AMC 8 1998 Problem 6
AMC8 1998 年第 6 题
面积与周长（综合）★★☆☆☆
Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is
点之间水平和垂直方向间距均为一个单位。该多边形围成的面积是多少平方单位？
- A.5
- B.6
- C.7
- D.8
- E.9
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
We could count the area contributed by each square on the 3×3 grid:
Top-left: 0
Top: Triangle with area
Top-right: 0
Left: Square with area 1
Center: Square with area 1
Right: Square with area 1
Bottom-left: Square with area 1
Bottom: Triangle with area
Bottom-right: Square with area 1
Adding all of these together, we get 6 which is the same as
By Pick's Theorem, we get the formula where is the number of lattice points in the interior and is the number of lattice points on the boundary. In this problem, we can see that I=1 and b=12. Substituting gives us . Thus, the answer is .
Notice that the extra triangle on the top with area 1/2 can be placed (like a jigsaw puzzle) at the bottom of the grid where there is a triangular hole, also with area 1/2. This creates a 2*3 rectangle, with a area of 6. The answer is
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