Tiêu đề 46 General Polygons.pdf ✅

MSTC 46: General Polygons 1. Polygons Polygons are figures with linear edges and are named based on the number of sides that they have. Name Number of Sides Triangle 3 Quadrilateral 4 Pentagon 5 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 Undecagon 11 Dodecagon 12 Icosagon 20 Hectogon 100 Chiliagon 1000 Myriagon 10000 Megagon 1000000 Other polygons not in this list have their own names but they come up very rarely. For a more specific number of sides n, the term n-gon is acceptable. A diagonal is any segment connecting the vertices of the polygon excluding the sides. To find the number of diagonals in a polygon with n sides, choose any two vertices to form a segment with. # of segments = ( n 2 ) To exclude the number of sides, n must be subtracted, # of diagonals = ( n 2 ) − n = n! 2! (n − 2)! − n = n(n − 1)(n − 2)! 2(n − 2)! − n = n 2 − n 2 − n = n 2 − n − 2n 2 = n 2 − 3n 2 # of diagonals = n(n − 3) 2 If all diagonals of a polygon are inside the polygon, the polygon is called convex. Otherwise, it is called concave.

3. Apothem Similar to triangles, polygons can also have inscribed circles. The radius of a circle inscribed in an equilateral (all sides of equal) is called the apothem. The angle subtended by one side is θ = 360° n , where n is the number of sides. From the figure, tan θ 2 = s 2 a a = s 2 cot 360° n 2 a = 1 2 s cot 180° n 4. Area and Perimeter The area of a shape is the region it bounds. The perimeter of a shape is the length of its boundary. For an equilateral polygon, its area can be found by dividing it into multiple triangles, sharing a vertex at the center of the polygon. From the figure, A = n ( 1 2 sa) A = 1 2 ns ( 1 2 s cot 180° n ) A = 1 4 ns 2 cot 180° n
For a non-equilateral polygon, its area can be computed by dividing it into multiple triangles whose vertices are the vertices of the polygon and adding the areas. For an equilateral polygon with n sides of length s, the perimeter is P = ns From A = n ( 1 2 sa), A = 1 2 Pa Number of Diagonals. # of diagonals = n(n − 3) 2 Summation of Interior Angles. sum of interior angles = 180°(n − 2) Interior Angles of Equiangular Polygons. interior angles = 180°(n − 2) n Exterior Angles of Equiangular Polygons. exterior angles = 360° n Apothem. a = 1 2 s cot 180° n Area of a Polygon. A = 1 4 ns 2 cot 180° n A = 1 2 Pa Perimeter of a Polygon. P = ns