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Pyramids are three-dimensional closed surfaces.
At this level, the focus will be on right rectangular pyramids and right triangular pyramids.

A pyramid is a three-dimensional solid with one base, which is a polygon, and lateral faces that are triangles converging to a single point at the top (called the vertex, or apex).

Right Square Pyramid

Regarding right pyramids: • The one and only base of the pyramid is a polygon (no circles or ovals). • 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. • Pyramids usually (but not always) sit on their bases. • All of the pyramids, at this level are right pyramids. • Pyramids are called polyhedra since their faces are polygons.

Pyramids are named for the shapes of their bases.

Triangular Base Triangular Pyramid

Square Base Square Pyramid

Hexagonal Base Hexagonal Pyramid

A right rectangular pyramid has a base that is a rectangle and lateral faces that are triangles. The height (altitude) is measured from the vertex (the top, apex) perpendicular to the base, intersecting at the center of the base. Slant height refers to the height (altitude) of each lateral face. If the base is a square (that is, a regular polygon with all sides (and angles) of equal measure), then the 4 lateral triangles will be congruent to one another.

A right triangular pyramid has a base that is a triangle and e lateral faces that are also triangles. The height is measured in the same manner as in the right rectangular pyramid above. If the base is an equilateral triangle, then the 3 lateral triangles will be congruent to one another (they may be isosceles or equilateral triangles).

All pyramids discussed will be right pyramids.
In a pyramid, that is a right pyramid, the altitude (height) is measured from the vertex perpendicular to base, where the intersection with the base is the center of the base. In a right, regular pyramid, the base is a regular polygon where all of the sides are of the same length.
FYI: Some pyramids can be "slanted" such that the height does not intersect with the center of the base.
These are called oblique pyramids. These will be studied in high school geometry.

The volume of a pyramid is one-third the area of its base times its height.   B = base of pyramid, h = height

We know that V = Bh is the formula for the volume of a prism. By examining the formula of a pyramid, we could state that the volume of a pyramid is exactly one third the volume of a prism with the same base and height.

Justification of formula by "pour and measure":
(For this discussion, our pyramid will be a right square pyramid.)
We can conduct an experiment to demonstrate that the volume of a pyramid is actually equal to one-third the volume of a prism with the same base and height. We will fill a right square pyramid (whose height happens to be equal to the side of its base: b = h) with water. When the water is poured into a prism (a cube) with the same base and height as the pyramid, the water fills one-third of the prism (cube).

• The base of the pyramid is a square, with an area of b2. • The base of the cube is a square, with an area of b2. • The height of the pyramid and the cube is h. • In this example, b = h.

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.)

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