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| 13 Theory slides |
| 9 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
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:
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 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.
The applet shows various three-dimensional shapes. Identify if the given 3D shape is 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
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.
B=10.825, h=12
Multiply
Calculate quotient
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!
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.
a=5.5, p=40
Multiply
b1⋅a=ba
Calculate quotient
V=1100, B=110
LHS/110=RHS/110
Rearrange equation
LHS⋅3=RHS⋅3
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
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.
Substitute values
Multiply
b1⋅a=ba
Calculate quotient
Add terms
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.
c=6, b=3
Calculate power
LHS−9=RHS−9
Rearrange equation
LHS=RHS
Split into factors
a⋅b=a⋅b
Calculate root
a>0
b=6, h=33
Multiply
b1⋅a=ba
Calculate quotient
Substitute values
LHS−93=RHS−93
b1⋅a=ba
Calculate quotient
Rearrange equation
LHS/9=RHS/9
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.
Substitute values
LHS−100=RHS−100
b1⋅a=ba
Calculate quotient
LHS/20=RHS/20
Rearrange equation
AC=19, BC=5
Calculate power
LHS−25=RHS−25
Rearrange equation
LHS=RHS
Split into factors
a⋅b=a⋅b
Calculate root
B=100, h=421
Multiply
b1⋅a=ba
Use a calculator
Round to nearest integer
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.
B=4, h=6
Multiply
b1⋅a=ba
Calculate quotient
Consider the following solid.
Let's begin by looking at the given solid.
The solid has a square base, and its triangular lateral faces meet at a vertex. This means that it is a pyramid. As the base is a square, the solid is a regular square pyramid. In addition, since a square is also a rectangle, the pyramid is also rectangular. However, the pyramid cannot be an hexagonal pyramid, as its base is not a hexagon.
Description | True or False? |
---|---|
The solid is a square pyramid. | True |
The solid is a regular pyramid. | True |
The solid is a rectangular pyramid. | True |
The solid is a hexagonal pyramid. | False |
We can see that three out of the four names are correct for the pyramid. The only not valid name for the pyramid is a hexagonal pyramid. Therefore, this name does not belong with the other three.
Find the volume of each pyramid. Round the answer to two decimal places if needed.
Let's look at the given pyramid.
The pyramid has a pentagonal base with an area of 34 square feet and a height of 23 feet. To calculate the volume of a pyramid, we divide the product of the base area B and the height h by 3. V=1/3B* h Let's substitute 34 for B and 23 for h into this formula to find the volume of the given pyramid.
The pyramid has a volume of 260.67 cubic feet.
Consider the given diagram.
This is a regular octagonal pyramid with a base area of 90 square inches and a height of 47 inches. Let's plug in these values into the formula for the volume of a pyramid to determine the pyramid's volume.
The volume of the pyramid is 1410 cubic inches.
Find the surface area of each pyramid given its slant height, the area, and perimeter of its base.
Let's look at the given pyramid.
The pyramid has a base with an area B of 36 square decimeters and a perimeter p of 24 decimeters. The slant height l of the pyramid is 8 decimeters. We can use the formula for the surface area of a pyramid to calculate the surface area of this pyramid. SA=1/2pl + B Let's plug in the given information into the formula.
The surface area of the pyramid is 132 square decimeters.
Consider the given pyramid.
This is a regular pentagonal pyramid with a slant height of 12 millimeters. The area and perimeter of its base are 110 square millimeters and 40 millimeters respectively. We can substitute these values into the formula for the surface area of a pyramid to determine the surface area of this pyramid.
The surface area of the pyramid is 350 square millimeters.
The base of a right regular pyramid with a slant height of 13 meters has a side length of 10 meters.
The volume of a pyramid can be determined using the following formula. V=1/3Bh Here, B is the area of the base and h is the height of the pyramid. However, before we can apply this formula, we need to determine both the area of the base and the height of the pyramid, as they are currently unknown.
The base of the pyramid is a square with a side length s of 10 meters. We will calculate its area by multiplying its side length by itself. B=10m*10m ⇔ B=100 m^2 The area of the base is 100 square meters.
Since we have a right pyramid, the vertex is located directly above the center of its base. This means that the distance between the midpoint of any side of the base and its center is equal to half the length of that side.
Moving on, we can see that the slant height of the pyramid is the hypotenuse of the right triangle ABC. We can use the Pythagorean Theorem to find AB, which corresponds to the height of the pyramid.
The height of the pyramid is 12 meters.
We have calculated the area of the base of the pyramid and also the height of the pyramid. Now, we will use these values in the formula for calculating the volume of the pyramid.
The volume of the pyramid is 400 cubic meters.
The base of a right regular pyramid with a height of 24 inches has a side length of 8 inches and an apothem of 7 inches.
The surface area of a pyramid can be found using the following formula. SA=1/2pl+B Here, p is the perimeter of the base, l is the slant height of the pyramid, and B is the area of the base. However, before we can apply this formula, we need to determine the perimeter and area of the base, as well as the slant height of the pyramid.
The base of the pyramid is a regular octagon with a side length of 8 inches. We will multiply the side length by 8 to get the perimeter of the base. p=8*8in ⇔ p=64in The perimeter of the base is 64 inches.
The area of a regular polygon is equal to half the product of the polygon's perimeter p and its apothem a. A=1/2ap In this case, we know that the perimeter of the base of the pyramid is 64 inches, and the apothem is 7 inches. Let's plug in these values into the formula to find the area of the base of the pyramid.
The area of the base is 224 square inches.
Since we have a right pyramid, its apex is located just above the center of its base. In this pyramid, the slant height, the height, and the apothem of the base form a right triangle.
Let's use the Pythagorean Theorem to find the slant height of the pyramid, given that we know the height and apothem of the base.
The slant height of the pyramid is 25 inches.
We have calculated the perimeter and area of the base of the pyramid. Additionally, we have determined the slant height of the pyramid. We can now substitute this information into the formula for the surface area of a pyramid to find the surface area of the given pyramid.
The surface area of the pyramid equals 1024 square inches.