الاثنين، 13 ديسمبر 2010

formulas for area of triangle

area =  ah/2

there is big rectangle made up by two smaller rectangles 

the area of the first = hd  and  area of the second = he

the first rectangle is divided by two triangles. area of these triangles = hd/2

the second rectangle is divided by two triangles. area of these triangles = he/2

thanks to "ElliottMK1" for Correction

since the big triangle is made up by two triangles whose areas are hd/2 and he/2 

then area of big triangle = area of the two triangles = hd/2 + he/2 = h/2 [ d + e] = ha/2


area = (ab/2) sin C

to proof this we know that area = ah/2 
also sin c = opposite / hypotenuse = h/b
then h = b sin C then area = (ab/2) sin C


by use vectors
area = 

to proof this Let vectors AB and AC point respectively from A to B and from A to C
then.the area of parallelogram ABDC is  

The area of triangle ABC is half of area of parallelogram ABDC

area of  triangle ABC

now we can write tha area of triangle ABC in dot product

now we will write vector AB = (x1,y1) and AC = (x2,y2) then the area will be


Using coordinates

let vertex A is located at the origin (0, 0) and the coordinates of the other two vertices are given by B = (xB, yB) and C = (xC, yC), then the area can be computed as ½ times the absolute value of the determinant

For three general vertices, the equation is:

In three dimensions, the area of a general triangle {A = (xA, yA, zA), B = (xB, yB, zB) and C = (xC, yC, zC)} is the Pythagorean sum of the areas of the respective projections on the three principal planes
 (i.e. x = 0, y = 0 and z = 0):


Heron's formula.

from Pythagoras's theorem a=d+e, b2=h2+d2, and c2=e2+h2. then 
d = (b2-h2) and and c2 =(a-d)2+h
then  c2  = a2-2 X a X d + b2 = a2-2 X a X √(b2-h2)+b
then h=(4 X aX b2-(a2+b2-c2)2)/(2 X a) 
then h = √((a+b+c) X (-a+b+c) X (a-b+c) X (a+b-c))/(2 X a).

area  = ah/2 = a X ((a+b+c) X (-a+b+c) X (a-b+c) X (a+b-c))/(2 X a) = 

((a+b+c) X (-a+b+c) X (a-b+c) X (a+b-c))/4 = 

(((a+b+c)/2) X ((-a+b+c)/2) X ((a-b+c)/2) X ((a+b-c)/2))

take s = (a+b+c)/2. then

area = √(s X (s-a) X (s-b) X (s-c))


Using Pick's Theorem

Pick's theorem for a technique for finding the area of any arbitrary lattice polygon.
The theorem states:

where I is the number of internal lattice points and B is the number of lattice points lying inline with the border of the polygon.

there are other area formulas i will show them later 


هناك 70 تعليقًا:

  1. With havin so much written content do you ever run into any problems of plagorism

    or copyright violation? My website has a lot of unique content

    I've either created myself or outsourced but it looks like a lot of it is popping it up

    all over the web without my authorization. Do you know any techniques to help prevent content from being ripped off? I'd

    definitely appreciate it.
    My weblog :: mohurs

  2. I'm also writing to

    make you know of the notable discovery my friend's girl undergone reading your webblog.

    even learned several pieces, which include how it is like

    to possess an excellent

    teaching mood to make certain people really

    easily grasp specific impossible

    topics. You truly exceeded my expectations.
    Many thanks for imparting these informative, trusted, informative and also fun guidance
    on your topic to Sandra.
    Here is my web blog http://puppylover8383.blogspot.ru/

  3. After I initially commented I clicked the -Notify me when
    new feedback are added-

    checkbox and now each time a comment is added I get
    four emails with the identical comment.

    Is there any approach you can take away me

    from that service? Thanks!
    Also visit my web page ... http://www.thespainforum.com

  4. I have been exploring for a little bit for any high-quality articles or
    blog posts

    on this sort of area . Exploring in Yahoo I at last stumbled
    upon this web site. Reading this

    info So i am happy to convey that I've an incredibly good uncanny feeling I

    discovered just what I needed. I most certainly will make sure to do not forget this site and give it a glance on a constant basis.
    Also see my website: http://www.thespainforum.com/f197

  5. I like what you guys are up too. Such intelligent work and reporting!

    Keep up the

    superb works guys I have incorporated you guys to my blogroll.

