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Surface Area and Volume of Pyramids

Surface area and volume of solids with a base and lateral faces that meet at a common vertex.

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Pyramids and Cones

The volume of a pyramid is given by \begin{align*}V=\frac{A_{Base} \cdot h}{3}\end{align*}. How does this formula help you find the formula for the volume of a cone?

Pyramids and Cones

Recall that a pyramid is a solid with a polygon base and triangular lateral faces that meet in a vertex. Pyramids are named by their base shape.

You have seen the formula for the volume of a pyramid before.

\begin{align*}Pyramid: V=\frac{A_{Base} \cdot h}{3}\end{align*}

Where does this formula come from? Recall that to find the volume of a prism or a cylinder, you need to find the area of the base and multiply by the height.

\begin{align*}Prism \text{ or } Cylinder: V=A_{Base} \cdot h\end{align*}

The difference between these two formulas is the division by 3. The key to understanding where the 3 comes from is remembering Cavalieri's principle and investigating a square based pyramid.

The two square pyramids below are each constructed within cubes of the same size. The pyramid on the left has a vertex at the center of a face of the cube. The pyramid on the right has a vertex at one of the vertices of the cube. Is the volume of each pyramid the same?

Because each pyramid is constructed within the same cube, the heights of the pyramids are the same and the areas of their bases are the same. When a plane parallel to the base of the pyramid is constructed through the center of each cube, the cross sections of each pyramid are the same.

Each cross section is a square. The length of the side of the square is half the length of an edge of the original cube, since the plane was constructed through the middle of the cube.

Because cross sections are the same area, heights are the same, and bases are the same, these pyramids must have the same volume due to Cavalieri's principle.

How many square based pyramids congruent to the one below would it take to fill the cube? Can you visualize this?

It will take exactly 3 congruent pyramids to fill the cube. The image below shows each pyramid being added to the cube.

Use the answers from the previous problems to explain why the volume of a pyramid is \begin{align*}V=\frac{A_{Base} \cdot h}{3}\end{align*}.

The volume of a rectangular prism is \begin{align*}A_{Base} \cdot h\end{align*}. This is because \begin{align*}A_{Base}\end{align*} gives the volume of one “layer”, and multiplying by the height scales that base volume by the number of “layers” of the prism. 

Three congruent pyramids fit inside the cube in Example B, so the volume of each pyramid must be \begin{align*}\frac{1}{3}\end{align*} the volume of the cube. Therefore, the volume of a pyramid is \begin{align*}\frac{A_{Base} \cdot h}{3}\end{align*}. Remember that pyramids with the same base area and height will have the same volume due to Cavalieri's principle, so both of the pyramids below will have a volume of \begin{align*}\frac{A_{Base} \cdot h}{3}\end{align*}.

Note: This is an informal argument for the formula for the volume of a pyramid. A rigorous derivation of the formula that considers pyramids of any base shape will be developed in calculus.

Examples

Example 1

Earlier, you were asked how does the formula \begin{align*}V=\frac{A_{Base} \cdot h}{3}\end{align*} help you find the formula for the volume of a cone. 

A cone is essentially a pyramid with a circular base. The volume of a pyramid is given by \begin{align*}V=\frac{A_{Base} \cdot h}{3}\end{align*}. Since the area of the base of a cone is \begin{align*}\pi r^2\end{align*}, the formula for the volume of a cone is \begin{align*}V=\frac{\pi r^2 h}{3}\end{align*}.

Example 2

The area of the base of the pyramid below is \begin{align*}40 \ cm^2\end{align*}. What is the volume of the pyramid?

 \begin{align*}V=\frac{A_{Base} \cdot h}{3}=\frac{(40)(6)}{3}=80 \ cm^3\end{align*}

Example 3

The apothem of a regular polygon is a perpendicular segment from the center point of the polygon to the midpoint of one of its sides (see image below). Find the volume of a pyramid with a height of 10 inches and a regular pentagon base with an apothem of 1 inch.

\begin{align*}V=\frac{A_{Base} \cdot h}{3}\end{align*}. To find the area of the base, divide the pentagon into five congruent triangles. The apothem is the height of each of these triangles. Use trigonometry to find the base length of each of these triangles.

\begin{align*}\tan 36 &= \frac{\frac{b}{2}}{1} \rightarrow b=2 \tan 36 \rightarrow b=1.453 \ in\\ A_{triangle} &= \frac{(1.453)(1)}{2}=0.7265 \ in^2\\ A_{pentagon} &= (0.7265)(5)=3.6325 \ in^2\\ V &= \frac{(3.6325)(10)}{3}=12.11 \ in^3 \end{align*}

Example 4

Find the volume of a cone with a height of 10 inches and a radius of 1 inch.

\begin{align*}V=\frac{\pi r^2 h}{3}=\frac{10 \pi}{3} \ in^3 \approx 10.47 \ in^3\end{align*}

Review

1. Explain the connections between a prism and a pyramid. Why do you divide by three when calculating the volume of a pyramid?

2. Explain the connections between a cone and a cylinder. Why do you divide by three when calculating the volume of a cone?

Find the volume of each solid based on its description.

3. A cone with a diameter of 4 inches and a height of 12 inches.

4. A pyramid with a height of 15 inches and a regular hexagon base with an apothem of 4 inches.

5. A cone with a radius of 8 centimeters and a height of 15 centimeters.

6. A square based pyramid with a vertex in the center of the square such that each triangular face has a base of 12 inches and a height of 10 inches.

An hourglass is created by placing two congruent cones inside of a cylinder with the same base area. The radius is 5 inches and the height of the cylinder is 20 inches.

7. Find the volume of one of the cones.

8. Find the volume of the cylinder.

9. Find the volume of the space between the cones and the cylinder.

10. You want to fill one of the cones with a thick liquid. If one cup of liquid has a volume of approximately \begin{align*}14.44 \ in^3\end{align*}, how much liquid will you need to fill one of the cones?

11. A cone and a square pyramid have the same volume and height. The volume of each solid is \begin{align*}100 \ cm^3\end{align*}. If the radius of the cone is 2.82 centimeters, what is the length of a side of the base of the pyramid?

12. The ratio of the area of the red circle to the area of the base is 1:9. If the height of the cone is 15 inches, what is the length of \begin{align*}\overline{AB}\end{align*}?

13. The height of the cone below is 10 inches. Find the length of \begin{align*}\overline{AB}\end{align*}.

14. A tetrahedron is a pyramid with four congruent equilateral faces. If the area of each of the faces of a tetrahedron is \begin{align*}9 \sqrt{3} \ in^2\end{align*}, what is the volume of the tetrahedron?

15. The length of each side of the triangular faces making up a tetrahedron is \begin{align*}s\end{align*}. What is the volume of the tetrahedron in terms of \begin{align*}s\end{align*}?

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 9.2. 

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Vocabulary

Cone

A cone is a solid three-dimensional figure with a circular base and one vertex.

Pyramid

A pyramid is a three-dimensional object with a base that is a polygon and triangular faces that meet at one vertex.

Vertex

A vertex is a point of intersection of the lines or rays that form an angle.

Volume

Volume is the amount of space inside the bounds of a three-dimensional object.

Cavalieri's Principle

States that if two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.

Base Edge

The base edge is the edge between the base and the lateral faces of a prism.

Slant Height

The slant height is the height of a lateral face of a pyramid.

Apothem

The apothem of a regular polygon is a perpendicular segment from the center point of the polygon to the midpoint of one of its sides.

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