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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.

Surface Area
Volume
Polyhedron
Rectangular Prism
Pythagorean Theorem

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

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

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 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

External credits: Thomas Ledl

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.

V = \frac{1}{3} B h

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.

V = \frac{1}{3} B h

$V = 245760$ , $h = 80$

245760 = \frac{1}{3} B \cdot 80

Solve for

B

$LHS \cdot 3 = RHS \cdot 3$

737280 = B \cdot 80

$LHS / 80 = RHS / 80$

9216 = B

Rearrange equation

B = 9216

The area of the rooftop is

9216

square meters.

Discussion

Surface Area of a Pyramid

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

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 $ℓ$ of the pyramid, which can be found by using the Pythagorean Theorem.

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,

b = \frac{3 0}{2} = 15

feet.

ℓ^{2} = b^{2} + h^{2}

$b = 15$ , $h = 8$

ℓ^{2} = 15^{2} + 8^{2}

Solve for

ℓ

Calculate power

ℓ^{2} = 225 + 64

Add terms

ℓ^{2} = 289

$LHS = RHS$

ℓ = 289

Calculate root

ℓ = 17

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.

p = 4 \cdot 30 = 120

Finally, the lateral area of the pyramid can be found by substituting

p = 120

and

ℓ = 17

into the formula.

L A = \frac{1}{2} p ℓ

$p = 120$ , $ℓ = 17$

L A = \frac{1}{2} (120) (17)

Evaluate right-hand side

Multiply

L A = \frac{1}{2} \cdot 2040

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

L A = \frac{2 0 4 0}{2}

Calculate quotient

L A = 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.

External credits: Eric Baetscher

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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