    I think it'll improve the value

    of my site :)
    My page - www.villasspainproperty.com

  6. Hi, i think that i saw you visited my web site thus i

    got here to “go back the want”.I'm trying to to find things to enhance my website!I suppose its

    adequate to make use of some of your ideas!!
    My website http://wiki.atlatlonline.com/

  7. My brother recommended I might like this blog.

    He was

    once totally right. This publish actually made my day.
    You can not

    imagine just how a lot time I had spent for this information!
    Here is my web site - www.almoradi.us

  8. There is evidently a lot to

    realize about this. I feel you made some nice points in

    features also.
    Also see my page - www.spainweather.org

  9. Hi there! I could have sworn I've been to this website

    before but after reading through some of the post I realized it's new to me.

    Anyhow, I'm definitely glad I found it and I'll be book-marking
    and checking back

    Also visit my webpage :: us vs spain online

  10. hi!,I like your writing very much! share we communicate more about your article on AOL?
    I need an expert on this area to solve my problem.
    Maybe that's you! Looking forward to see you.
    my site > history spanish language spain

  11. Thanks - Enjoyed this update, is

    there any way I can receive an email

    sent to me when there is a new post?
    Stop by my blog ... xylitols

  12. I'm not sure why but this weblog is loading incredibly slow for

    me. Is anyone else having this issue or is it a issue on my end? I'll check back later and

    see if the problem still exists.
    Also see my site :: elcheproperty.net

  13. My brother recommended I would possibly like this blog. He used to be
    entirely right. This submit truly made my day. You cann't

    consider simply how so much time I had spent for this information! Thank

    Here is my homepage ... costa del sol spain Photos

  14. Thanks , I have recently been searching for information

    about this subject for a long time and yours is the best I have came

    upon so far. But, what concerning the conclusion? Are you positive in regards to the source?
    Feel free to visit my homepage :: houses rent nerja spain

  15. Neat blog! Is your theme custom made or did you download it from

    somewhere? A theme like yours with a few simple tweeks would really make my blog

    stand out. Please let me know where you got your theme.
    Thanks a lot
    Also see my page: golfspain.us

  16. Hello there, You have done a fantastic job.

    I’ll certainly digg it and for my part recommend

    to my friends. I am sure they'll be benefited from this web site.
    My page: www.televisionspain.net

  17. Hello there, I found your website by the use of Google

    at the same time as looking for a comparable

    subject, your web site came up, it seems to be good.
    I've bookmarked it in my google bookmarks.
    My webpage ; http://en.wikipilipinas.org/index.php?title=User:SaraiLopez81

  18. Hello, you used to write fantastic, but the last few posts

    been kinda boring… I miss your super writings.
    Past several posts are just a little bit out of
    track! come on!
    Here is my blog post :: http://www.charthunt.com/thespainforum.com

  19. What i do not realize is if truth be told how you're

    no longer really a lot more neatly-favored than you might be now. You're so intelligent.
    You know therefore considerably in the case of this subject, made

    individually consider it from a lot

    of various angles. Its like men and women don't seem to be

    fascinated until it is one thing to accomplish with Girl gaga! Your own stuffs outstanding.

    At all times maintain it up!
    Here is my site : http://tejji.com/ip/url-to-ip-address.aspx?domain=thespainforum.com

  20. Thank you for the good writeup. It in fact was a amusement account it.

    Look advanced to far added

    agreeable from you! By the way, how could we communicate?
    Also visit my web page www.alboxproperty.com

  21. Heya i am for the first time here. I came across this board and I find
    It really useful & it helped

    me out much. I hope to give something back and
    help others like you aided me.
    Feel free to visit my weblog ; jobsinspain.org

  22. Hi are using Wordpress for your blog platform? I'm new to the

    blog world but I'm trying to get started and set up my
    own. Do you require any html coding

    knowledge to make your own blog? Any help would be greatly appreciated!
    My website :: fiacres

