11.5: Pyramids
What if you wanted to know the volume of an Egyptian Pyramid? The Khafre Pyramid is the second largest pyramid of the Ancient Egyptian Pyramids in Giza. It is a square pyramid with a base edge of 706 feet and an original height of 407.5 feet. What was the original volume of the Khafre Pyramid? After completing this Concept, you'll be able to answer this question.
Watch This
CK12 Foundation: Chapter11PyramidsA
Brightstorm: Surface Area of Pyramids
Brightstorm: Volume of Pyramids
Guidance
A pyramid has one base and all the lateral faces meet at a common vertex. The edges between the lateral faces are lateral edges. The edges between the base and the lateral faces are called base edges. If we were to draw the height of the pyramid to the right, it would be off to the left side.
When a pyramid has a height that is directly in the center of the base, the pyramid is said to be regular. These pyramids have a regular polygon as the base. All regular pyramids also have a slant height that is the height of a lateral face. Because of the nature of regular pyramids, all slant heights are congruent. A nonregular pyramid does not have a slant height.
Surface Area
Using the slant height, which is usually labeled \begin{align*}l\end{align*}
Surface Area of a Regular Pyramid: If \begin{align*}B\end{align*}
If you ever forget this formula, use the net. Each triangular face is congruent, plus the area of the base. This way, you do not have to remember a formula, just a process, which is the same as finding the area of a prism.
Volume
Recall that the volume of a prism is \begin{align*}Bh\end{align*}
Investigation: Finding the Volume of a Pyramid
Tools needed: pencil, paper, scissors, tape, ruler, dry rice or sand.
 Make an open net (omit one base) of a cube, with 2 inch sides.
 Cut out the net and tape up the sides to form an open cube.
 Make an open net (no base) of a square pyramid, with lateral edges of 2.45 inches and base edges of 2 inches. This will make the overall height 2 inches.
 Cut out the net and tape up the sides to form an open pyramid.
 Fill the pyramid with dry rice. Then, dump the rice into the open cube. How many times do you have to repeat this to fill the cube?
Volume of a Pyramid: If \begin{align*}B\end{align*}
The investigation showed us that you would need to repeat this process three times to fill the cube. This means that the pyramid is onethird the volume of a prism with the same base.
Example A
Find the slant height of the square pyramid.
Notice that the slant height is the hypotenuse of a right triangle formed by the height and half the base length. Use the Pythagorean Theorem.
\begin{align*}8^2+24^2&=l^2\\
64+576&=l^2\\
640&=l^2\\
l&= \sqrt{640} = 8 \sqrt{10}\end{align*}
Example B
Find the surface area of the pyramid from Example A.
The surface area of the four triangular faces are \begin{align*}4\left ( \frac{1}{2} bl \right ) =2(16)\left ( 8 \sqrt{10} \right )=256 \sqrt{10}\end{align*}
Example C
Find the volume of the pyramid.
\begin{align*}V=\frac{1}{3} (12^2)12=576 \ units^3\end{align*}
Watch this video for help with the Examples above.
CK12 Foundation: Chapter11PyramidsB
Concept Problem Revisited
The original volume of the pyramid is \begin{align*}\frac{1}{3} (706^2)(407.5)\approx 67,704,223.33 \ ft^3\end{align*}
Vocabulary
A pyramid is a solid with one base and lateral faces that meet at a common vertex. The edges between the lateral faces are lateral edges. The edges between the base and the lateral faces are base edges.
A regular pyramid is a pyramid where the base is a regular polygon. All regular pyramids also have a slant height, which is the height of a lateral face.
Surface area is a twodimensional measurement that is the total area of all surfaces that bound a solid. Volume is a threedimensional measurement that is a measure of how much threedimensional space a solid occupies.
Guided Practice
1. Find the area of the regular triangular pyramid.
2. If the lateral surface area of a square pyramid is \begin{align*}72 \ ft^2\end{align*}
3. Find the area of the regular hexagonal pyramid below.
4. Find the volume of the pyramid.
5. Find the volume of the pyramid.
6. A rectangular pyramid has a base area of \begin{align*}56 \ cm^2\end{align*}
Answers:
1. The area of the base is \begin{align*}A=\frac{1}{4} s^2 \sqrt{3}\end{align*}
\begin{align*}B & =\frac{1}{4} 8^2 \sqrt{3}=16 \sqrt{3}\\
SA& = 16 \sqrt{3}+\frac{1}{2} (24)(18)=16 \sqrt{3}+216 \approx 243.71\end{align*}
2. In the formula for surface area, the lateral surface area is \begin{align*}\frac{1}{2} Pl\end{align*}
\begin{align*}\frac{1}{2} nbl & = 72 \ ft^2\\
\frac{1}{2} (4) b^2 & = 72\\
2b^2 & = 72\\
b^2 & =36\\
b & = 6\end{align*}
Therefore, the base edges are all 6 units and the slant height is also 6 units.
3. To find the area of the base, we need to find the apothem. If the base edges are 10 units, then the apothem is \begin{align*}5 \sqrt{3}\end{align*}
\begin{align*}SA& = 150 \sqrt{3}+\frac{1}{2}(6)(10)(22)\\
& = 150 \sqrt{3}+660 \approx 919.81 \ units^2\end{align*}
4. In this example, we are given the slant height. For volume, we need the height, so we need to use the Pythagorean Theorem to find it.
\begin{align*}7^2+h^2&=25^2\\
h^2&=576\\
h&=24\end{align*}
Using the height, the volume is \begin{align*}\frac{1}{3} (14^2 )(24)=1568 \ units^3\end{align*}
5. The base of this pyramid is a right triangle. So, the area of the base is \begin{align*}\frac{1}{2} (14)(8)=56 \ units^2\end{align*}
\begin{align*}V=\frac{1}{3} (56)(17) \approx 317.33 \ units^3\end{align*}
6. The formula for the volume of a pyramid works for any pyramid, as long as you can find the area of the base.
\begin{align*}224&=56h\\
4& = h\end{align*}
Practice
Fill in the blanks about the diagram below.

