Quadrilaterals Paper

In this paper, using a computer programming language, we determine the number of integer quadrilaterals that can be formed by using a stick of a given length, say n units, n being a positive integer and also given sum of any two opposite angles less than 180.

Keywords: quadrilateral, area of a quadrilateral.

INTRODUCTION

Generally, one can form a quadrilateral by so many ways. In this paper, we form all possible quadrilaterals, for any such n [see 1,2]. First cut this stick at three places to form 4 parts of the stick. Let a, b, c, d be the lengths of the four parts of the stick and assume that a, b, c, d are positive integers. Hence we have the basic relation a + b + c + d = n. Here number n is given but a, b, c, d are variable numbers. For formation of a

RESULT ANALYSIS: We have to break a stick into three places then we get four parts. Then write all possible combinations and to display those lengths. And we have to display the areas of the quadrilaterals formed by those combinations with a given sum of the opposite angels.

This can be achieved by the following steps.

Step-1: Write all the possible combinations that can be obtained by breaking a stick.

Step-2: Calculated the area of a quadrilateral which can be formed by those combinations with the given sum of opposite angels by Bretschneiders formula.

To illustrate how this work let us perform this process with a stick length of 8 and with sum of opposite angels is 145.

Step 1: Write all the possible combinations that can be obtained by breaking a stick of length 8 into four parts.

(2, 2, 2, 2), (3, 2, 2, 1), (3, 3, 1, 1), (4, 2, 1, 1), (5,1,1,1)

Step 2: : Calculated the area of a quadrilateral which can be formed by those combinations with the given sum of opposite angels is 142.

(2, 2, 2, 2) area = 3.782075.

(3, 2, 2, 1) area = 3.275373

(3, 3, 1, 1) area = 2.836556

More examples are shown below.