  23. I appreciate, cause I found just what I was looking for.
    You've ended my four day long hunt! God

    Bless you man. Have a great day. Bye
    My web blog ; daisymaebelle.com

  24. F*ckin’ tremendous things here. I’m very glad to see
    your post. Thanks a lot

    and i am looking forward to contact you. Will you please drop me a mail?
    Also see my web site :: schaf-foren.de

  25. Thanks a bunch for sharing this with all folks you actually know what
    you're talking approximately! Bookmarked. Kindly

    also consult with my website =). We

    will have a hyperlink trade contract between us!
    Here is my web site ; spanabox replacement parts

  26. Hi there, just turned into aware of your weblog via Google,
    and located that it's truly informative. I am going to

    watch out for brussels. I’ll be grateful when you proceed this in future. A lot of other people will be benefited

    out of your writing. Cheers!
    My site > www.prstars.co.uk

  27. Hello! I simply would like to give an

    enormous thumbs up for the nice information you've gotten right here on this post. I will probably be coming back to your blog for more soon.
    My blog post : Holiday packages

  28. I was wondering if you ever thought of changing the structure of your blog?

    Its very well written; I love what youve got to say. But maybe you could a little more in
    the way of content

    so people could connect with it better. Youve got an awful lot of
    text for only having one or 2 images. Maybe you could space it out better?
    My website - myechochamber.blogspot.ru

  29. Hey There. I found your blog using msn. That is an extremely
    neatly written article. I’ll be sure to bookmark it and

    come back to learn more of your useful info. Thank you for the

    post. I’ll certainly return.
    Also visit my web blog ; live football match italy vs spain

  30. I together with my guys appeared

    to be viewing the excellent strategies located on your
    web site then all of the sudden I had an awful suspicion
    I never expressed respect to

    the web blog owner for

    those tips. Most of the ladies

    ended up so excited to read through

    them and have in truth been taking advantage of these things.
    Many thanks for

    genuinely so helpful as well as for

    choosing some remarkable resources most people are really desirous to

    know about. Our own sincere regret for

    not saying thanks to you sooner.
    Also visit my web-site :: http://maggiesfarmrevisited.Blogspot.ru

  31. You actually make it appear so easy together with

    presentation however I in finding this matter to be actually something

    which I feel I'd by no means understand. It sort of

    feels too complex and very vast for me. I am

    taking a look ahead to your next post, I’ll try to get the cling of it!
    Here is my web-site herd-it.net

  32. As I site possessor I believe the content matter here
    is rattling great , appreciate it for your
    hard work. You should keep it up forever! Good Luck.
    my website :: sported

  33. Wow! This can be one particular of the most beneficial blogs We've ever arrive across

    on this subject. Actually Fantastic. I am also an expert in this topic so I can understand your hard work.
    Here is my webpage - http://info.org.do/index.php?title=Usuario:CatriceRose75

  34. I cherished as much as you'll receive carried out

    proper here. The comic strip is tasteful, your authored subject matter

    stylish. however, you command get bought an edginess

    over that you want be handing over the following. in poor health no doubt come more before once more

    since precisely the same just about a lot regularly inside of case you protect this increase.
    Also visit my web blog ; http://txninspain.blogspot.ru/2007/12/christmas-in-spain.html

  35. Hi! Do you know if they make any plugins to help with

    Search Engine Optimization? I'm trying to get my blog to rank for some targeted keywords but I'm not seeing very good

    success. If you know of any please share. Kudos!
    Also visit my website news-asahi.com

  36. What’s Going down i'm new to this, I stumbled upon this I've found It

    positively useful and it has aided me out loads.

    I hope to give a

    contribution & assist different users like its aided me.

