Pole jest pewną liczbą, która może zostać przyporządkowana
każdej figurze geometrycznej. W celu
określania pola często stosuje się tzw. sieci kwadratowe, gdzie jednostką jest
kwadrat o boku długości 1. W celu dokładnego obliczenia pola, dzielimy daną
figurę właśnie na takie kwadraty jednostkowe, a w przypadku jeszcze
dokładniejszego przybliżenia dzielimy te kwadraty na jeszcze mniejsze.
Twierdzenie 1 (własności pola)- Pole figury jest
liczbą nieujemną.
- Pola figur przystających, wyznaczone przy tej samej
jednostce, są równe.
- Jeśli figura F składa się z dwóch figur
i
, mających pola i wnętrzami
rozłącznych, to pole figury F jest równe sumie pól figur
i
(przy tej samej jednostce). - Kwadrat o boku jednostkowym ma pole równe 1.
Przykład 1
Policz pole figury przedstawionej poniżej.
![](data:image/png;base64,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)
Możemy policzyć wszystkie kwadraty tworzące tą figurę lub
podzielić ją na dwa prostokąty o polach równych 77 i 30. Pole tej figury jest zatem równe 107.
Zadania do zrobienia
1. Bok kwadratu ABCD ma długość 12 cm. Punkty K,
L, M, N należą odpowiednio
do boków AB, BC, CD, AD. Wiedząc, że |AK| : |KB| = 1 : 2, |BL| : |LC| = 1 : 5, |DM| : |MC| = 7 : 5 oraz |DN| = |AN|, oblicz pole
czworokąta KLMN.
2. Pole jednej kratki jest równe 1.
Oblicz pola poniższych figur
a)
![](data:image/png;base64,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)
b)
![](data:image/png;base64,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)
Odp. a)
20,5
b)
15,5