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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

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?

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b What is the volume of each pyramid-shaped container?

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c Is there enough potion to fill both pyramid-shaped containers?

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Discussion

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

The applet shows various three-dimensional shapes. Identify if the given $3D$ shape is a pyramid.

Discussion

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

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.

$V=31 Bh$

Example

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. ### Hint

### Solution

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 $5$ feet and a height of $4.33$ feet. The height of the pyramid is $12$ feet. Help them solve the mysterious riddle.

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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.

The base of the pyramid is an equilateral triangle that has a side length of $5$ feet and a height of $4.33$ feet. The height of the pyramid is $12$ feet.
The base area $B$ is $10.825$ square feet. With this information and the height of the pyramid being $12$ feet, they can use the volume formula to calculate the volume of the pyramid.
The volume of the pyramid is $43.3$ cubic feet. That is great news! LaShay and Vincenzo can proceed to the next challenge now.

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 $B$ and the height $h$ of the pyramid divided by $3$ or multiplied by a third.

$V=31 B⋅h $

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. $B=2bh $

Here, $b$ is the base of the triangle and $h$ is the height of the triangle. In this case, the triangular base has a base of $5$ feet and a height of $4.33$ feet. Plug this information into the formula to find the base area $B.$ $B=2b⋅h $

SubstituteII

$b=5$, $h=4.33$

$B=25⋅4.33 $

Multiply

Multiply

$B=221.65 $

UseCalc

Use a calculator

$B=10.825$

$V=3B⋅h $

SubstituteII

$B=10.825$, $h=12$

$V=310.825⋅12 $

Multiply

Multiply

$V=3129.29 $

CalcQuot

Calculate quotient

$V=43.3$

Example

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!

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Calculate the perimeter $p$ of the base of the vessel. Then, find the area of the base using the formula $A=21 a⋅p.$ 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 $h.$

The volume of a pyramid is given by one third of the product of its base area $B$ and its height $h.$
Finally, substitute the area of the vessel and its volume into the formula for the volume of a pyramid and solve for $h.$ 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.

$V=31 B⋅h $

In this scenario, the volume of the pyramid is known while the area of the base $B$ and the height $h$ 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. $A=21 a⋅p $

In this formula, $p$ is the perimeter of the base and $a$ is its apothem. Find the perimeter of the base by multiplying its side length of $8$ centimeters by the number of sides $5.$
$Perimeter of the Base5⋅8=40cm $

The perimeter of the base is $40$ 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 $5.5$ centimeters. $B=21 a⋅p$

SubstituteII

$a=5.5$, $p=40$

$B=21 ⋅5.5⋅40$

Multiply

Multiply

$B=21 ⋅220$

MoveRightFacToNumOne

$b1 ⋅a=ba $

$B=2220 $

CalcQuot

Calculate quotient

$B=110$

$V=31 B⋅h$

SubstituteII

$V=1100$, $B=110$

$1100=31 ⋅110⋅h$

DivEqn

$LHS/110=RHS/110$

$10=31 h$

RearrangeEqn

Rearrange equation

$31 h=10$

MultEqn

$LHS⋅3=RHS⋅3$

$h=30$

Discussion

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

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

$SA=21 pℓ+B$

Example

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.
### Hint

### Solution

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 $15$ meters, and its slant height is $22$ meters. Help them calculate the surface area of the inverted pyramid.

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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.

Start by looking at the given diagram.
LaShay and Vincenzo entered the code $885.$ 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.

The surface area of a pyramid is calculated by using the following formula.

