A Shortcut Formula in Finding the Area between an Even-Sided Regular Polygon (Hexagon to Dodecagon) and an Inscribed and Circumscribed Circle

Abstract

Geometry is a branch of mathematics and considered one of the oldest sciences. It is referred to as the third toughest branch of mathematics. For every 100 students, 88 of them hate mathematics and 12% of them considered geometry as the hardest branch. One topic in geometry is getting the area of the region between a polygon inscribing and/or circumscribing a circle. It takes a lot of time to compute it if this formula is used. This study aimed to derive a shortcut formula for the area of the region. Illustrations were drawn to indicate the region that was being used to compute for its area. Derivation of the formula was conducted by computing for the constant relationship between the measure of the side of the polygon and the radius of the circle. The constructed formulas were in the form A=cs^2, wherein c is the constant while s is the measure of the side of the polygon. Two tests were conducted including equivalence and duration test for the original and derived formulas. The results were analyzed using the Two One-Sided Test (TOST) and t-test. The results of the TOST analysis showed that the values produced by the derived formula and the original formula showed equivalence. For the duration test, the average time (in seconds) for the computation using the original formula and the derived formula were 43.13 and 3.32 respectively. For the results of t-test, it showed that there was a significant difference between the time of computing the area of the region using the derived and original formula. The derived formulas were accurate to solve the mathematical problems about getting the area between an even-sided regular polygon and an inscribed and circumscribed circle. It was also proven that they were able to compute for the answer in a shorter period. This study also showed that there were constant relationships between a circle and a hexagon, octagon, decagon, or dodecagon. These values could be used to generate a constant value which is applicable for all kinds of polygons.