3. Volume and Surface Area of Pyramids
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Chapter 10
3.
Volume and Surface Area of Pyramids
Pyramids are common shapes in geometry and are used in many real-life applications such as architecture and design. Calculating the volume and surface area of a pyramid is important for tasks like building structures, designing objects, and estimating materials. The volume of a pyramid tells how much space it can hold, while the surface area measures the total outer area, helping to determine how much material is needed to cover it. These calculations are crucial for anyone involved in construction or design, ensuring the right amount of material is used and that structures are built efficiently. Understanding these concepts makes it easier to solve practical problems and make informed decisions in various fields that rely on geometric analysis.
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Lesson Settings & Tools
| | 13 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Volume and Surface Area of Pyramids
Slide of 13
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?
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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
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 3D 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 B is the area of the polygon opposite the vertex of the pyramid, and the height h is measured perpendicular to the base.
V=31Bh
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 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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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 5 feet and a height of 4.33 feet. The height of the pyramid is 12 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 B and the height h of the pyramid divided by 3 or multiplied by a third.
The volume of the pyramid is 43.3 cubic feet. That is great news! LaShay and Vincenzo can proceed to the next challenge now.
V=31B⋅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.
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. V=3B⋅h
SubstituteII
B=10.825, h=12
V=310.825⋅12
Multiply
Multiply
V=3129.9
CalcQuot
Calculate quotient
V=43.3
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!
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Hint
Calculate the perimeter p of the base of the vessel. Then, find the area of the base using the formula A=21a⋅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.
Solution
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=31B⋅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=21a⋅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=40 cm
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=21a⋅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=31B⋅h
SubstituteII
V=1100, B=110
1100=31⋅110⋅h
DivEqn
LHS/110=RHS/110
10=31h
RearrangeEqn
Rearrange equation
31h=10
MultEqn
LHS⋅3=RHS⋅3
h=30
Discussion
Surface Area of a Pyramid
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=21pℓ+B
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 15 meters, and its slant height is 22 meters. Help them calculate the surface area of the inverted pyramid.
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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.
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.
SA=21pℓ+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=60 m
By similar logic, the area of the base can be determined by squaring its side length.
Area of the Base152=225 m2
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=21pℓ+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
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 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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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 ℓ.
Solution
Begin by considering the given diagram of the trellises. 
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. 
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.
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.
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.
SA=21pℓ+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 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.Perimeter of the Base6⋅3=18 meters
Area of the Base
The area of a triangle is calculated as half the product of its base and height.A=21bh
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. c2=a2+b2
SubstituteII
c=6, b=3
62=a2+32
▼
Solve for a
CalcPow
Calculate power
36=a2+9
SubEqn
LHS−9=RHS−9
27=a2
RearrangeEqn
Rearrange equation
a2=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=21b⋅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
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.SA=21p⋅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
Practice Finding the Surface Area and Volume of Pyramids
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 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. 
Note that the slant height of the pyramid is the hypotenuse of the right triangle ABC. Apply the Pythagorean Theorem to find AB, which corresponds to the height of the pyramid.
The height of the pyramid is 421 meters.
Therefore, the volume of the map is about 611 cubic meters. The two friends have successfully deciphered the map's prophecy and can now relax and gaze into the stunning star constellations.
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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 s of the base.Side Length of the Base100=10 meters
The side length of the base is 10 meters. Perimeter of the Base
Now that the side length of the base is known multiply it by 4 to find its perimeter p.Perimeter of the Base4⋅10=40 meters
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.SA=21pℓ+B
SubstituteValues
Substitute values
480=21⋅40ℓ+100
SubEqn
LHS−100=RHS−100
380=21⋅40ℓ
MoveRightFacToNumOne
b1⋅a=ba
380=240ℓ
CalcQuot
Calculate quotient
380=20ℓ
DivEqn
LHS/20=RHS/20
19=ℓ
RearrangeEqn
Rearrange equation
ℓ=19
Height of the Pyramid
Based on the information gathered so far, the height of the pyramid can be now determined. Since it is a right pyramid, the vertex is located above the center of its base. This also means that the distance between the center and the midpoint of any side length of the base is equivalent to half the length of that side.AC2=BC2+AB2
SubstituteII
AC=19, BC=5
192=52+AB2
▼
Solve for AB
CalcPow
Calculate power
361=25+AB2
SubEqn
LHS−25=RHS−25
336=AB2
RearrangeEqn
Rearrange equation
AB2=336
SqrtEqn
LHS=RHS
AB=336
SplitIntoFactors
Split into factors
AB=16⋅21
SqrtProd
a⋅b=a⋅b
AB=16⋅21
CalcRoot
Calculate root
AB=421
Volume of the Pyramid
Plug in the height of the pyramid and the area of its base into the formula for the volume of a pyramid to determine the volume of the Celestial Observatory.V=31Bh
SubstituteII
B=100, h=421
V=31⋅100⋅421
Multiply
Multiply
V=31⋅40021
MoveRightFacToNumOne
b1⋅a=ba
V=3400⋅21
UseCalc
Use a calculator
V=611.010092…
RoundInt
Round to nearest integer
V=611
Closure
The Apex of Discovery
After completing the last adventure of the Pyramidium House, LaShay and Vincenzo enter the heart of the structure. They recall their initial challenge of needing to pour a potion from a prism container into two smaller pyramid containers.
