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I would like to know if there is a term kind of "straight pyramid"? I mean, is there a word for a pyramid such that the line segment from the apex to the center of the base is perpendicular to the base.

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    The quality of mercy is not strained – Will Jagy Feb 05 '15 at 21:47
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    @WillJagy That sounded pretty deep. – Daniel W. Farlow Feb 05 '15 at 21:49
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    It droppeth as the gentle rain from heaven Upon the place beneath. It is twice blessed: It blesseth him that gives and him that takes. 'Tis mightiest in the mightiest. It becomes The thronèd monarch better than his crown. – Will Jagy Feb 05 '15 at 21:55
  • @WillJagy Merchant of Venice baby. Right on. – Daniel W. Farlow Feb 05 '15 at 21:55
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    My father had this complicated joke about Mercy Hospital in Australia, which served a beverage with bits of herbs and nuts still in it, and named after a local animal; the conclusion was that The Koala Tea of Mercy is not strained. – Will Jagy Feb 05 '15 at 21:58

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I believe the term you are looking for is right.

From mathwords.com:

right pyramid: A pyramid that has its apex aligned directly above the center of the base.