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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.