    Have a look at my homepage joe montana

  37. Hi are using Wordpress for your site platform? I'm new to the blog world but I'm trying
    to get started and create my own. Do you require any html coding expertise to make your own blog?
    Any help would be greatly appreciated!
    Also visit my web page ; Water wastage

  38. You are so cool! I do not think I've read through a single thing like this before. So wonderful to discover someone with some genuine thoughts on this topic. Seriously.. thank you for starting this up. This web site is something that is needed on the internet, someone with a little originality!
    Look at my page Http://Weekpedia.Myht.Org/

  39. Very good article. I certainly appreciate this site. Keep it up!

    My web blog shopping online

  40. I like what you guys are up also. Such intelligent work and reporting!
    Keep up the

    excellent works guys I have incorporated you guys to my blogroll.
    I think it will improve the value

    of my web site :)
    My blog post :: wiki.brg-schoren.ac.at

  41. Hey! I know this is kinda off topic however I'd figured I'd ask.
    Would you be interested in trading links or maybe guest

    authoring a blog post or vice-versa? My site goes over a

    lot of the same topics as yours and I believe we could greatly benefit from each other.
    If you're interested feel free to send me an email. I look forward to

    hearing from you! Excellent blog by the way!
    Feel free to visit my homepage ; robotc.mx

  42. Just want to say your article is as surprising. The

    clarity in your publish is simply cool and i could assume you are

    knowledgeable on this subject. Well along with your permission

    allow me to snatch your RSS feed to stay up to date

    with drawing close post. Thank you 1,000,000 and
    please keep up the gratifying work.
    Also see my page: ctrlauto.org

  43. Does your blog have a contact page? I'm having a tough time locating it but, I'd like to

    send you an email. I've got some ideas for your blog you

    might be interested in hearing. Either way, great website and I look forward to seeing it

    expand over time.
    My site ... http://km.sisat.ac.th/

  44. I think other web site proprietors should take
    this site as an model, very clean

    and fantastic user friendly style and design, as well as the

    content. You are an expert in this topic!
    Also visit my page - bttclubedechaves.net

  45. I like the valuable information you provide in your articles.
    I will bookmark your blog

    and check again here regularly. I'm quite certain I will learn lots of new stuff right here! Good luck for the next!
    My webpage Counterstrikefever.blogspot.fr

  46. I am 100% convinced that my 8yo daughter has Pandas Syndrome. On a recent trip to California two.5 weeks ago, she broke out in the rash on her upper torso. Assuming it was both warmth rash or a reaction to sunscreen, I treated it with Benadryl and returned home the next day. On our past vacation night, my daughter experienced a strange response to obtaining the covers pulled over her. Her reaction was one of terror, and she claimed that the bed was dirty. I wondered aloud to my husband if possibly there was some Autism in her diagnosis that experienced been skipped (she's Down Syndrome, mild retardation).

    Have a look at my website; http://dollamurmartialarts.com/profile/FerminEly

  47. The major pгoblеmѕ I've with abx procedure with strep are: 1. Docs typically give without having culturing 2. Abx are held hostage by docs wanting to make a buck. My spouse and children does not have insurance, so when now we have sore throats with low to no grade fever, we just have to hold out it out. I cannot afford $120 for each household member being analyzed. There ought to be OTC tests and then, meds could be released. I won't be able tο tell you hоw a lot of ѕituаtіons we've got been in a very doc office for about 10 min test and afterwards left that has a $300 bill for three folks. It can be outrageous.

    my site - www.fitnesshappiness.Com

  48. Priceу, thanks foг your perѕonal comment.
    cοnsumerѕ are so brаin-washeԁ in recent
    times with the Alopathic mеdicine, they believe
    antіbiotics are wanted in virtually evегy case.
    I am usіng a strep throat noω,
    and in aԁdition tаking an oregano οil capsules.
    It togetheг gave me an huge aid. Ιndeed,
    I am sick wіth the 6-th working ԁay now, сοntinue to usіng a fever, but i do experiencе consideгably
    іmρroved. Just one just shoulԁ feel in one's individual ability to appear outside of the disorder. I am also breast-feeding, so are unable to just take antibiotics in any case.

    Check out my web page; ukrainegay.org

  49. I was recommended this blog by my cousin. I
    am not sure whether this post is written

    by him as nobody else know such detailed about my difficulty.

    You're incredible! Thanks!

