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A pyramid is a polyhedron with one base, which is a polygon, and lateral faces that are triangles converging to a single point at the top.
pyramidcrosssection
Similar Cross Sections
(parallel to base)
•
The one and only base of the pyramid is a polygon (no circles or ovals).
•
A pyramid is a polyhedron since its faces are polygons.
•
The lateral faces are always triangles with a common vertex.
•
The vertex of a pyramid (the point, the apex) is not in the same plane as the base.
•
All cross sections of a pyramid parallel to the base will be similar to the base.
pyramidrightside
Right Square Pyramid

Pyramids are named for the shapes of their bases.
pyramid2a
Triangular Base
Triangular Pyramid
pyramid1
Square Base
Square Pyramid
hexagonpyramid
Hexagonal Base
Hexagonal Pyramid

Regarding heights:
The most commonly seen pyramid is a regular pyramid, which is a right pyramid whose base is a regular polygon and whose lateral edges are congruent. In a regular, right pyramid, the height(altitude) is measured from the vertex (the top) perpendicular to the base. The point of intersection with the base will be the center of the base. Slant height refers to the height (altitude) of each lateral face.
 pyramidheights
Regarding lateral faces:
In a regular pyramid, the lateral faces are congruent isosceles triangles.
 pyramidisosceles
Oblique pyramids:
If a pyramid is oblique, its height (altitude) is also measured from the vertex perpendicular to the base. In this case, however, the point of intersection with the base will not be the center of the base. It may even be the case that the height is outside of the pyramid.
obliquepyramid

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Volume of a Pyramid (for both right and oblique pyramids):

The volume of a pyramid is one-third the area of its base
times its height.    
pyramidformula2


Example:
Find the volume of this right, square pyramid.

Solution:
• Find the area of the base.
Since the base is a square, the A = s2.
            A = 62 = 36 sq. units.
• The height = 9 units.
• The volume formula is volumepyramidcone.
            vpy

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Justification of formula by "dissecting a cube":
Since we know the formula for the volume of a cube (V = b3), and the cube is an easy solid with which to work, let's start with the cube and a regular square pyramid.

Regular Pyramid
Square Base
pyamidNB
The base is congruent to the base of the cube.
The height is half the height of the cube.
pyramidcubeNB

How many square pyramids will fit inside the cube when they have the same base as the cube and half of the height?
cubeTB cubeLR cubeFB

A total of 6 pyramids can fit inside this cube, as long as the pyramids' bases are the same as the base of the cube, and the heights of the pyramids are half the height of the cube (b). So the volume of one pyramid is one-sixth the volume of the cube.    volume16

This dissecting a cube into 6 congruent pyramids only works because the height of the pyramid is half the height of the cube. What happens if the height is not half the height of the cube? We will need the formula to contain a variable to deal with the height of the pyramid.

Since h = ½ b, we have 2h = b. Using substitution, we get:formulalong.

This new formula, , says the Volume = one-third (the height) x (area of the base).

This formula can be generalized to the statement:

"the volume of a pyramid is equal to one-third the volume of a prism (V = Bh)
with the same base and height as the pyramid."

So, the volume of a pyramid formula generalizes to formulalong2
formulacage

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Surface Area of a Pyramid:

The surface area of a pyramid is the sum of the area of the base plus the areas of the lateral faces. (The sum of the areas of all the faces.)


Right, Regular, Square Pyramid
pyramidprenet
Surface Area, S, of a regular pyramid:
S = B + ½ps
B = area of pyramid's base
p = perimeter of pyramid's base
s = slant height (height of lateral side)
pyramidnet
A net of the pyramid shows the "surfaces" whose areas, when added, comprise the surface area.

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A frustum is a portion of a solid (usually a pyramid or a cone) which remains after its upper part has been cut off by a plane parallel to its base.

frustumP

To find the volume of a frustum,
find the volume of the entire large pyramid
and subtract the volume of the smaller pyramid being cut off of the top.
The amount that remains, is the the volume of the frustum.
A frustum (in relation to pyramids) can also be called a truncated pyramid.
("truncated" means without its top or end section)


See applications of pyramids under Modeling.


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