Interactive: A Cube within a Hemisphere

# A cube is placed within a hemisphere so that the corners of the cube touch the surface of the hemisphere; Observe numerically the ratio of the volume of the cube and the volume of the hemisphere. # Lauri Ruotsalainen, 2010 x, y, z = var("x,y,z") @interact def _(size = slider(0.5, 1, label="The Edge Length x:")): hemisphere_graph = implicit_plot3d(x^2+y^2+z^2==1, (x, -1, 1), (y, -1, 1), (z, 0, 1), color="green", opacity=0.4) cube_graph = cube(size=size, opacity=0.9, color="red", frame_thickness=1).translate((0, 0, size/2)) surface_graph = plot3d(0, (x, -1.2, 1.2),(y, -1.2, 1.2), color="lightblue", opacity=0.6) show(hemisphere_graph + cube_graph + surface_graph, aspect_ratio=1) V_c = size^3 V_hs = 4*pi*1^3/6 html("$\\text{Volume of the Cube: }V_{cube} = x^3 = %s^3 = %s" % (N(size, digits=5), N(V_c, digits=5))) html("$\\text{Volume of the Hemisphere: }V_{hemisphere} = \\frac{4\pi r^3}{3}:2 = \\frac{4\pi 1^3}{3}:2 = %s$" % N(V_hs, digits=5)) html("$\\text{Ratio: }V_{cube}/V_{hemisphere} = %s/%s = %s$" % (N(V_c, digits=5), N(V_hs, digits=5), N(V_c/V_hs, digits=5)))