    Look at my website :: http://www.oahuislandnews.com/index.php/member/84463/

  50. This really answered my problem, thank you!
    Here is my page - wikibello.cl

  51. I have to thank you for the efforts you have put in writing this site.
    I really hope to see the same high-grade blog posts from you later on as well.
    In truth, your creative writing abilities has inspired me to get my
    very own site now ;)

    Stop by my website - coffee flavor

  52. Great blog here! Also your website loads up fast! What host are

    using? Can I get your affiliate link to your host?
    I wish my web site loaded up as quickly as yours


    Here is my weblog - www.seofriendly.net

  53. I haven’t checked in here for some time because I thought it was getting boring, but the

    last few posts are good quality so I guess I will add you back to my daily bloglist.

    You deserve it my friend :)

    Feel free to surf to my web site - design.lorea.org

  54. Howdy! This post could not be written any better!

    Reading this post reminds me of my old room mate! He always kept chatting

    about this. I will forward this article to him. Fairly certain he will have a good read.

    Thanks for sharing!

    My weblog: beta.crowdsquare.net

  55. I have read several just right stuff here. Definitely

    value bookmarking for revisiting. I surprise how a lot attempt
    you put to make one of these

    wonderful informative web site.

    My web blog drupal.familylifeassemblyofgod.com

  56. Excellent goods from you, man. I have understand your stuff previous
    to and

    you're just too great. I really like what

    you have acquired here, really like what you are saying and the way in which you

    say it. You make it entertaining and you still care for to keep it smart. I

    can't wait to read much more from you. This is actually a

    tremendous site.

    Also visit my site :: http://Gayloveme.com/

  57. We are a group of volunteers and starting a new scheme in our
    community. Your web site

    offered us with valuable info to work on.
    You have done a formidable job

    and our entire community will be grateful to you.

    Have a look at my webpage: boevye-iskusstva.bosa.org.ua

  58. Well I sincerely enjoyed reading it. This tip provided by you is very effective for correct planning.

    Here is my web site :: http://www.shiatsu-massagechairs.org

  59. Hey there, You've performed an excellent job.

    I’ll definitely digg it and for my part suggest

    to my friends. I am sure they'll be benefited from
    this web site.

    Look at my blog post; www.cesja.pl

  60. Very great post. I just stumbled upon your weblog and wanted to say

    that I have really loved surfing around your blog posts.
    In any case I will be subscribing on your feed and I hope

    you write once more soon!

    My web-site - http://Ourmapa.org/?q=node/7378/

  61. I'm still learning from you, while I'm trying to achieve my goals.
    I certainly liked reading everything

    that is written on your blog.Keep the posts coming.


    enjoyed it!

    my web page :: ciz1.com

  62. Great website. Lots of useful info here. I am
    sending it to several friends ans also sharing in delicious.
    And obviously, thank you to your effort!

    Also visit my blog post: http://code6studio.com/?p=7

  63. I think other website proprietors should take this website as an
    model, very clean

    and fantastic user friendly style and design, as well as the

    content. You're an expert in this topic!

    my blog post :: spotlightmywork.com

  64. Thanks in favor of sharing such a pleasant thinking, piece of writing is
    fastidious, thats why i have read it entirely

    Feel free to surf to my web page http://www.efatum.com

  65. Great post.

    Review my web-site ... Modcloth coupon

  66. I needed to thank you for this wonderful read!
    ! I absolutely loved every little bit of it. I have got you book-marked
    to check out new stuff you post…

    Feel free to visit my blog :: http://azonbundlereview.com/great-designer-dog-clothing-gift-ideas

  67. Hey, I think your site might be having browser compatibility issues.
    When I look at your website in Ie, it looks fine but when opening in Internet
    Explorer, it has some overlapping. I just
    wanted to give you a quick heads up! Other then that, great blog!

    Here is my page 40sl733g.de

  68. Εxcellent post. I was checκing constantly thіs ωеblog and I am impressed!
    Very helpful іnfo speсially the ultimate sеction
    :) I maintain such informаtion a lοt. I used to
    be ѕeeking this certaіn іnformation for a verу lengthy time.
    Thank you and best of luck.

    Chеck out my webѕite - fb tetris battle Cheat codes

  69. It's an awesome article designed for all the web people; they will obtain benefit from it I am sure.

    My web site supplemental resources

  70. My brother recommended I might like this web site. He was once totally right.
    This submit truly made my day. You can not believe
    simply how much time I had spent for this info!
    Thank you!

    Feel free to visit my weblog: myVEGAS Hack