What is a Rectangular Pyramid? - Definition & Formula

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  • 0:05 Defining a Rectangular Pyramid
  • 1:02 Finding Volume:…
  • 2:08 Finding Surface Area
  • 4:44 Surface Area When the…
  • 6:20 Lesson Summary
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Lesson Transcript
Instructor: Eric Istre

Eric has taught high school mathematics for more than 20 years and has a master's degree in educational administration.

This lesson is about detailing the characteristics of a rectangular pyramid and how to find its surface area and volume. Examples of each are included along with the step by step calculations and sample pyramid diagrams to help with visualization.

Defining a Rectangular Pyramid

Imagine that you lived several thousand years ago. The pharaoh has asked you to find out how much treasure he can possibly store inside one of his pyramids. He also wants to seal the surface of the pyramid so that it will last forever, and he wants to know how much surface he will need to seal. He will not accept failure! What will you do? How are you going to be able to do what he wants?

The first and most important thing to know is exactly what is the shape of a pyramid. Pyramids are named by the type of base that they have. A rectangular pyramid has a rectangle for a base with triangles for the other faces. Sometimes the rectangle at the base is actually a square. The pyramids in Egypt are rectangular pyramids. A pyramid culminates at a peak, sometimes called the apex. A pyramid can be a right pyramid in which the apex is directly over the center of the base or an oblique pyramid in which the apex is not directly over the center of the base.

Finding Volume: Rectangular Pyramids

Think about the first request the pharaoh made. The pharaoh wants to know how much space is inside the pyramid. This is known as the volume. Volume is the amount of space inside a three-dimensional object, like a pyramid, and it must always be given in cubed units. The Great Pyramid in Egypt is 455 feet tall and each side of the base is 756 feet long, making it a square pyramid with the length and width both being the aforementioned 756 feet.

The volume of a pyramid, right or oblique, has the following formula:

Volume = (length * width * height) / 3

To solve for the volume, plug the value of each of the dimensions into the formula and calculate.

Volume = (756 ft * 756 ft * 455 ft) / 3

Volume = 86,682,960 ft3

The pharaoh will have to have a lot of treasure to fill all of that space up!

Finding Surface Area

The pharaoh's second request was what would it take to seal the pyramid? To answer this question, the surface area of the pyramid needs to be determined. Surface area, or the total area of all the outside faces of the object, can be a bit trickier to calculate than volume. The formula for surface area is:

Surface area = (1/2 * base perimeter * slant height) + base area

The slant height is needed to solve this equation, which was not required for the volume calculation. If the slant height is unknown, use the Pythagorean theorem to figure it out. The Pythagorean theorem is:

a2 + b2 = c2

For example, without the slant height of the pharaoh's pyramid, construct a right triangle with the pyramid's height and half the length of the base as the legs of the right triangle. The longest side, or hypotenuse, of the triangle would be your slant height. Using the Pythagorean theorem, follow these steps:

c2 = (455 ft)2 + (756 / 2 ft)2

c2 = 349,909 ft2

c = √349,909ft2 = 591.5 ft

That's how to find slant height in case it's not given.

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