Pyramids
Topic Index | Geometry Index | Regents Exam Prep Center

 

Pyramid

We will be working with regular pyramids unless otherwise stated.

Pyramids are three-dimensional closed surfaces.

   The one base of the pyramid is a polygon and the lateral faces are always triangles with a common vertex.  The vertex of a pyramid (the point, or apex) is not in the same plane as the base. 

Pyramids are also called polyhedra since their faces are  polygons.  

The most common pyramids are regular pyramids.  A regular pyramid has a regular polygon for a base and its height meets the base at its center.  The slant height is the height (altitude) of each lateral face.

In a regular pyramid, the lateral edges are congruent.
Since the base is a regular polygon, whose sides are all congruent, we know that the lateral faces of a regular pyramid are congruent isosceles triangles.

Pyramids are named for the shape of their base.

      
        Triangular pyramid                                    Square pyramid              

The volume of a pyramid is one-third the product of the base area times the height of the pyramid.



 

(Volume of a pyramid:  B = base area,  h = height)


h
= height (altitude) from vertex to base
B = area of base

 

The surface area of a pyramid is the sum of the area of the base plus the areas of the lateral faces.
This simply means the sum of the areas of all faces.
 
The surface area, S, of a regular pyramid can be found using the formula .
B = area of base, p = perimeter of base, s = slant height.

 

A net is a two-dimensional figure
that can be cut out and folded up to
make a three-dimensional solid.

 


The Great Pyramid of Egypt

The Great Pyramid of Khufu, at Giza, Egypt, is 751 feet long on each side at the base, is 450 feet high, and is composed of approximately 2 million blocks of stone, each weighing more than 2 tons.  The maximum error between side lengths is less than 0.1%.

The sloping angle of its sides is 51°51'.  Each side is oriented with the compass points of north, south, east, and west.  Each cross section of the pyramid (parallel to the base) is a square.

 


Artist's rendering of the pyramids.

Until the 19th century, the Great Pyramid at Giza was the tallest building in the world.  At over 4500 years in age, it is the only one of the famous Seven Wonders of the Ancient World that remains standing.

According to the Greek historian Herodotus, the Great Pyramid was built as a tomb for the Pharaoh Khufu.