Pyramid

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This is an AoPSWiki Word of the Week for Sep 6-12

A pyramid is a 3-dimensional geometric solid. It consists of a base that is a polygon and a vertex not on the plane of the polygon. The edges of the pyramid are the sides of the polygonal base together with line segments connected the vertex of the pyramid to each vertex of the polygon.

The volume of a pyramid is given by the formula $\frac13bh$, where $b$ is the area of the base and $h$ is the height.

Some well-known pyramids include the tetrahedron, which has a triangle for its base. (A regular tetrahedron has all edges of equal length, and is one of the Platonic solids). Another is the regular square pyramid. Two of these with their bases joined form an octahedron, which is another Platonic solid.

If the base of the pyramid has $\displaystyle n$ sides, then the pyramid has $\displaystyle 2n$ edges, $\displaystyle n+1$ vertices, and $\displaystyle n+1$ faces (of which $\displaystyle n$ are triangular, and the remaining one is the base).

Problems

Introductory

Intermediate

  • In an unit cube, the vertices are sliced off with 8 cuts such that each of the faces now is a regular octagon. Find the volume of the resulting solid. (2007 AMC 12A, #15?)
  • In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. Find the ratio of the volume of the smaller tetrahedron to that of the larger. (2003 AIME II, #4)
  • A square pyramid with base ABCD and vertex E has eight edges of length 4. A plane passes through the midpoints of AE, BC, and CD. Find the area of the plane's intersection with the pyramid. (2007 AIME I, #13)
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