MathDB

Each cell of a beehive is constructed from a right regular 6-angled prism, open at the bottom and closed on the top by a regular 3-sided pyramidical mantle. The edges of this pyramid are connected to three of the rising edges of the prism and its apex

T

is on the perpendicular line through the center

O

of the base of the prism (see figure). Let

s

denote the side of the base,

h

the height of the cell and

\theta

the angle between the line

TO

and

TV

. (a) Prove that the surface of the cell consists of 6 congruent trapezoids and 3 congruent rhombi. (b) the total surface area of the cell is given by the formula

6sh - \dfrac{9s^2}{2\tan\theta} + \dfrac{s^2 3\sqrt{3}}{2\sin\theta}

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Problem Statement