Online calculator to calculate the total and lateral surface areas and the volume of a regular polygon frustum given the altitude and sides of the polygons or the radii of the circumcircle of the polygons..
In figure A) below is shown the regular polygon frustum with similar regular polygons as bases. The upper base has side of length \( a \) and the lower base has side of length \( b \) and the two polygons have the same number of sides.
In figure B) is shown one face of the frustum which is an isosceles trapezoid where whose altitude is \( H \) which is also called the slant height of the frustum.
In figure C) is shown a top view of the frustum including the circumcircle of each polygon and with radii \( r_1 \) for the upper polygon and \( r_2 \) for the lower polygon.
In figure D) is shown the trapezoid made up of one edge of the frustum of length \( c \), the height of the frustum \( h \) and the radii \( r_1 \) and \( r_2 \).
Let \( A_1 \) be the area of the upper (regular polygon) base and \( A_2 \) be the area of the lower (regular polygon) base given by 1
\( A_1 = \dfrac{1}{4} \; n \; a^2 \cot\left(\dfrac{180^{\circ}}{n}\right) \) and \( A_2 = \dfrac{1}{4} \; n \; b^2 \cot\left(\dfrac{180^{\circ}}{n}\right) \)
The volume \( V \) of the regular polygon frustum is given by 1
\( V = \dfrac{1}{3} (A_1+A_2 +\sqrt{A_1 A_2}) h\)
The area \(A_L\)of the lateral surface is given by
\(A_L = \dfrac{1}{2} n (a+b) H \) ,
where
\( H^2 = c^2 - \left(\dfrac{b-a}{2}\right)^2 \)
\( c^2 = h^2 + (r_2-r_1)^2 \)
\( r_1 = \dfrac{a}{2 \sin(\alpha/2)}\) and \( r_2 = \dfrac{b}{2 \sin(\alpha/2)}\)
\( \alpha = \dfrac{360^{\circ}}{n} \)