Topics
7. 3D Geometry
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United States of AmericaVirginia
Grade 8

7.02 Volume of square-based pyramids

Lesson
Practice

Volume of square-based pyramids

Volume

The amount a container holds.

Recall that a rectangular prism is a polyhedron that has a congruent pair of parallel rectangular bases and four faces that are rectangles. To calculate the volume of a rectangular prism, we used the formula V=Bh. We can use this idea to help us calculate the volume of pyramids.

Exploration

Drag the sliders to fold the pyramids in to the cube and change the size of the figure.

Loading interactive...

  1. How many pyramids fit into the prism?

  2. Knowing the formula of volume for a prism, how do you think we can find the volume of a pyramid?

The volume of a pyramid is found by using the formula

\displaystyle V=\dfrac{1}{3}Bh
\bm{B}
Area of the base
\bm{h}
Height

The formulas for determining the volume of pyramids and rectangular prisms are related. For pyramids, the volume is \dfrac{1}{3} of the volume of the rectangular prism with the same base and height.

Examples

Example 1

The Pyramid of Giza is a square pyramid, that is 280 Egyptian royal cubits high and has a base length of 440 Egyptian royal cubits. What is the volume of the Pyramid of Giza?

A pyramid.
Worked Solution
Create a strategy

First we find the area of the base, then we can use that to find the volume. Since this solid is a square pyramid, the base is a square.

Apply the idea

Finding the area of the base, we have:

\displaystyle B\displaystyle =\displaystyle \text{side length}^{2}Area of a square
\displaystyle =\displaystyle 440^{2}Substitute the side length
\displaystyle =\displaystyle 193600Simplify

We can now use this to calculate the volume:

\displaystyle V\displaystyle =\displaystyle \dfrac{1}{3}BhVolume of a pyramid
\displaystyle =\displaystyle \dfrac{1}{3} \cdot 193600 \cdot 280Substitute known values
\displaystyle =\displaystyle 18069333 \frac{1}{3}Simplify

So the volume of the Pyramid of Giza is 18 \, 069 \, 333 \dfrac{1}{3} cubic Egyptian royal cubits.

Example 2

A small square pyramid of height 4 \operatorname{ cm} was removed from the top of a large square pyramid of height 8 \operatorname{ cm} forming the solid shown. Find the exact volume of the solid.

A square-based pyramid with a base side length of 4 centimeter. The vertical height of the pyramid is divided into two equal segments of 4 centimeter each, totaling 8 centimeter. Dotted lines indicate the height of the pyramid and the diagonals of the base and a mid-section square. Two dots denote the center of the base and the center of the mid-section square.
Worked Solution
Create a strategy

Subtract the volume of the smaller pyramid from the volume of the larger pyramid.

Apply the idea

Start by calculating the volume of the larger pyramid. We have

\displaystyle V\displaystyle =\displaystyle \dfrac{1}{3}BhVolume of a pyramid
\displaystyle =\displaystyle \dfrac{1}{3} \left(64 \right) \left(8 \right)Substitute B=64 for the area of the base and h=8 for the height of the larger pyramid
\displaystyle =\displaystyle \dfrac{512}{3}Evaluate the multiplication

The volume of the larger pyramid is exactly \dfrac{512}{3} \operatorname{ cm}^{3}.

Now we can calculate the volume of the smaller pyramid. We have

\displaystyle V\displaystyle =\displaystyle \dfrac{1}{3}BhVolume of a pyramid
\displaystyle =\displaystyle \dfrac{1}{3} \left(16 \right) \left(4 \right)Substitute B=16 for the area of the base and h=4 for the height of the smaller pyramid
\displaystyle =\displaystyle \dfrac{64}{3}Evaluate the multiplication

The volume of the smaller pyramid is exactly \dfrac{64}{3} \operatorname{ cm}^{3}.

The volume of the solid after removing the smaller pyramid is \dfrac{512}{3} \operatorname{ cm}^{3} - \dfrac{64}{3} \operatorname{ cm}^{3} = \dfrac{448}{3} \operatorname{ cm}^{3}.

Reflect and check

We could choose to leave the solution in exact form, since the directions do not specify rounding requirements, or we could simplify the solution and round to a measure we find appropriate.

Idea summary

The volume of a pyramid can be found by taking one-third the volume of a prism with the same base area and height. The formula for the volume of a pyramid is given by:

\displaystyle V = \frac{1}{3}Bh
\bm{B}
area of the base
\bm{h}
perpendicular height of the apex from the base

Outcomes

8.MG.2

The student will investigate and determine the surface area of square-based pyramids and the volume of cones and square-based pyramids.

8.MG.2b

Determine the volume of cones and square-based pyramids, using concrete objects, diagrams, and formulas.

8.MG.2c

Examine and explain the relationship between the volume of cones and cylinders, and the volume of rectangular prisms and square based pyramids.

8.MG.2d

Solve problems in context involving volume of cones and square-based pyramids and the surface area of square-based pyramids.

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