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volume of solid whose distance is at most t, from a given parallelepiped

Source: Austrian-Polish 1997

May 24, 2019
geometrysolid geometry3D geometryVolume

Problem Statement

Given a parallelepiped PP, let VPV_P be its volume, SPS_P the area of its surface and LPL_P the sum of the lengths of its edges. For a real number t≥0t \ge 0, let PtP_t be the solid consisting of all points XX whose distance from some point of PP is at most tt. Prove that the volume of the solid PtP_t is given by the formula V(Pt)=VP+SPt+π4LPt2+4π3t3V(P_t) =V_P + S_Pt + \frac{\pi}{4} L_P t^2 + \frac{4\pi}{3} t^3.
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