How do you find the volume of a composite space figure formed by a rectangular pyramid sitting on top of a rectangular prism?

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Understanding Volumes of Composite Figures

A composite space figure, like a rectangular pyramid resting on a rectangular prism, can be analyzed by calculating the volumes of its individual components and then summing them up. To determine the volume of the entire figure, we must first know the dimensions of both the rectangular prism and the rectangular pyramid.

Step 1: Calculate the Volume of the Rectangular Prism

The volume (V) of a rectangular prism can be calculated using the formula:

V = length × width × height

Where:

  • length is the length of the base of the prism,
  • width is the width of the base of the prism, and
  • height is the height of the prism.

Make sure to plug in the appropriate measurements for the dimensions of the rectangular prism to obtain its volume.

Step 2: Calculate the Volume of the Rectangular Pyramid

The volume (V) of a rectangular pyramid is given by the formula:

V = (1/3) × base area × height

Where the base area can be calculated as:

base area = length × width

And:

  • The length and width are the same dimensions as the base of the prism if the pyramid is directly sitting atop it, and
  • height is the vertical height of the pyramid from its base to the apex.

After calculating the base area, substitute this value into the pyramid’s volume formula to find the volume of the pyramid.

Step 3: Sum the Volumes

Once you have both volumes, the total volume of the composite figure can be determined by simply adding the volume of the rectangular prism and the volume of the rectangular pyramid:

Total Volume = Volume of Prism + Volume of Pyramid

Example Calculation

To illustrate, consider a prism with a length of 4 units, a width of 3 units, and a height of 5 units. The volume of the prism would be:

Vprism = 4 × 3 × 5 = 60 cubic units

Now, for a pyramid with the same base dimensions (4 units and 3 units) and a height of 6 units, the base area is:

base area = 4 × 3 = 12 square units

Thus, the volume of the pyramid would be:

Vpyramid = (1/3) × 12 × 6 = 24 cubic units

Finally, the total volume of the composite figure is:

Total Volume = 60 + 24 = 84 cubic units

This method not only helps to visualize the individual sections of the shape but also provides a clear path to finding the combined volume.

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