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Three-dimensional shapes with a flat base and sides that slant upwards to meet at a point are called pyramids. Known for their stability, they are seen in ancient Egyptian monuments and modern designs like France's Louvre Pyramid. This lesson is focused on the properties of pyramids and the calculation of their volume and surface area.

### Catch-Up and Review

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

Types and Properties of Three-Dimensional Figures:

Geometric Measures of Three-Dimensional Figures:

Challenge

## The Potion Challenge

LaShay and Vincenzo step through the ancient, engraved doors of the Pyramidium House. As they enter, a guidebook enveloped in a mystical glow drifts toward them. It flips open to a page titled The Potion Challenge, marking the beginning of their enthralling quest.

The book lays out a fascinating challenge: LaShay and Vincenzo are to craft a magical potion, transferring it from a prism-shaped container into two smaller pyramid-shaped vessels.

To escape the enigmatic confines of the Pyramidium House, they must unravel a series of enigmatic challenges, starting with this one. Help them answer the following questions to solve this great challenge.

a What is the volume of the prism-shaped container?
b What is the volume of each pyramid-shaped container?
c Is there enough potion to fill both pyramid-shaped containers?
Discussion

## Pyramid

A pyramid is a polyhedron that has a base, which can be any polygon, and faces that are triangular and meet at a vertex called the apex. The triangular faces are called lateral faces. The altitude of a pyramid is the perpendicular segment that connects the apex to the base, similar to the altitude of a triangle.

The length of the altitude is the height of the pyramid. If a pyramid has a regular polygon as its base and congruent, isosceles triangles as its lateral faces, it is called a regular pyramid. The altitude of each lateral face in a regular pyramid is also known as the slant height of the pyramid.

If the apex of the pyramid is over the center of its base, it is called a right pyramid. Otherwise, it is called an oblique pyramid.

Pop Quiz

## Is It a Pyramid?

The applet shows various three-dimensional shapes. Identify if the given shape is a pyramid.

Discussion

## Volume of a Pyramid

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

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

Example

## The Secret Chamber's Pyramid

LaShay and Vincenzo walk into a dimly lit room within the Pyramidium House. They discover a secret chamber decorated with ancient hieroglyphs. At the center of the room awaits a mysterious pyramid emitting a faint glow. It seems that a riddle is inscribed on the walls. They take a closer look.
LaShay and Vincenzo need to calculate the volume of the Guardian Pyramid to unlock the next part of their adventure. The pyramid is a regular triangular pyramid. The base has a side length of feet and a height of feet. The height of the pyramid is feet. Help them solve the mysterious riddle.

### Hint

Find the area of the base using the formula for the area of a triangle. Then, use the formula for the volume of a pyramid to calculate its volume.

### Solution

The base of the pyramid is an equilateral triangle that has a side length of feet and a height of feet. The height of the pyramid is feet.
LaShay and Vincenzo must find the volume of this pyramid. The formula for calculating the volume of a pyramid is the product of the base area and the height of the pyramid divided by or multiplied by a third.
However, before they can apply this formula, they need to determine the base area of the pyramid. The base is a triangle, so they can use the formula for the area of a triangle to find the base area.
Here, is the base of the triangle and is the height of the triangle. In this case, the triangular base has a base of feet and a height of feet. Plug this information into the formula to find the base area
The base area is square feet. With this information and the height of the pyramid being feet, they can use the volume formula to calculate the volume of the pyramid.
The volume of the pyramid is cubic feet. That is great news! LaShay and Vincenzo can proceed to the next challenge now.
Example

## The Puzzle of the Shifting Walls

After successfully solving the riddle of the Secret Chamber's Pyramid, LaShay and Vincenzo enter a room where the walls are constantly moving and reshaping the space. Amidst this changing environment, they notice several geometrically shaped vessels that are morphing in shape and size. One vessel, in the shape of a pentagonal pyramid, catches their attention.

The volume of the vessel, the side length of the base, and its apothem are known. However, the height of the vessel cannot be seen due to the shifting walls. LaShay and Vincenzo need to calculate the height of the vessel to stabilize the room. Help LaShay and Vincenzo solve it!

### Hint

Calculate the perimeter of the base of the vessel. Then, find the area of the base using the formula Finally, substitute the area of the base of the vessel and its volume into the formula for the volume of a pyramid and solve for

### Solution

The volume of a pyramid is given by one third of the product of its base area and its height
In this scenario, the volume of the pyramid is known while the area of the base and the height are unknown. LaShay and Vincenzo need to find the height to stabilize the room. They know the side length of the base and its apothem, which can be used to determine the area of the base using the following formula.
In this formula, is the perimeter of the base and is its apothem. Find the perimeter of the base by multiplying its side length of centimeters by the number of sides
The perimeter of the base is centimeters. Now that the perimeter is known, plug its value into the formula for the area of the base to determine the area of the vessel's base. Recall that the apothem is centimeters.
Finally, substitute the area of the vessel and its volume into the formula for the volume of a pyramid and solve for This will give the height of the vessel.
Once LaShay and Vincenzo determine the height of the vessel, the magic in the room dissipates. Suddenly, a hidden doorway is revealed. They step through the doorway only to stumble upon their next challenge.
Discussion

## Surface Area of a Pyramid

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

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

In this formula, is the perimeter of the base, 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.
Example

## The Secret Code

LaShay and Vincenzo have entered a room with a unique inverted square pyramid structure. The vertex of the pyramid is fixed to the ground, and its base extends upwards. This intriguing structure holds the key to their progress.
The door to the next room is locked with a numerical code. LaShay and Vincenzo discover that the code is the total surface area of the inverted pyramid. The side length of the base of the pyramid is meters, and its slant height is meters. Help them calculate the surface area of the inverted pyramid.

