How to find the area of a polygon?
- There is no universal formula, because an arbitrary polygon
not a "rigid" figure, its shape and area can be changed by changing
angles. Only by triangulation.
- In general, at work, I was confused with my own deduction of the square ... and wondered how much for more number of sides and the same perimeter - the area increases.
I decided to derive the formula of the polygon S = Pr ^ 2 / (4N * tg (180 / n)), where Pr is the perimeter, and N is the number of sides. So I found out that N * tg (180 / N) -gt; PI, ie, tends to the number of PI, for infinitely high N, thereby increasing the area to the area of the ideal circle.
Transforming S = Pr ^ 2 / 4PI.
- divide it into triangles and add up the sum of the areas of all triangles
- In the first case it turns out to be a triangle, and one can use one of the formulas: S = 1 / 2 * a * н, where a is the side, and the height to it; S = 1 / 2 * a * in * sin (A), where a, to the sides of the triangle, A is the angle between the known sides; S = (p * (p - a) * (p - c) * (p - c)), where, on the side of the triangle, to the already designated two, p is the semiperimeter, that is, the sum of all three sides divided by two.
- Only along the lengths of the sides it is impossible. The coordinates of the vertices can be:
- Any or any regular polygon? ?
The area of any regular polygon can be calculated from formula
S = n * a ^ 2 / (4tg 180 / n) where a is the length of the polygon side, n is the number of sides
- Can you draw a triangle or quadrilateral in which ALL corners are stupid?
- no such. For polygons with number of sides there are formulas, but they include angles or diagonals. for example, the brahmagupta formula for the 4-square or the formulas 5 and 6-gons
(the link is blocked by the decision of the project administration)
- However, if we are talking about an arbitrary convex ("no concavities inward") polygon, then it is not necessary that the corners are only obtuse; any convex polygon can have three sharp inner corners.
There is no universal formula for the area of arbitrary convex polygons. The only possible way was indicated by Ivantrs. For a regular polygon - M. Mamishev.