Question 1191007
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Could you help me with this problem, please? I greatly appreciate your time and effort.
The cube shown has a side length of 6. Find the volume of the pyramid that has triangle BDE as its base and A as its vertex.
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<pre>
Let the length of an edge of the cube be  "a".


To calculate the volume, consider triangle ABE as a base of a pyramid BDEA 

and edge AD as the height of the pyramid (same as its altitude).



Use the formula for the volume of a pyramide

    volume = 1/3 of the base area times the height = {{{(1/3)*(1/2)*a*a*a}}} = {{{(1/6)*a^3}}} = {{{(1/6)*6*6*6}}} = 36 cubic units.



Thus the volume of the pyramid BDEA is 1/6 of the volume of the cube  {{{a^3}}} = {{{6^3}}} = 216 cubic units.



<U>ANSWER</U>.  The volume of the pyramid BDEA is  36  cubic units, which is  1/6  of the volume of the cube.
</pre>

Solved.



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<U>Comment from student</U> : &nbsp;&nbsp;Hello, &nbsp;I was going through your solution again and realized you are asking 
to consider the triangle &nbsp;ABE &nbsp;as a base of a pyramid but the question specifies that the base of pyramid is triangle &nbsp;BDE. 
Could you please help me taking into consideration that base is triangle &nbsp;BDE &nbsp;and not &nbsp;ABE?



<U>My response</U> : &nbsp;&nbsp;In this problem, &nbsp;you can consider &nbsp;(and can call) &nbsp;ANY &nbsp;FACE &nbsp;with vertex &nbsp;A &nbsp;of the pyramid &nbsp;BDEA &nbsp;as a base of the pyramid.


Then the corresponding altitude will be the edge perpendicular to this base.