The volume formula for any pyramid relates the volume, V, to the area of the base, B, and the height of the prism, h.

The way that you might choose to approach a volume problem involving a pyramid depends on the shape of the base. In this section, you will focus on rectangular pyramids, or pyramids with rectangular bases.

Cheops Pyramid in Egypt

Source: Flickr - Gaspa - Giza, la piramide grande, Francesco Gasparetti, Wikimedia Commons

The Cheops Pyramid in Egypt is a square pyramid, or a pyramid whose base is a square. Ancient Egyptians constructed the Cheops Pyramid.

Kukulcan Pyramid in Mexico

Source: Kukulcan, Chichén Itzá, Kyle Simourd, Wikimedia Commons

The Kukulcan Pyramid at Chichen Itza, Mexico, is also an example of a square pyramid. Ancient Mayans constructed the Kukulcan Pyramid.

pyramid of fluorite and a blue laser

Source: Fluorite pyramid and blue laser, Mike Lewinski, Flickr Commons

A pyramid of fluorite, a mineral found throughout the world, can be used to refract the light from a blue laser and create special visual effects!

Rectangular pyramids appear frequently in art and architecture. Because they are so common, it is important to be able to calculate the volume of a rectangular pyramid.

A problem-solving process may help you when you encounter problems related to a rectangular pyramid.

four-step problem solving process

Consider the problem shown below.

Interactive exercise. Assistance may be required. Click on the Begin button to begin the interactive below, and see how Sabrina used a problem-solving model to solve this problem.

Pause and Reflect

If you were solving the same problem as Sabrina, what solution strategy would you have selected?

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Check Your Answer

One way to solve the problem is to use what you know about rectangular prisms, and calculate the volume of a prism with congruent bases and heights. Then, you can divide the volume of the prism by 3.Close Pop Up

Sabrina used rounding to check her answer for reasonableness. What is another way to know that your answer is reasonable?

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Check Your Answer

Use compatible numbers. Since the volume formula involves 1 over 3 1 3 , round all dimensions up to the next multiple of 3 to calculate a maximum volume. Then, round all dimensions down to the next multiple of 3 to calculate a minimum volume. When you do so, you know that the actual volume should be between 72 cubic inches and 243 cubic inches.Close Pop Up

Practice

  1. Calculate the volume of the paperweight below that is in the shape of a rectangular prism.
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    Need a hint?

    Use the formula, V = 1 over 3 1 3 Bh. What are the dimensions of the base of the paperweight?Close Pop Up
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    Check Your Answer

    First, determine the area of the base, B.
    B = lw = (51 over 2 1 2 in.)(4 in.) = 22 in.2

    Next, determine the volume of the pyramid, V.
    V = 1 over 3 1 3 Bh = 1 over 3 1 3 (22 in.2)(33 over 4 3 4 in.) = 271 over 2 1 2 in.3

    Close Pop Up
  2. temple near Delhi, India

    Source: A Hindu temple at Chhattarpur temple complex, Delhi, Alicia Nijdam, Wikimedia Commons

    The Chhatarpur Temple near Delhi, India, has a central temple that is in a shape approximating a square pyramid. The pyramid portion of the temple has a height of about 90 feet and a base edge of about 45 feet. What is the approximate volume of the pyramid portion of the temple?

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    Need a hint?

    Use the formula, V = 1 over 3 1 3 Bh.
    What are the dimensions of the pyramid portion of the temple?Close Pop Up

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    Check Your Answer

    First, determine the area of the base, B.
    B = s2 = (45 ft)2 = 2,025 ft2

    Next, determine the volume of the pyramid, V.
    V = 1 over 3 1 3 Bh = 1 over 3 1 3 (2,025 ft2)(90 ft) = 60,750 ft3

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  3. An attic space on top of a building is in the shape of a rectangular pyramid. The dimensions of the attic space are shown in the figure below.

    What is the volume of the interior of the attic space?

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    Need a hint?

    Use the formula, V = 1 over 3 1 3 Bh. What are the dimensions of the base of the attic space?Close Pop Up
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    Check Your Answer

    First, determine the area of the base, B.
    B = lw = (17.5 ft)(32 ft) = 560 ft2

    Next, determine the volume of the pyramid, V.
    V = 1 over 3 1 3 Bh = 1 over 3 1 3 (560 ft2)(24 ft) = 4,480 ft3

    Close Pop Up