$SA=21 pℓ+B $

In this formula, $p$ is the perimeter of the base, $B$ is the base area, and $ℓ$ is the slant height. Since the base of the pyramid is a square with a side length of $15$ meters, its perimeter can be calculated by adding all sides, or simply multiplying its side four times — that is, $15$ by $4.$
$Perimeter of the Base4⋅15=60m $

By similar logic, the area of the base can be determined by squaring its side length.
$Area of the Base15_{2}=225m_{2} $

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.
$SA=21 pℓ+B$

SubstituteValues

Substitute values

$SA=21 (60)(22)+225$

Multiply

Multiply

$SA=21 ⋅1320+225$

MoveRightFacToNumOne

$b1 ⋅a=ba $

$SA=21320 +225$

CalcQuot

Calculate quotient

$SA=660+225$

AddTerms

Add terms

$SA=885$

Example

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.
### Hint

### Solution

Each trellis is a right regular triangular pyramid with a base that has a side length of $6$ meters and a surface area of $117+93 $ 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.

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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 $ℓ.$

Begin by considering the given diagram of the trellises. #### Perimeter of the Base

Since the base is an equilateral triangle with a side length of $6$ meters, multiply this side length by $3$ 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.
The value of $a$ corresponds to the height of the equilateral triangle. Then, height $h$ of the equilateral triangle is $33 $ 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 $93 $ square meters, which corresponds to the area of the base $B$ 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 $ℓ$ to find the slant height of the trellises.
The slant height measures $13$ meters. This indicates that LaShay and Vincenzo must choose door number $13$ to access their final challenge — the Celestial Observatory.

Each trellis is shaped like a pyramid and has a surface area of $117+93 $ square meters. The base of the trellis is an equilateral triangle with sides of $6$ meters each. The surface area $SA$ of a pyramid can be found using the following formula.

$SA=21 pℓ+B $

In this formula, $B$ is the base area, $p$ is the base perimeter, and $ℓ$ 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 Base6⋅3=18meters $

$A=21 bh $

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 $3$ meters, which is half the length of the equilateral triangle's side. Substituting $c=6$ and $b=3$ into the Pythagorean Theorem, the missing side $a$ of these right triangles can be determined.
$c_{2}=a_{2}+b_{2}$

SubstituteII

$c=6$, $b=3$

$6_{2}=a_{2}+3_{2}$

▼

Solve for $a$

CalcPow

Calculate power

$36=a_{2}+9$

SubEqn

$LHS−9=RHS−9$

$27=a_{2}$

RearrangeEqn

Rearrange equation

$a_{2}=27$

SqrtEqn

$LHS =RHS $

$a=±27 $

SplitIntoFactors

Split into factors

$a=±9⋅3 $

SqrtProd

$a⋅b =a ⋅b $

$a=±9 ⋅3 $

CalcRoot

Calculate root

$a=±33 $

$a>0$

$a=33 $

$A=21 b⋅h$

SubstituteII

$b=6$, $h=33 $

$A=21 ⋅6⋅33 $

Multiply

Multiply

$A=21 ⋅183 $

MoveRightFacToNumOne

$b1 ⋅a=ba $

$A=2183 $

CalcQuot

Calculate quotient

$A=93 $

$SA=21 p⋅ℓ+B$

SubstituteValues

Substitute values

$117+93 =21 ⋅18⋅ℓ+93 $

SubEqn

$LHS−93 =RHS−93 $

$117=21 ⋅18ℓ$

MoveRightFacToNumOne

$b1 ⋅a=ba $

$117=218ℓ $

CalcQuot

Calculate quotient

$117=9ℓ$

RearrangeEqn

Rearrange equation

$9ℓ=117$

DivEqn

$LHS/9=RHS/9$

$ℓ=13$

Pop Quiz

Example

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.
### Hint

### Solution

#### Side Length of the Base

Since the base is a square, take the square root of the base area to find the side length $s$ of the base.
#### Perimeter of the Base

Now that the side length of the base is known multiply it by $4$ to find its perimeter $p.$
#### 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 $ℓ.$

The area of the base of the regular pyramid is $100$ square meters and the surface area is $480$ square meters. Help the two friends unlock this challenge by determining the volume of the pyramid. Round the volume to the nearest integer.

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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.

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 Base100 =10meters $

The side length of the base is $10$ meters. $Perimeter of the Base4⋅10=40meters $