Prisms and Pyramids
The amazing structure of the honey bee cells has attracted the attention of humans for centuries.
Bees collect nectar and pollen from flowers to make honey for their colony. Honey provides bees the energy they need to survive and reproduce, as well as to build their homes.
These homes are called honeycombs. All the bees on earth are using the same shape to create their honeycombs.
But why do bees create honeycombs in the shape they do?
Honeycomb cells are actually a special kind of polyhedra called
A prism is a type of polyhedron with two congruent polygonal faces that are parallel to each other.
Those identical faces are called bases. They are often used to indicate the top and bottom of the prism. Actually, prisms are named according to the shape of their bases. That’s why honeycomb shapes are called
Imagine slicing a prism into lots of thin layers parallel to its base.
All cross sections will be exactly the same shape and size. The cross sections will be congruent.
Think of a loaf of sliced bread, each bread slice has the same shape and size. The shape of the bread is a prism.
There are all kinds of prisms like triangular prisms, rectangular prisms, pentagonal prisms, and so on. The cuboids we’ve seen in the previous chapter are also prisms:
Honeycomb cells are always horizontally aligned. They share walls with the neighbor cell to decrease the amount of wax used to build each cell.
Those shared walls are the lateral faces of the hexagonal prisms. The lateral faces are
In all kinds of prisms, the bases are connected by a set of rectangles (or sometimes parallelograms) regardless of the type of the base.
The
A hexagonal prism has
Why do bees choose hexagonal prisms over the other prisms?
Hexagonal prisms can be aligned next to each other without any gaps or overlaps but also so can cuboids and triangular prisms.
Bees also need to have maximum storage for the honey without wasting more wax than necessary. This means that they need to use as little wax as possible to create their comb which can store as much honey as possible.
Recall the method we use to calculate the volume of the cuboids.
Volume =
Can we use the same formula for all prisms?
The reason that the volume of a cuboid is calculated as
Let’s look at these prisms to see if they are all made up of the multiple layers of the same polygon that they have as a base.
Rotate the prisms to select their bases.
Let's look at the different slices of the prism.
All the triangular layers are the
All the hexagonal layers are the
Since all prisms are made up from the multiple layers of their base, we
Now, use the slider to try the different polygons as the base of the prism that could be used as a honeycomb. Average cell width and length for a honeycomb is approximately 4mm and the depth is 10 mm.
Triangular Prism
Since the base of the prism an equilateral triangle, Base Area can be found by the
Square Prism
Since the base of the prism a square, Base Area can be found by squaring the side length
Hexagonal Prism
Since the base of the prism a hexagon,
Base Area is the six equilateral triangle's area that the hexagon is made of.
Compared to the other prisms that leave no gaps or overlaps (such as triangular and square), the hexagon prism creates a comb with the maximum volume.
Remember, bees also need to use the least amount of wax possible to construct these combs. Since they can only produce 1 oz of wax by using 8 oz of honey, the wax is very precious for them too. They cannot spend more wax than necessary.
For finding the amount of wax needed to build walls of the combs, we need to find the
Using
We have already found the base area of the prisms.
Triangular Prism
There are
So the sum of all the areas of the faces will be
Square Prism
There are
Hexagonal Prism
There are
So the sum of all the areas of the faces will be
Although it seems that the surface area of the hexagonal prism higher, this amount of wax has to be produced to create a volume of 420
Let’s look at the different shaped honeycombs to see how many cells are needed to create a volume of 420
To create 420
To create 420
To create the same volume for storage, bees need to use more wax as the surface area of the triangular or square prisms. That’s why (Neglecting the closed ends of the combs), honeycombs are in the shape of hexagonal prisms.
The closed ends of the honeycomb cells are a bit more complicated.
They are composed of three flat planes to ensure the back-to-back ends of cells fit against each other and still the overall shape of the honeycomb cell minimizes surface area for a given volume.
Nowadays, It is also being argued that bees build cylindrical cells that later transform into hexagonal prisms through a process that is still debated like physical forces and mechanical shaping.
