Algebraic estimation of apothems in regular polygons using linear regression and bootstrap optimization. - Line, Sergio Roberto Peres; Martin, Walter Gonçales (C.Q.D. - Revista Eletrônica Paulista de Matemática, 2025)
Estimating the apothem of regular polygons may be required in academical and real-life situations such as for calculating the area of regular polygons and the volume of prisms and pyramids. The calculus of apothem requires the use and knowledge of the tangent function. In the present article we describe an algebraic approximation procedure for estimation of the apothem of any regular polygon given only the length of one side and the number of sides. An initial approximation was obtained based on nearly linear correlation between the product of polygon sides number and length with the squared root of the sum of the squared apothem and squared hypotenuse of one of the inner right-angled triangles of the polygon. In a second step the bootstrap method was to use to obtain the coefficients of a second-degree polynomial. The resulting equation predicts the apothem with a minimum accuracy of 0.9997. In polygons of 3, 4, 6 and with more than 25 sides the accuracy was higher than 0.9999.