### Hint

Start by finding the perimeter and the base area of the pyramid. Then, substitute this information jointly with the slant height of the pyramid into the formula for the surface area of a pyramid to find the surface area of the inverted pyramid.

### Solution

Start by looking at the given diagram.
The surface area of a pyramid is calculated by using the following formula.
In this formula, is the perimeter of the base, is the base area, and is the slant height. Since the base of the pyramid is a square with a side length of meters, its perimeter can be calculated by adding all sides, or simply multiplying its side four times — that is, by
By similar logic, the area of the base can be determined by squaring its side length.
All of the information needed to apply the formula for the surface area of a pyramid is now known. Substitute the values into the formula to find the surface area of the inverted pyramid.
LaShay and Vincenzo entered the code Nothing happens. Did they make a mistake? Actually, Vincenzo just forgot to hit Enter. The door opens and the two friends can continue exploring and unraveling the mysteries inside of the Pyramidium House.
Example

## The Enchanted Garden Challenge

Vincenzo and LaShay's eyes are widened as they enter the Pyramidium House's Enchanted Garden. A sense of mystery sends a tingle across their forearms. They are greeted by a maze of vibrant, magical plants and pyramidal trellises, each adorned with rare, sparkling flowers seemingly reaching for the skies. These mystical structures hold a crucial secret to their next challenge.
Each trellis is a right regular triangular pyramid with a base that has a side length of meters and a surface area of square meters. They must find the door number that will take them to their final adventure. This door number corresponds to the slant height of the trellises.

### Hint

Start by finding the area and the perimeter of the base. Then, plug these values and the surface area into the formula for the surface area of a pyramid. Solve the obtained equation for the slant height

### Solution

Begin by considering the given diagram of the trellises.
Each trellis is shaped like a pyramid and has a surface area of square meters. The base of the trellis is an equilateral triangle with sides of meters each. The surface area of a pyramid can be found using the following formula.
In this formula, is the base area, is the base perimeter, and is the slant height of the pyramid. The perimeter and area of the base must be found first to calculate the slant height of the trellis.

#### Perimeter of the Base

Since the base is an equilateral triangle with a side length of meters, multiply this side length by to get the perimeter of the base.

#### Area of the Base

The area of a triangle is calculated as half the product of its base and height.
This means that the height of the triangle must be determined first. The height of an equilateral triangle is a perpendicular segment that extends from one vertex to the opposite side, effectively bisecting that side.
When drawing the height of this triangle, two right triangles are formed. Each of these right triangles has a base of meters, which is half the length of the equilateral triangle's side. Substituting and into the Pythagorean Theorem, the missing side of these right triangles can be determined.
Solve for

The value of corresponds to the height of the equilateral triangle. Then, height of the equilateral triangle is meters. Next, substitute the base and height of the equilateral triangle into the formula for the area of a triangle to determine its area.
The area of this triangle is square meters, which corresponds to the area of the base of the pyramid.

#### Calculating the Slant Height

The base area, the base perimeter, and surface area can now be substituted into the formula for the surface area of a pyramid. Then, the obtained equation can be solved for to find the slant height of the trellises.
The slant height measures meters. This indicates that LaShay and Vincenzo must choose door number to access their final challenge — the Celestial Observatory.
Pop Quiz

## Practice Finding the Surface Area and Volume of Pyramids

The applet displays right pyramids with different regular polygonal bases. Use the given information to answer the question, and round the answer to two decimal places if necessary.

Example

## The Celestial Observatory

LaShay and Vincenzo finally reach the last passage of the Pyramidium House — the Celestial Observatory. They are awe-stricken by a pyramid-shaped map that displays the constellations in a three-dimensional layout. To activate the map's prophecy, they must find the volume of the map.
The area of the base of the regular pyramid is square meters and the surface area is square meters. Help the two friends unlock this challenge by determining the volume of the pyramid. Round the volume to the nearest integer.

### Hint

Find the side length and the perimeter of the base. Then substitute these values and the surface area in the formula for the surface area of a pyramid and solve for the slant height. Finally, use the Pythagorean Theorem to find the pyramid's height.

### Solution

The height and base area of the pyramid are required to calculate its volume. Here, only the surface area and base area are given. In that case, start by determining the side length of the base, the perimeter, the slant height, and finally the height of the pyramid. Then the volume of the pyramid can be determined.

#### Side Length of the Base

Since the base is a square, take the square root of the base area to find the side length of the base.
The side length of the base is meters.

#### Perimeter of the Base

Now that the side length of the base is known multiply it by to find its perimeter

#### Slant Height of the Pyramid

The perimeter and area of the base of the pyramid are known, as well as its surface area. Plug in this information in the formula for the surface area of a pyramid and solve for the slant height