\begin{align*}x\end{align*}
x is the ___________.  The slant height is ________.

\begin{align*}y\end{align*}
y is the ___________.  The height is ________.
 The base is _______.
 The base edge is ________.
Find the area of a lateral face and the volume of the regular pyramid. Leave your answer in simplest radical form.
Find the surface area and volume of the regular pyramids. Round your answers to 2 decimal places.
 A regular tetrahedron has four equilateral triangles as its faces. Find the surface area of a regular tetrahedron with edge length of 6 units.
 Using the formula for the area of an equilateral triangle, what is the surface area of a regular tetrahedron, with edge length \begin{align*}s\end{align*}
s ?
For questions 1315 consider a square with diagonal length \begin{align*}10\sqrt{2} \ in\end{align*}
 What is the length of a side of the square?
 If this square is the base of a right pyramid with height 12, what is the slant height of the pyramid?
 What is the surface area of the pyramid?
A regular tetrahedron has four equilateral triangles as its faces. Use the diagram to answer questions 1619.
 What is the area of the base of this regular tetrahedron?
 What is the height of this figure? Be careful!
 Find the volume. Leave your answer in simplest radical form.

Challenge If the sides are length \begin{align*}s\end{align*}
s , what is the volume?
A regular octahedron has eight equilateral triangles as its faces. Use the diagram to answer questions 2022.
 Describe how you would find the volume of this figure.
 Find the volume. Leave your answer in simplest radical form.

Challenge If the sides are length \begin{align*}s\end{align*}
s , what is the volume?
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Show More 
lateral edges
Edges between the lateral faces of a prism.lateral faces
The nonbase faces of a prism.Cone
A cone is a solid threedimensional figure with a circular base and one vertex.Pyramid
A pyramid is a threedimensional 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 threedimensional object.Cavalieri's Principle
States that if two solids have the same height and the same crosssectional 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.Image Attributions
Here you'll learn how to calculate the surface area and volume of a pyramid.