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| 11 Theory slides |
| 8 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.
Paulina enjoys creating origami objects. After she made the three congruent origami pyramids shown below, she noticed that they could form a cube.
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.
When the base area and the height of a pyramid are known, its volume can be calculated.
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
Consider a pyramid and a prism that have the same base area and height.
A pyramid can be modeled as a stack of prisms. The sum of the volumes of the small prisms will be greater than the pyramid's volume. However, as the number of prisms increases and they get thinner, the sum will approximate the volume of the pyramid.
Furthermore, the ratio of the sum of the volumes of each small prism to the volume of the prism will be approximated to 31.
Number of Layers | Volume of PrismSum of Thin Prisms’ Volumes |
---|---|
4 | ≈0.469 |
16 | ≈0.365 |
64 | ≈0.341 |
256 | ≈0.335 |
1024 | ≈0.334 |
4096 | ≈0.333 |
∞ | 31 |
Therefore, the volume of a pyramid is one third of the prism with the same base area and height.
Tadeo is getting ready to go camping. He has a pyramid-shaped tent with a regular pentagonal base.
The tent has a height of 1.6 meters and its base has side lengths of 1.4 meters. Find the volume of the tent. Round the answer to the two decimal places.Find the perimeter p and the apothem a of the base. Then substitute their values in the formula A=21ap to find the area of the base. To find the apothem, use the tangent ratio of half the central angle of regular pentagons.
Start by finding the area of the base of the tent. Then, the formula for the volume of a pyramid will be used.
The apothem is perpendicular to any side of the polygon and bisects it. As a result, a right triangle with a leg of 0.7 meters is formed.
The value of a can be written in terms of the tangent ratio of 36∘.a=tan36∘0.7, p=7
ca⋅b=ca⋅b
Multiply fractions
Use a calculator
Round to 2 decimal place(s)
B=3.37, h=1.6
Multiply
b1⋅a=ba
Calculate quotient
Round to 2 decimal place(s)
Architects might enjoy turning things upside down. Maya is interested in architecture and follows some online magazines about it. After reading an article about the Slovak Radio Building in Bratislava, Slovakia, she wonders about the area of the square rooftop.
If the height of the building is 80 meters and its volume is about 245760 cubic meters, find the area of the rooftop of the building.
The volume of a pyramid is one third of the product of its base area and height.
Designers and inventors also benefit from pyramids. An object attracts Mark's attention on a school trip to a maritime museum. The guide explains that it is called a deck prism, which was invented to illuminate the cabins below deck before electric lighting. Mark buys a replica of the deck prism, which is composed of a prism and pyramid, each with a regular hexagonal base.
Find the volume of the deck. Round the answer to the two decimal places.The formula for the area of a regular hexagon with side lengths a is B=23a23. Apply the rounding in the last step.
The deck is composed of two solids:
Therefore, the volume of the deck is the sum of the volumes of the above solids. The volume of each solid will be found one at a time.
B=2273, h=4
ca⋅b=ca⋅b
Multiply fractions
Simplify quotient
The surface area of a pyramid is just as important as its volume.
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
Maya's father decides to cover the roof of their house with waterproof insulation material. Maya's father asks Maya to calculate how many square feet of insulation material is needed.
The roof is a square pyramid with a height of 8 feet and base side length of 30 feet. Help Maya calculate the area.
To find the slant height, use the Pythagorean Theorem. Note that only the lateral area of the pyramid is needed.
Maya needs to calculate the lateral area of the square pyramid. To do so, she first needs to calculate the slant height ℓ of the pyramid, which can be found by using the Pythagorean Theorem.
The height h of the pyramid is the distance between the vertex and the center of the base, and b is half the base side length. Therefore, b=230=15 feet. Since a negative value does not make sense in this context, only the principal root is considered. Therefore, the slant height is 17 feet. Next, the perimeter of the base will be found. Since the base is a square, its perimeter p is 4 times the base side length.The applet shows some right pyramids with different regular polygonal bases. Use the given information to answer the question. If necessary, round the answer to two decimal places.
Find the volume of the pyramid.
To calculate the volume of a pyramid, we have to multiply the base area B by the height of the pyramid h, then divide the product by 3. V=1/3Bh First we need to calculate the base area.
Examining the diagram, we see that the base is a rectangle with a length and width of 8 and 6 meters, respectively. With this information, we can calculate the base area.
The base area is 48 square meters.
Now we can calculate the volume of the pyramid by using the formula.
The volume of the pyramid is 64 cubic meters.
As in the previous section, we will first calculate the base area.
The base area is a rectangle with a length and width of 8 and 3 meters, respectively.
The base area is 24 square meters.
Now we can calculate the volume of the pyramid by using the same formula from earlier.
The volume of the pyramid is 24 cubic meters.
In this case, we have a triangular base. The formula for calculating the pyramid's volume is the same as in previous parts, but the base area is calculated with a different formula.
The pyramid has a base which is a right triangle. This triangle has legs of 3 and 4 meters. With this information, we can calculate the base area.
The base area is 6 square meters.
Now we can calculate the pyramid's volume.
The volume of the pyramid is 10 cubic meters.
Kriz missed the math lesson about pyramids but has asked their friend Davontay to write down the homework they received in class. Davontay wrote down the following exercises for Kriz. Since Kriz missed today's lesson, they need some help.
We have been given the volume and height of a pyramid with a square base. Volume:& 270cubic inches Height:& 10inches To find the length of the base's side, we will substitute the given volume V and height h into the formula for calculating the volume of a pyramid, then solve for B. V=1/3Bh
We can find the base area by substituting the given volume and height into the formula and solving for B.
The base area is 81 square inches.
Let s be the side length of the square base.
To find the side length of the base, we can use the formula for the area of a square.
The side length of the square base is 9 inches.
As before, we are given the volume of the pyramid. We also know the pyramid's height and the width of its rectangular base. Volume:& 420cubic inches Height:& 9inches Length:& 14inches
Again we can find the base area by using the formula for calculating the volume of a pyramid.
The area of the rectangular base is 140 square inches.
Let w and l be the width and length of the rectangular base. We already know that l =14 inches.
To find the length, we will substitute the rectangle's area and length into the formula for calculating a rectangle's area and then solve for w.
The width of the base is 10 inches.
To determine the volume of a pyramid, we multiply its base area by its height and divide by 3. V=1/3Bh In this case, the base is a square with a side length of 35.4 meters. By using the formula for the area of a square, we can identify the base area B. B=(35.4)^2 With this information, we can determine the volume.
The Louvre has a volume of about 9023 cubic meters.
What is the surface area of the following pyramid if it has a square base?
The surface area of a pyramid can be calculated by adding the base area B to the product of the perimeter p, the slant height l, and 12. S=1/2Pl+B
The base of the pyramid is a square with a side length of 14 inches. With this information, we can calculate the area by squaring the side length. A= 14^2 ⇒ A=196 in^2 The base area is 196 square inches.
To calculate the perimeter, we will multiply the side length by 4. p=4( 14) ⇒ p=56 in The perimeter is 56 inches.
Now that we know the perimeter and the base area, we can calculate the surface area by substituting p= 56, l= 10 and B= 196 into the formula and evaluate.
The surface area of the pyramid is 476 square inches.