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Extending to Three Dimensions

Properties of Pyramids

Pyramid-shaped structures can be found in different countries such as Egypt and Mexico. In this lesson, the formulas for the volume and surface area of a pyramid will be used to explore some of these real-life locations.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Relating the Volumes of Pyramids and Cubes

Paulina enjoys creating origami objects. After she made the three congruent origami pyramids shown below, she noticed that they could form a cube.

Three pyramids
If the volume of a cube is the perfect cube of its side length, what is the volume of each pyramid? Can any three identical pyramids form a cube?

Discussion

Definition of a Pyramid

A pyramid is a polyhedron in which one face (the base) can be any polygon, and the other faces (the lateral faces) are triangles that meet at a common vertex called the vertex of the pyramid. Similar to an altitude of a triangle, the altitude of a pyramid is a segment from the vertex that is perpendicular to the plane of the base.

Pyramid

The length of the altitude is called the height of a pyramid. A pyramid is regular if its base is a regular polygon and its lateral faces are congruent, isosceles triangles. In a regular pyramid, the length of the altitude of each lateral face is sometimes referred to as the slant height of a pyramid.

Height and Slant Height of Pyramid

If the vertex of a pyramid is not over the midpoint of its base, it is called an oblique pyramid.

Oblique Pyramid

When the base area and the height of a pyramid are known, its volume can be calculated.

Rule

Volume of a Pyramid

The volume of a pyramid is one third of the product of its base area and height.

Base and height of a pyramid

The base area B is the area of the polygon opposite the vertex of the pyramid, and the height h is measured perpendicular to the base.

Proof

Informal Justification

Consider a pyramid and a prism that have the same base area and height.

A pyramid can be modeled as a stack of prisms. The sum of the volumes of the small prisms will be greater than the pyramid's volume. However, as the number of prisms increases and they get thinner, the sum will approximate the volume of the pyramid.

Furthermore, the ratio of the sum of the volumes of each small prism to the volume of the prism will be approximated to

Number of Layers
4
16
64
256
1024
4096

Therefore, the volume of a pyramid is one third of the prism with the same base area and height.

By Cavalieri's principle, two pyramids have the same volume — regardless of where the vertex of the pyramid is located — so long as the pyramids have the same base and the same height.

Example

Finding the Volume of a Pentagonal Pyramid

Tadeo is getting ready to go camping. He has a pyramid-shaped tent with a regular pentagonal base.

Pentagonal pyramid tent
The tent has a height of 1.6 meters and its base has side lengths of 1.4 meters. Find the volume of the tent. Round the answer to the two decimal places.

Hint

Find the perimeter p and the apothem a of the base. Then substitute their values in the formula to find the area of the base. To find the apothem, use the tangent ratio of half the central angle of regular pentagons.

Solution

Start by finding the area of the base of the tent. Then, the formula for the volume of a pyramid will be used.

Area of the Base

The base of the tent is a regular pentagon with side lengths 1.4 meters. Therefore, its perimeter p is 5 times 1.4.
Now, draw the apothem of the pentagonal base. The measure of each central angle of a regular pentagon is and is bisected by the apothem.
Apothem of pentagonal base

The apothem is perpendicular to any side of the polygon and bisects it. As a result, a right triangle with a leg of 0.7 meters is formed.

Right triangle
The value of a can be written in terms of the tangent ratio of
Finally, to find the area of the base B, substitute and p=7 into the area formula for regular polygons.
Evaluate right-hand side
B=3.37
The area of the base is about 3.37 square meters.

Volume of the Tent

Now that the base area and height are known, substitute these values into the formula for the volume of a pyramid.
Evaluate right-hand side
The volume of the tent is about 1.80 cubic meters.

Example

Finding Height and Volume of the Walter Pyramid

Dominika goes to watch a basketball match at the Walter Pyramid in Long Beach, California. She is amazed by the appearance of the arena. She finds out that the arena was built on a square base with side lengths of 345 feet.

Walter Pyramid

If the Walter Pyramid is a right pyramid and its slant height is about 258 feet, help Dominika answer the following questions.

a What is the height of the pyramid? Round the answer to the nearest foot.
b What is the volume of the pyramid?

Hint

a To find the height of the pyramid, use the Pythagorean Theorem.
b The volume of a pyramid is one third of the product of its base area and height.

Solution

a To find the height of the pyramid, the Pythagorean Theorem will be used. Since the pyramid is a right pyramid, the vertex of the pyramid is over the center of its base. Subsequently, the distance between the center and a side is half its side length,
Pyramid
As can be seen, the slant height of the pyramid is the hypotenuse of the right triangle ABC. Now, the height of the pyramid, or AB, can be found using the Pythagorean Theorem.
AC2=BC2+AB2
2582=172.52+AB2
Solve for AB
66564=29756.25+AB2
36807.75=AB2
AB2=36807.75
The height of the pyramid is approximately 192 feet.
b Since the base of the pyramid is a square, its area is the square of its side length.
B=a2
B=3452
B=119025
The base is 119025 square feet. Now that the base area and height are known, the formula for the volume of a pyramid can be used to find the volume.
Evaluate right-hand side
V=7617600
The volume of the pyramid is 7617600 cubic feet.

