1983 AIME Problems/Problem 11

Problem

The solid shown has a square base of side length . The upper edge is parallel to the base and has length . All other edges have length . Given that , what is the volume of the solid?

size(180);
import three; pathpen = black+linewidth(0.65); pointpen = black;
currentprojection = perspective(30,-20,10);
real s = 6 * 2^.5;
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
D(A--B--C--D--A--E--D); D(B--F--C); D(E--F);
MP("A",A);MP("B",B);MP("C",C);MP("D",D);MP("E",E,N);MP("F",F,N);
 (Error making remote request. Unknown error_msg)

Solution 1

First, we find the height of the figure by drawing a perpendicular from the midpoint of to . The hypotenuse of the triangle is the median of equilateral triangle one of the legs is . We apply the Pythagorean Theorem to find that the height is equal to .

size(180);
import three; pathpen = black+linewidth(0.65); pointpen = black; pen d = linewidth(0.65); pen l = linewidth(0.5);
currentprojection = perspective(30,-20,10);
real s = 6 * 2^.5;
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
triple Aa=(E.x,0,0),Ba=(F.x,0,0),Ca=(F.x,s,0),Da=(E.x,s,0);
D(A--B--C--D--A--E--D); D(B--F--C); D(E--F); 
D(B--Ba--Ca--C,dashed+d);D(A--Aa--Da--D,dashed+d);D(E--(E.x,E.y,0),dashed+l);D(F--(F.x,F.y,0),dashed+l);
D(Aa--E--Da,dashed+d); D(Ba--F--Ca,dashed+d);
MP("A",A);MP("B",B);MP("C",C);MP("D",D);MP("E",E,N);MP("F",F,N);MP("12\sqrt{2}",(E+F)/2,N);MP("6\sqrt{2}",(A+B)/2);MP("6",(3*s/2,s/2,3),ENE);
 (Error making remote request. Unknown error_msg)

Next, we complete the figure into a triangular prism, and find the area, which is .

Now, we subtract off the two extra pyramids that we included, whose combined area is .

Thus, our answer is .

Solution 2

Extend and to meet at , and and to meet at . now, we have a regular tetrahedron , which has twice the volume of our original solid. This tetrahedron has side length . Using the formula for the volume of a regular tetrahedron, which is , where S is the side length of the tetrahedron, the volume of our original solid is: