Mathematics Dictionary Dr. K. G. Shih

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Polygons

Subjects

Symbol Defintion Example : LT = Less Than

GE 05 00 | - GE 05 01 | - Various names of polygons GE 05 02 | - Angles GE 05 03 | - Symmetrical axese of polygon GE 05 04 | - Adothem of regualar polygon GE 05 05 | - Area of regular hexagon GE 05 06 | - Change pentagon to an equal area triangle GE 05 07 | - GE 05 08 | - GE 05 09 | - GE 05 10 | -

Answers

Q01. Various names of polygon Defintions Study | Names of polygons Diagrams Subjects | Definition : Program 05 01 Method to use Start the program Click Menu Click section 5 in upper box Click program 01 in lower box and will see one poygon Click replot or press Enter key on heyboard and will see next Go to Begin Q02. Angles Triangle Sum of internal angles = 180 degrees = pi radians Sum od external angles = 360 degrees = 2*pi radians Square Sum of internal angles = 180 degrees = pi radians Sum od external angles = 360 degrees = 2*pi radians Polygons Sum of internal angles = (n-2)*180 degrees = (n-2)*pi radians Sum od external angles = 360 degrees = 2*pi radians Go to Begin Q03. Symmetrical axese Equilateral triangle Three bisector are the three symmetrical axes Hence it has three symmetrical axese Square It has two diagonals which are two symmetrical axes It has two bisectors to bisect the opposite sides which are two symmetrical axes Hence it has four symmetrical axes Regular polygon has n sides (n = odd) It has n diagonals which are symmetrical axes Hence it has n symmetrical axes Regular polygon has n sides (n = even) It has n/2 diagonals which are symmetrical axes It has n/2 bisectors to bisect the opposite sides which are symmetrical axes Hence it has n symmetrical axes Go to Begin GE 05 04. Adothem of regular polygon Defintion Distance from Center of regular polygon to the side is called adothem It is biector of the side of polygon Find adothem of regular polygon Let the each side is a Let the center be O One side is AB = a Let OD is the adothem and D is mid point of AB Hence AD = a/2 Since triangle OAB is equiangular triangle, hence each angle = 60 degrees Hence and OA = a Right triangle AOD, we have OD^2 = OA^2 - AD^2 (Pythagorean law) Hence OD = Sqr(a^2 - (a/2)^2) = (Sqr(3)*a)/2 Go to Begin GE 05 05. Area of regular hexagon Area of equilateral triangle Regular hexagon has six equilateral triangle Area of each triangle Let side AB = a Adothem = OD = (Sqr(3)*a)/2 Area of triangle OAB = AB*OD/2 = (Sqr(3)*a)/4 Regular hexagon inscribed a circle of radius a. Find area bounded by polygon and circle The area = area of circle - area of hexagon = pi*a^2 - (Sqr(3)*a)/4 Go to Begin GE 05 06. Change a pentagon to an equal area triangle 1. Change pentagon to equal area quadrilteral Let pentagon be ABCDE Draw EQ parallel to AD Produce CD and meet EQ at P Area of qudrilateral ABCP is equal area of pentagon ABCDE The proof is similar as example in GE 06 10 2. Change quadrilateral ABCP to an equal area triangle The method and proof are given in GE 06 06 Go to Begin GE 05 07. Answer Go to Begin GE 05 08. Answer Go to Begin GE 05 Q09. Answer Go to Begin GE 05 10. Answer Go to Begin GE 05 00. Outlines Angles of polygons Sum of internal angles is (n - 2)*pi Sum of external angles is 2*pi Adothem of regular polygon It is the bisector of each side from the center Adothem of regular hexagon is (Sqr(3)*a)/2 where a is one side Go to Begin

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