and also $D(13,8)$. We let $x_1=13,y_1=8,x_2=9,y_2=6,x_3=4,y_3=2,x_4=8,y_4=4$ and the location is offered by $$A=frac 12|13cdot 6+9cdot 2+4cdot 4+8cdot 8-9cdot 8-4cdot 6-8cdot 2-13cdot 4|\=frac 12cdot 12=6$$


Share
Cite
Follow
edited Oct 13 "16 at 12:24
answered Oct 13 "16 at 0:01
*

FrankFrank
5,69411 gold badge1212 silver badges4242 bronze badges
$endgroup$
2
Add a comment |
19
$egingroup$
The absolute value of the cross product of 2 vectors $veca, vecb in occupychristmas.orgbbR^3$ extending the parallelogram is its area:

$$A_ extparallelogram= left|veca imesvecb ight|$$

So in your case we have to compose the points in $occupychristmas.orgbbR^2$ as vectors in $occupychristmas.orgbbR^3$ and also apply the formula:

$vecAB = eginpmatrix8\4\0endpmatrix -eginpmatrix4\2\0endpmatrix =eginpmatrix4\2\0endpmatrix$

$vecAD = eginpmatrix13\8\0endpmatrix -eginpmatrix4\2\0endpmatrix =eginpmatrix9\6\0endpmatrix$

$A_ extparallelogram= left|vecAB imesvecAD ight| = left| eginpmatrix4\2\0endpmatrix imes eginpmatrix9\6\0endpmatrix ight| = left|eginpmatrix0\0\6endpmatrix ight| = 6$

You can have actually noticed that this simplifies to

$$A_ extparallelogram= (b_1 - a_1)(d_2-a_2)-(b_2-a_2)(d_1-a_1)$$$$= (8 - 4)(8-2)-(4-2)(13-4)=-24-(-18)=6$$


Share
Cite
Follow
edited Oct 13 "16 at 14:40
answered Oct 12 "16 at 23:52
*

adjanadjan
5,5331515 silver badges3737 bronze badges
$endgroup$
Add a comment |
9
$egingroup$
Tbelow are plenty of ways, such as the Shoelace Theorem and also Pick"s Theorem.

If you have actually a graph, you deserve to additionally simply draw a rectangle approximately the form and subtract the parts you do not want.


Share
Cite
Follow
answered Oct 12 "16 at 23:51
*

suomynonAsuomynonA
6,69422 gold badges3030 silver badges6464 bronze badges
$endgroup$
Add a comment |
1
$egingroup$
I think this is a special situation of shoelace theorem. A quad is made up of two triangle and location of a triangle is

$$1over 2$$

Or you have the right to usage distance formula

$$distance = sqrt(x_1 - x_2)^2 + (y_1 - y_2)^2$$

and then heron"s formula

$$A = 1over 2sqrts(s-a)(s-b)(s-c))$$ Where s is the semi-perimeter of the triangle and also, a,b,c are the size of its sides.

See more: How To Make A Glass Bottle In Minecraft, Glass Bottle


Share
Cite
Follow
answered Oct 13 "16 at 8:10
user377111user377111
2511 silver badge77 bronze badges
$endgroup$
Add a comment |
1
$egingroup$
For any type of quadrilateral the area is one-fifty percent the magnitude of the cross product of the two diagonal vectors.


Share
Cite
Follow
answered Oct 13 "16 at 16:15
chowdahchowdah
1111 bronze badge
$endgroup$
Add a comment |
-2
$egingroup$
I am simply giving you via the easiest shortcut to doing this

Pick the first three points A(4,2), B(8, 4) and also C(9, 6)

Negate point A to gain (-4, -2) and also add to the various other two points B and also C. Add x"s and also y"s so you have a new point

(4, 2)(5, 4) Now use determinant to uncover the area.

16-10 = 6sq unit

Neglect any type of negative authorize that arises


Share
Cite
Follow
answered Jan 25 "19 at 4:28
Daniel WastyDaniel Wasty
1
$endgroup$
2
Add a comment |

Not the answer you're looking for? Browse other inquiries tagged geomeattempt analytic-geomeattempt location or ask your own question.


Featured on Meta
12 votes · comment · stats
Linked
1
Soft question on the area of a parallelogram using vectors
Related
73
Why determinant of a 2 by 2 matrix is the area of a parallelogram?
4
Area of a parallelogram, vertices $(-1,-1), (4,1), (5,3), (10,5)$.
1
Find the maximum area of a parallelogram given that you know the perimeter
0
Finding the location of a parallelogram provided vertices
1
Find the location of a quadrilateral in a parallelogram.
1
Given a parallelogram $ABCD$, if $B=(-3,-4)$, $C=(-7,-7)$, and $A=(0,0)$, what is the area of the parallelogram?
1
Find area of parallelogram given location amount of four segmented quadrilaterals
0
Analytical strategy to find location of parallelogram
3
How do you find the location of parallelogram from just the sides of a equation
Hot Network Questions even more hot concerns
occupychristmas.orgematics
Company
Stack Exadjust Netjob-related
site architecture / logo design © 2021 Stack Exadjust Inc; user contributions licensed under cc by-sa. rev2021.9.15.40218


occupychristmas.orgematics Stack Exreadjust works ideal with JavaScript allowed
*

Your privacy

By clicking “Accept all cookies”, you agree Stack Exreadjust can save cookies on your device and also discshed indevelopment in accordance through our Cookie Policy.