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Linenn - Jeux et Mathematiques - <br /> <b>Deprecated</b>: Non-static method plxUtils::strCheck() should not be called statically in <b>/homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.show.php</b> on line <b>127</b><br /> Théorème de Pick - linenn


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Théorème de Pick

Par
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jp, le
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vendredi 15 janvier 2010 à
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01:10 |
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Géométrie | Page principale Jeux et Mathématiques

Activités pour enfants.
Faites découvrir sur des figures simples et progressivement, l'aire d'un polygone du plan et le théorème de Georg Alexander Pick qui date de 1899.

Téléchargez et imprimez au format A4, à la page http://jeux-et-mathematiques.davalan.org/divers/a4/, les plans pointés qui vous permettent de dessiner les figures de cette page ou d'autres.

Grille carrée

Pick
Carrés

    Tracez, comme sur le schéma, un polygone tel que
  1. tous ses sommets soient des points de la grille
  2. sa frontière soit une ligne fermée simple (qui ne se recoupe pas) connexe (d'un seul tenant, en un seul morceau)
  3. i soit le nombre de points intérieurs (en bleu) et b le nombre de points du bord (ou de la frontière, en rouge)
Alors l'aire (en carreaux unités) de ce polygone est

A = i + b/2 -1

Grille triangulaire

Pick
Carrés

Si l'unité est un petit losange unité, la formule est bien entendu la même que précédemment.

A = i + b/2 -1

Si l'unité est celle d'un petit triangle, alors la valeur est double :

T = 2i + b - 2

Pick
Carrés

Téléchargez et imprimez les feuillets au format A4 à la page http://jeux-et-mathematiques.davalan.org/divers/a4/.

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