Question

一个凸六边形最多可以有多少个锐角？

What is the largest number of acute angles that a convex hexagon can have?

$2$

$3$

$4$

$5$

$6$

答案：B

知识点：角度和极限情形界定

难度评级：1310

小提示：

凸多边形的外角和为 $360^\circ$ 。

The exterior angles of a convex polygon sum to $360^\circ$

大提示：

一个锐内角对应一个大于 $90^\circ$ 的外角。

An acute interior angle gives an exterior angle greater than $90^\circ$

解答：

每个锐内角对应一个大于 $90^\circ$ 的外角。因为凸多边形的外角和为 $360^\circ$ ，其中最多有三个外角能大于 $90^\circ$ 。这个上界可以达到：取一个等边六边形，其外角交替为 $100^\circ$ 和 $20^\circ$ 。它的边方向分成两组三条，每组内相差 $120^\circ$ ，所以图形闭合；内角交替为 $80^\circ$ 与 $160^\circ$ 。因此最大可能数为三。

所以正确答案是 B。

Each acute interior angle corresponds to an exterior angle greater than $90^\circ.$ Since the exterior angles of a convex polygon sum to $360^\circ,$ at most three of them can exceed $90^\circ.$ This bound is attainable: take a hexagon with equal side lengths and exterior angles alternating $100^\circ$ and $20^\circ.$ Its edge directions split into two triples $120^\circ$ apart, so it closes, and its interior angles alternate between $80^\circ$ and $160^\circ.$ Hence the largest possible number is three.

Thus, the correct answer is B.

1999 AMC 12 第 7 题

答案：B

难度评级：1310

小提示：

大提示：

解答：

其他年份的第 7 题