Architectural Design imitates nature when seeking solutions of sustainability and efficiency. All kinds of prisms are regularly used in architecture.
Hexagonal Cabins
The Flat Iron building in New York City, a 22-story
The Baltimore World Trade Center, a 30-story
The Seagram Building in XXX, a 38-story
Let’s have a look at the tallest buildings in the world.
When the buildings become taller and taller they start losing their prism-like shapes and become more triangular.
Pyramids
Pyramids are a particular type of architecture developed since ancient times and still used today for modern buildings. The first pyramids were built in Mesopotamia, but the most famous pyramids are the Egyptian and Mayan pyramids.
Egyptians knew vertical walls got less stable as they got taller, that’s why they first tried stacked bricks at an incline. They realized that a pyramid gets you the most stability for the least material.
Thanks to the stability of the triangular structure The Great Pyramid of Giza remained the tallest building of the world for 4000 years until the Eiffel Tower was built in 1889. The Great Pyramid is the oldest monument on the list of the Seven Wonders of the Ancient World, built almost 4600 years ago.
Can you imagine the number of stones needed to build these giant ancient wonders?
The Great Pyramid of Giza along with the Pyramid of Menkaure and the Pyramid of Khafre
Like prisms, pyramids are polyhedra too. But unlike prisms, pyramids only have
There are lots of different kinds of pyramids, depending on the shape of their base.
Just like prisms, Pyramids are named for the shape of their base.
For example, if the base is a square, then it is called a “
Regardless of the shape of its base, a pyramid always has
Before starting to work on a pyramid, Egyptian builders had to calculate its volume in order to acquire the right amount of stone needed to build the pyramid.
Imagine slicing a pyramid into lots of thin layers parallel to its base.
All cross sections will be exactly the same shape but different in size. When you go up, the size of the slices decreases proportionally to the base of the pyramid. The cross sections of a pyramid parallel to its base are similar but not
Since pyramids are made up of decreasing the size of layers of the same base, we
This is sure a great advantage during the construction, but it requires a different method to calculate the volume of the pyramids.
{.fixme} We should add a paragraph (and interactive diagram) that shows that all cross sections parallel to the bases are similar (but not congruent).
{.fixme} When I teach this in school, we have a set of many different prisms and pyramids with the same base and same height. Students predict how many pyramids it will take to fill up the related prism. Then, students use rice or pasta or water to check their hypothesis. They fill up the pyramid and pour it into the prism and count how many it takes to fill it up. I found this a really powerful and effective lesson. I wonder how some of that could be incorporated here? I image the GIF on the first slide has some animation of that, but I wonder if there is a way to make students do it? There could be an empty square based pyramid and an empty cuboid and the fill up the pyramid with water by placing it under a sink and then they have to drag it over to the cuboid and fill up the cube. Then, they'd actually have to do it 3 times. Not quite as powerful as doing it in person I think, but if they do it 2-3 times, they helps build understanding of the formula. | I totally agree - I was doing the same experiment with my students with the set of geo solids.. I was planning to talk with Philipp about the animations, pills, hover targets in this section. This part has to be really interactive for kids to grab the formula | Jumping in to agree with you both here. An interactive element that really lets students explore here will take pressure off of the language/text. Since this is such an important, and sometimes challenging, concept. It may be worthwhile to require as little effort on reading as possible.And I think all of the slides help lay a foundation for the interactive!
Experiments
Decomposition
Cubes
Now we are able to calculate the number of stones needed to build the Great Pyramid of Giza.
The height of the Great Pyramid of Giza is 146.7 m.
Be sure you do not confuse the slant height of a pyramid with its solid height.
Slant height is a measure along a triangular face. It is the height of the lateral face.
Solid height is an internal measure from the apex to the **center **of the base.
To be able to calculate the volume of a pyramid, you need to know the
The base of the Great Pyramid is a square with each side measuring 230 m and covering an area of
How much is that? Imagine a football field. Nearly 10 football fields could fit within the base of the Great Pyramid.
