# Difference between revisions of "1983 AIME Problems/Problem 11"

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== Problem == | == Problem == | ||

The solid shown has a square base of side length <math>s</math>. The upper edge is parallel to the base and has length <math>2s</math>. All edges have length <math>s</math>. Given that <math>s=6\sqrt{2}</math>, what is the volume of the solid? | The solid shown has a square base of side length <math>s</math>. The upper edge is parallel to the base and has length <math>2s</math>. All edges have length <math>s</math>. Given that <math>s=6\sqrt{2}</math>, what is the volume of the solid? | ||

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== Solution == | == Solution == |

## Revision as of 01:40, 21 January 2007

## Problem

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

## Solution

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 .

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 .