The look on the inner corner, only to see:
And angles of a triangle is 180 °;
It is quadrilateral and interior angle 360 °;
Pentagonal and interior angle is 540 °;
…………
n-gon and the interior angle is (n-2) × 180 °.
I found a formula to calculate the angle sum. Formula appeared a number of sides n.
If you look at the outer corner of it?
The outer corner is triangular and 360 °;
Are quadrilateral and the outer corner 360 °; and the outer corner of the pentagon is 360 °;
…………
Any n-gon are outer corners and 360 °.
This brings a variety of situations with a very simple conclusion profiles up. With a constant independent of n and n replaces the formulas relating to find a more general rule.
Ant contemplated circle (FIG. 1) on the boundary of the polygon. After each vertex, will change the direction of its advance time, change the angle of the exterior angle is exactly at the apex. Climb a circle, back to the original place, the same direction and when starting, and the amounts of angular change of course is exactly 360 °.
figure 1
See it that way, not only to the "polygon exterior angle equal to 360 °" This universal law find an explanation on the intuitive and immediately our eyes towards the wider world.
A convex closed curve - oval, not to mention what the outer corner and inner corner and. But when ants crawling above, it is also constantly changing direction. It lap climb, angle change amounts of still and 360 ° (FIG. 2).
figure 2
"Exterior angles of 360 °", this rule applies to closed curve! However, the narrative together, use the "change of direction and the amount of" instead of "the outer corner and" work.
For concave polygon, it should "sum of the direction change amount" to "the direction of the algebraic sum of the amount of change" (FIG. 3). Wish conventions: counterclockwise rotation angle is a positive angle, clockwise angle is a negative angle. When the ants in the illustrated quadrilateral recess boundary crawling in A₁, A₂, A₄, the change in direction by the angle formed by a positive angle: ∠1, ∠2, ∠4; A₃ in place, the direction changing the angle formed by a negative angle: ∠3. If you look carefully calculated, against the four corners negative, algebra, and precisely 360 °.
image 3
Is the situation on the plane, the situation on the surface and how kind it says is it? The Earth is round. If you have been moving forward along the equator, you can circle around the Earth come full circle. But measuring where you are going on the ground, but it did not change at any time. In other words: you around the equator week, the sum of the amount of change in direction is 0 °!
Smaller circle, you walk around in the room, the amount of change of direction still appears to be 360 °.
不大不小的圈子又怎么样呢？如果让蚂蚁沿着地球仪上的北回归线绕一圈，它自己感到的（也就是在地球仪表面上测量到的）方向的改变量应当是多少呢？
用一个圆锥面罩着北极，使圆锥面与地球仪表面相切的点的轨迹恰好是北回归线（图4）。这样，蚂蚁在球面上的方向的改变量和在锥面上方向的改变量是一样的。把锥面展开成扇形，便可以看出，蚂蚁绕一圈，方向改变量的总和，正好等于这个扇形的圆心角（图5）：
图4
图5
要弄清楚这里面的奥妙，不妨看看蚂蚁在金字塔上沿正方形爬一周的情形（图6）。
图6
它的方向在拐角处改变了多大角度？把金字塔表面摊平了一看便知：在B处改变量是180°-（∠1+∠2）；绕一圈，改变量是
4×180°-（∠1+∠2+∠3+∠4+∠5+∠6+∠7+∠8）
=∠AOB+∠BOC+∠COD+∠DOA
这个和，正是锥面展形后的“扇形角”（图7）！
图7
早在2000多年前，欧几里德时代，人们就已经知道三角形内角和是180°。到了19世纪，德国数学家、被称为“数学之王”的高斯，在对大地测量的研究中，找到了球面上由大圆弧构成的三角形内角和的公式。又经过几代数学家的努力，直到1944年，陈省身教授找到了一般曲面上封闭曲线方向改变量总和的公式（高斯—比内—陈公式），把几何学引入了新的天地。由此发展出来的“陈氏类”理论，被誉为划时代的贡献， 在理论物理学上有重要的应用。
从普通的、众所周知的事实出发，步步深入、推广，挖掘出广泛适用的深刻规律。从这里显示出数学家透彻、犀利的目光，也表现了数学家穷追不舍、孜孜以求的探索真理的精神。
作者：张景中