Recall that the volume of the pyramid is
So the volume is:
It is plenty of room for the Pharaoh and his belongings. It is estimated 2.3 million stone blocks each weigh an average of 2.5 to 15 tons were used to build a pyramid that size. Legend has it that the structure was constructed in just 20 years' time, meaning that a block had to have been moved into place about every 5 minutes of each day and night.
A not very well known fact about The Great Pyramid Giza is that it was covered with the smooth white Tura limestone casing, which now only exists on the upper cap. This stone covered the outer layer of the lateral faces of the pyramid to make them completely smooth.
It was polished until it shone so that the pyramid would have gleamed in the sun.
Think about how many limestone blocks were needed only to cover the Great Pyramid. To determine how many blocks were needed, we need to know the
Recall that all pyramids also have a slant height, which is the height of its
The slant height is used to calculate the
Slant Height of the Great Pyramid is 186.6m. Each triangle face’s area is
Since we have
Each block of limestone covered about 1.25 square meters. So the Egyptians need almost 70 000 extra white limestones to cover the lateral faces of the pyramid.
While it still stays as a mystery how Egyptians build the pyramids, it is no longer a mystery for us how many stones are used to build or cover them.
Surface Area calculations in prisms and pyramids have longer steps than finding their volume.
In surface area calculations, nets allow us to see all the faces of the solid at once. While calculating surface area, base or lateral area, instead of working on a picture of the solid, drawing nets helps us to visualize the hidden faces.
Do you know that?
What is the most efficient way to stack fruits?
On market stalls or street vendor carts, fruits like apples, and oranges are mostly arranged as
In 1611 Johannes Kepler stated that putting each fruit on top of a gap in the layer of fruit underneath is the most efficient arrangement of spherical objects.
After almost 400 years, in 1998, Thomas Hales presented a proof of the Kepler Conjecture. The proof was 300 pages long and it took a computer to verify its correctness.
Apparently, when it comes to stacking stones and fruits or piling cannonballs on ships, pyramids are always the most convenient shape!
Nets of Prisms and Pyramids
Remember, nets are composed of
Here you have the nets of different prisms or pyramids.
Match each net with the solid it belongs to.
A rectangular prism has
A triangular pyramid has
An octagonal prism has
A hexagonal pyramid has
A cube has
A triangular prism has
Nets of the polyhedra can give you a lot of information about the faces and the other characteristics of the solid. Properties of the nets can help us to compare and contrast the prisms and pyramids.
Swipe each property to the kind of net which it is associated with.
In the previous chapter, we have also seen that there are many nets for a cube. What about the prisms and pyramids?
Is there only one possible net for them?
The polygons can be arranged in
You can arrange these triangles and rectangles in different ways to fold up a triangular prism.
Use the grid to draw one of the possible nets of a triangular prism.
We may not build a pyramid or the tallest building of the world yet, but we can start planning for a treehouse (Playhouse / tent…).
Like Egyptians do, before we start building one, we need to calculate the amount of material we are going to use for the outer surface of our model.
A house-shaped prism is created by attaching a
These types of solids are called Composite solids.
A composite solid is a figure that is made up of more than one solid.
The base of the house-shaped prism consists of a
We may find the area of the rectangle and triangle separately and then add them up to find the
Rectangle’s areais
So each base area is
Now we can find the lateral area of the model;
Is there a way to simplify the calculations?
Imagine unfolding the prism into a net. Use the slider to see the net of the house-shaped prism.
Sometimes we can simplify the process by combining the lateral faces and finding the area of the combined region.
We can use one large rectangle instead of separate smaller ones.
We can treat the prism-like house as having three parts: two identical bases, and one long rectangle that has been taped along the edges of the bases.
The rectangle has the
In fact, for all the prisms, you can combine each rectangular lateral face and find the entire area by multiplying the height of the prism by the perimeter of the base.
So the area of the big rectangle that covers all the lateral face of the prism is
Now, all we are going to do is to add two
So the surface area is
What about the volume?
The Base Area was 9
Therefore, after completing the construction, your treehouse will have a volume of
You may have different designs for your treehouse but, as long as they are prism-like shapes, their volume will always be calculated as