MathDB
Flattened Polyhedron

Source:

January 18, 2009
geometry3D geometrypyramid

Problem Statement

In the figure, polygons A A, E E, and F F are isosceles right triangles; B B, C C, and D D are squares with sides of length 1 1; and G G is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is
<spanclass=latexbold>(A)</span> 1/2<spanclass=latexbold>(B)</span> 2/3<spanclass=latexbold>(C)</span> 3/4<spanclass=latexbold>(D)</span> 5/6<spanclass=latexbold>(E)</span> 4/3 <span class='latex-bold'>(A)</span>\ 1/2\qquad <span class='latex-bold'>(B)</span>\ 2/3\qquad <span class='latex-bold'>(C)</span>\ 3/4\qquad <span class='latex-bold'>(D)</span>\ 5/6\qquad <span class='latex-bold'>(E)</span>\ 4/3 [asy] size(180); defaultpen(linewidth(.7pt)+fontsize(10pt));
draw((-1,1)--(2,1)); draw((-1,0)--(1,0)); draw((-1,1)--(-1,0)); draw((0,-1)--(0,3)); draw((1,2)--(1,0)); draw((-1,1)--(1,1)); draw((0,2)--(1,2)); draw((0,3)--(1,2)); draw((0,-1)--(2,1)); draw((0,-1)--((0,-1) + sqrt(2)*dir(-15))); draw(((0,-1) + sqrt(2)*dir(-15))--(1,0));
label("<spanclass=latexbold>A</span><span class='latex-bold'>A</span>",foot((0,2),(0,3),(1,2)),SW); label("<spanclass=latexbold>B</span><span class='latex-bold'>B</span>",midpoint((0,1)--(1,2))); label("<spanclass=latexbold>C</span><span class='latex-bold'>C</span>",midpoint((-1,0)--(0,1))); label("<spanclass=latexbold>D</span><span class='latex-bold'>D</span>",midpoint((0,0)--(1,1))); label("<spanclass=latexbold>E</span><span class='latex-bold'>E</span>",midpoint((1,0)--(2,1)),NW); label("<spanclass=latexbold>F</span><span class='latex-bold'>F</span>",midpoint((0,-1)--(1,0)),NW); label("<spanclass=latexbold>G</span><span class='latex-bold'>G</span>",midpoint((0,-1)--(1,0)),2SE);[/asy]