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AMC 12 Mock Exam 4 Problem 12

Problem 12 of 25

A pyramid has a square base with side length 88, and each of its four lateral edges (the edges from the apex to the vertices of the base) has length 99, as shown. What is the volume of the pyramid?

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Solution

The apex is directly above the center OO of the base, since it is equidistant from the four base vertices. The distance from OO to a base vertex is half the base diagonal, 424\sqrt2, so the height is h=92−(42)2=81−32=7h=\sqrt{9^2-(4\sqrt2)^2}=\sqrt{81-32}=7. The volume is 13⋅64⋅7=4483\frac13\cdot64\cdot7=\boxed{\tfrac{448}{3}}. (Forgetting the factor 13\frac13 gives 448448.)

Tagged: solid geometry: square pyramid (base edge 8, lateral edge 9)