Using their newfound knowledge and the given dimensions, they can calculate whether the portion in the prism is enough to fill both pyramids.
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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Hint
a Use the formula for the volume of a prism to find the volume of the prism container.
b Use the formula for the volume of a pyramid to find the volume of each pyramid container.
c Find the total volume of both pyramids combined and compare it with the total volume of the prism container.
Solution
a LaShay and Vincenzo need to determine the volume of a prism-shaped container. The formula for calculating the volume of a prism is the product of the area of its base B and its height h.
V=Bh
In this case, the base of the prism is a square with a side length of 4 millimeters. The two friends need to square the side length of the base to find the area of the base.
Area of the Base42=16 mm2
The area of the base is 16 square millimeters. The height of the prism is 10 millimeters. Multiply the area of the base by the height of the prism to determine its volume.
Volume of the Prism16⋅10=160 mm3
Therefore, the prism-shaped container has a volume of 160 cubic millimeters of potion.
b Consider the formula for the volume of a prism.
V=31B⋅h
Each pyramid-shaped container has a base of 2 millimeters. The area of the base is found by squaring the side length of the base since the base is a square. Area of the Base22=4 mm2
The area of the base of each pyramid container is 4 square millimeters. The height of the pyramid is 6 millimeters. Substitute this information into the formula for the volume of a pyramid to find the volume of each pyramid. V=31B⋅h
SubstituteII
B=4, h=6
V=31⋅4⋅6
Multiply
Multiply
V=31⋅24
MoveRightFacToNumOne
b1⋅a=ba
V=324
CalcQuot
Calculate quotient
V=8
c Consider the volume of the prism container and the volume of each pyramid container.
Prism Container160 mm3Pyramid Container8 mm3
Each pyramid container requires 8 cubic millimeters of potion. Therefore, the two pyramid containers require a total of 16 cubic millimeters of potion, which is significantly less than the 160 cubic millimeters that the prism container can hold. Therefore, there is enough potion to fill both pyramid-shaped containers.
Volume and Surface Area of Pyramids
Exercise 3.1
A square pyramid has a base side length of 10 meters, a height of 12 meters, and a slant height of 13 meters.
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We want to determine the increase in volume of a pyramid when its dimensions are doubled. To do so, we will first calculate the volume of the original pyramid and then calculate the volume of the pyramid with doubled dimensions. Finally, we will calculate the difference between the volumes.
Volume of the Original Pyramid
We are given that the side length of a square pyramid is 10 meters. We need to square the side length to calculate the base area. Area of the Base 10^2=100m^2 The base area is 100 square meters. The height of the pyramid is known to be 12 meters. We can substitute these values into the formula for the volume of a pyramid to find its volume.
The volume of the original pyramid is 400 cubic meters.
Volume of the Pyramid with Doubled Dimensions
We are given that all dimensions of the pyramid have been doubled. The original pyramid has a side length of 10 meters and a height of 12 meters. We can represent the new side length of the pyramid with the variable s and its height with the variable h. Let's calculate the values of s and h. s=2* 10 ⇔ s=20m h=2* 12 ⇔ h=24m The side length of the base of the modified pyramid is 20 square meters. Let's square it to find its base area. Area of the Base 20^2=400m^2 We can then use the base area and height of the modified pyramid to find its volume.
The volume of the modified pyramid is 3200 cubic meters.
Difference
We have determined that the volume of the initial pyramid is 400 cubic meters, while the volume of the pyramid with doubled dimensions is 3200 cubic meters. We will now calculate the difference between these volumes to determine by how much the new volume increases compared to the original volume. Difference Between Volumes 3200-400=2800m^3 The new volume increases by 2800 cubic meters compared to the original volume.
Let's follow a similar process to determine the increase in surface area of the pyramid when its dimensions are doubled. First, we will calculate the surface area of the original pyramid, and then we will calculate the surface area of the modified pyramid.
Surface Area of the Original Pyramid
The surface area of a pyramid is calculated with the following formula. SA=1/2pl+B Previously, we have found that the base area of the pyramid is 100 square centimeters. However, we still need to calculate the base perimeter. To do so, we will multiply the side length of 10 meters by 4. Perimeter of the Base 4*10=40m The base perimeter is 40 meters. The slant height of the pyramid is 13 meters. We can calculate the surface area of the pyramid by substituting these values into the formula for the surface area.
The surface area of the original pyramid is 360 square meters.
Surface Area of the Pyramid with Doubled Dimensions
We previously discovered that doubling the dimensions of a pyramid results in a new side length of 20 meters. Additionally, the base area of the modified pyramid is 400 square meters. However, we still need to determine the new perimeter and slant height of the modified pyramid. Let's calculate them. p=4* 20 ⇔ p=80m l=2* 13 ⇔ l=26m We can now use the values we have found to calculate the surface area of the modified pyramid.
The surface area of the modified pyramid is 1440 square meters.
Comparison
We have calculated that the surface area of the initial pyramid is 360 square meters, and the surface area of the modified pyramid is 1440 square meters. Let's calculate the difference between the new surface area and the initial surface area to find out by how much the new surface area increases compared to the original surface area. Difference Between Surface Areas 1440-360=1080m^2 The new surface area increases by 1080 square meters compared to the original surface area.
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Volume and Surface Area of Pyramids
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