# Characteristics of Rectangular Pyramids

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A pyramid is a three-dimensional object consisting of a base and triangular faces that meet at a common vertex. A pyramid is classified as a polyhedron – a three-dimensional shape made of polygons – and it is made up of plane faces, or faces that are level two-dimensional surfaces. A rectangular pyramid possesses specific characteristics that make finding volume and area possible with certain formulas. Different types of pyramids might have different shapes of polygon bases and configurations, but they will all have triangular faces.

## Faces

A rectangular pyramid consists of five faces: one rectangular-shaped base and four triangular-shaped faces. Each triangular face is congruent to the opposite face in a right rectangular pyramid.

For example, on a rectangular pyramid where the edges of the rectangular base are labeled A, B, C and D, the triangular faces on edges A and C are congruent, while those on edges B and D are congruent. Since the triangular faces are attached to the sides of the base, instead of the top or bottom, they are called lateral faces.

## Height

When dealing with rectangular pyramid formulas, the height plays a key role in calculating area, volume, and many other metrics of the pyramid. Oftentimes, the height can be related to the length of a slant and volume through fractions and exponents, as we will see later.

## Vertices

A rectangular pyramid consists of five vertices, or points at which edges intersect. One vertex is located at the top of the pyramid, where the four triangular faces meet. The remaining four vertices are located on each corner of the rectangular base. According to MathsTeacher.com, the pyramid becomes a right pyramid when the top vertex is "directly above the center of the base."

For many of the following calculations and definitions, we assume a right rectangular pyramid to help simplify the calculations. When the vertex does not lie above the center, it is known as an oblique rectangular pyramid. In such a case, the opposite triangular faces are not always equal, and it makes the geometry much more complex when dealing with oblique pyramids.

## Edges

A rectangular pyramid consists of eight edges, or sharp sides "formed by the intersection of two surfaces," as defined by Word Net Web. Four edges are located on the rectangular base, while four edges form the upward slope to create the top vertex of the pyramid.

## Surface Area

The surface area of a rectangular pyramid depends on the area of the base and the area of each of the four lateral faces. To find this, we need to break the problem into two parts. First, we can calculate the area of the base which is simply length (l) times width (w):

l \cdot w

The area for the faces of the pyramid is more complex. The formula, given base length (l), base width (w), and height of the pyramid (h), can be written as:

l\sqrt{ \left(\frac{w}{2}\right)^2 + h^2} \ + w\sqrt{ \left(\frac{l}{2}\right)^2 + h^2}

The formula for total surface area of a closed right rectangular pyramid is the sum of the areas:

l \cdot w \ + \ l\sqrt{ \left(\frac{w}{2}\right)^2 + h^2} \ + w\sqrt{ \left(\frac{l}{2}\right)^2 + h^2}

#### Tips

• The square root terms in this formula use the Pythagorean Theorem to convert the vertical height to slant height and calculate the area for each lateral surface.

## Volume of a Rectangular Pyramid

When given a rectangular pyramid, the volume of the pyramid can be given in terms of the height (h), length (l), and width of the rectangular base (w):

V = \frac{l\cdot w\cdot h}{3}

While the three in the denominator may seem out of place, it can be explained with the help of a regular rectangular prism. The volume of a rectangular prism is length times width time height; when compared to a rectangular pyramid with the same base, the negative space around the pyramid actually adds up to two more pyramids of the same volume. So the volume of the rectangular prism is equivalent to three times the volume of a rectangular pyramid, this is why we divide by three.

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