Example

Finding the Base Area of the Slovak Radio Building

Architects might enjoy turning things upside down. Maya is interested in architecture and follows some online magazines about it. After reading an article about the Slovak Radio Building in Bratislava, Slovakia, she wonders about the area of the square rooftop.

Slovak Radio Building

If the height of the building is 80 meters and its volume is about 245760 cubic meters, find the area of the rooftop of the building.

Hint

The volume of a pyramid is one third of the product of its base area and height.

Solution

The formula for the volume of a pyramid will be used to find the area of the square rooftop.
In the formula, B is the area of the base and h is the height of the pyramid. Since h=80 meters and V=245760 cubic meters, the area of the base B can be found by substituting these values into the formula.
Solve for B
737280=B80
9216=B
B=9216
The area of the rooftop is 9216 square meters.

Example

Finding the Volume of a Composite Solid That Includes a Pyramid

Designers and inventors also benefit from pyramids. An object attracts Mark's attention on a school trip to a maritime museum. The guide explains that it is called a deck prism, which was invented to illuminate the cabins below deck before electric lighting. Mark buys a replica of the deck prism, which is composed of a prism and pyramid, each with a regular hexagonal base.

Deck prism
Find the volume of the deck. Round the answer to the two decimal places.

Hint

The formula for the area of a regular hexagon with side lengths a is Apply the rounding in the last step.

Solution

The deck is composed of two solids:

  • a regular hexagonal prism with an edge length of 3.5cm and a height of 2cm, and
  • a regular hexagonal pyramid with an edge length of 3cm and a height of 4cm.

Therefore, the volume of the deck is the sum of the volumes of the above solids. The volume of each solid will be found one at a time.

Prism's Volume

The base of the prism is a regular hexagon with a side length of 3.5cm. Recall the formula for the area of a regular hexagon with side lengths a.
Substitute 3.5 for a into the formula and evaluate its value.
Evaluate right-hand side
Now the volume can be found by multiplying the base area by the height.

Pyramid's Volume

The base of the pyramid is a regular hexagon with side lengths 3cm. Recall the formula for the area of a regular hexagon with side lengths a.
Substitute 3 for a into the formula and evaluate its value.
Evaluate right-hand side
Now the volume can be found. Recall that the volume of a pyramid is one third of the product of its base area and height.
Evaluate right-hand side

Deck Prism's Volume

The sum of the volumes found will give the volume of the deck prism.
V=V1+V2
Evaluate right-hand side
The deck prism has a volume of about 94.83 cubic centimeters.

Discussion

Surface Area of a Pyramid

The surface area of a pyramid is just as important as its volume.

Consider a regular pyramid with an edge length s and a slant height

A square pyramid

The surface area of a regular pyramid can be calculated using the following formula.

In this formula, p is the perimeter of the base, B is the base area, and is the slant height. In the case that the pyramid is not regular, the area of each lateral face has to be calculated one by one and then added to the area of the base.

Proof

A regular pyramid's surface area can be seen as two separate parts: the lateral area and the base.
Since for a regular pyramid, its base can be any n-sided regular polygon, the lateral area is the sum of the area of n congruent triangles. For example, consider a regular hexagonal pyramid with an edge length s and a slant height Take a look at its net.
Net of a pyramid
As can be seen, the area of each lateral face is Therefore, the lateral area will be 6 times because the lateral faces are congruent.
Notice that 6s is the perimeter of the base, which can be denoted by p. Then, the lateral area can be expressed as follows.
Therefore, the formula for the surface area is obtained.

Example

Finding the Lateral Area of a Roof

Maya's father decides to cover the roof of their house with waterproof insulation material. Maya's father asks Maya to calculate how many square feet of insulation material is needed.

House

The roof is a square pyramid with a height of 8 feet and base side length of 30 feet. Help Maya calculate the area.

Hint

To find the slant height, use the Pythagorean Theorem. Note that only the lateral area of the pyramid is needed.

Solution

Maya needs to calculate the lateral area of the square pyramid. To do so, she first needs to calculate the slant height of the pyramid, which can be found by using the Pythagorean Theorem.

House
The height h of the pyramid is the distance between the vertex and the center of the base, and b is half the base side length. Therefore, feet.
Solve for
Since a negative value does not make sense in this context, only the principal root is considered. Therefore, the slant height is 17 feet. Next, the perimeter of the base will be found. Since the base is a square, its perimeter p is 4 times the base side length.
Finally, the lateral area of the pyramid can be found by substituting p=120 and into the formula.
Evaluate right-hand side
LA=1020
The amount of material needed to cover the roof is 1020 square feet.

Pop Quiz

Practice Finding the Surface Area and Volume of Pyramids

The applet shows some right pyramids with different regular polygonal bases. Use the given information to answer the question. If necessary, round the answer to two decimal places.

Closure

Pyramid-Shaped Structures in the Real World

Pyramid-shaped structures can be seen in many countries around the world. The Aztecs, Mayans, and ancient Egyptians were some of the earliest civilizations to build pyramid-shaped structures. The Aztecs and Mayans built their pyramids with tiered steps and a flat top, whereas the pyramids built by the Egyptians fit the mathematical definition of a pyramid.
Civilizations used these pyramids for different purposes. For example, the pyramids in Mexico were used as places of human sacrifice. Conversely, the Egyptian pyramids were built to be the tombs of pharaohs.
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