3D SolidsIntroduction
So far, we’ve only looked at the geometry of flat, 2-dimensional objects like circles or rectangles – but we live in a three-dimensional world. Every object we see or touch has three dimensions.
A point has zero dimensions. A point just specifies a location but it has no size.
A line has one dimension, we can measure its length (l), but it has no width or thickness.
A square has two dimensions, we can measure its length (l) and, perpendicular to that, its width (w).
A cube has three dimensions, we can measure its length (l); width (w); and perpendicular to both of those dimensions, we have its height (h).
Plane Geometry deals with points, lines, and two-dimensional (2D) shapes like polygons and circles which do not have any thickness. Polygons are plane figures that have only length and width.
Solid Geometry is the study of three-dimensional (3D) shapes that have depth or thickness as well as length and width. Even a paper has a thickness and is a 3D shape, but since its thickness is so small, we often neglect it.
As soon as humankind began constructing complex buildings, we needed solid geometry. In some Babylonian and Egyptian texts, you can see that they are concerned about the problems related to the sizes of the pyramids and other structures they have made.
Since then, architects have been using 3D geometry to define the spatial form of a building. Nowadays, the design of buildings is continually advancing by combining different 3D shapes and their properties.
Barcelona Cube-Ball-Pyramid roof
London Skyline
Just like we’ve classified flat geometric shapes into a few categories like polygons or quadrilaterals, we can classify 3D Solids into a few different types:
A polyhedron is a three-dimensional geometric solid with flat sides. The plural of polyhedron is “polyhedra”. The word polyhedron comes from the Classical Greek as poly (many) + hedron (base, face)
Polyhedra have many different shapes and sizes like polygons. They can be as simple as a cube or a pyramid, or as complex as a star polyhedron with lots of sides.
FACE: The polygons that make up its surface of the polyhedron. EDGE: The line segments where two of its faces are connected. VERTEX: The “corners” of a polyhedron are called its vertices.
Which of these solids are polyhedra?
Even though we live in a 3D world, grasping 3D shapes and their properties may be a challenge.
The Flatland is an animated film based on Edwin Abbott’s book Flatland: A Romance of Many Dimensions. The movie revolves around the inability of 2D shapes to grasp the third dimension. Characters are circles, triangles, and squares living in a 2D world. Their reality is shattered when a sphere from the 3D World comes to visit.
Without realising, you will have seen many different types of polyhedra before:
IMAGE: Dice
A football consists of
There are some things impossible in the 2D world that become possible in the 3D.
Try this puzzle now.
Can you create 4 congruent triangles by using 6 toothpicks without bending or cutting them?
The trick is again thinking in 3D instead of 2D.
Cubes
One of the most common types of 3D solids is a cube.
A cube is a 3d shape that has 6 faces all of which are
For instance, The Rubik's Cube is a classic toy invented in 1974 by Hungarian architecture and design professor Erno Rubik. Each of the faces of a Rubik’s cube contains 9 colored squares.
Although one of the smaller cubes (the central one) is not exposed, if we count carefully, we see that the Rubik’s cube is made up of
A cube has all its sides of the same length. If the length of all its sides is 1 unit, then it is called a unit cube. Unit cubes can be stacked together to create different 3D solids.
The 3D solids below consist of unit cubes. Can we find out the number of cubes used in each one including the hidden cubes?
Rotate the solids below to find the number of cubes that used to build them.
Cuboids
As with a square being rectangle, a cube is a cuboid.
A cuboid is a box-shaped, where all six sides are rectangles. For example, pizza, cereal or shoe boxes are all cuboids.
Cuboids are some of the most common polyhedra we use every day.
Basically, a cuboid is a polyhedron with six rectangular faces. The opposite faces of cuboids are identical and parallel to each other. Cuboids have
An average pizza box has dimensions as 33 cm x 33 cm x 4 cm. It means it has 33 cm in length, 33 cm in width, and 4 cm in height or depth. Sometimes a cuboid has two square faces and four rectangular faces like most of the pizza boxes.
An average cereal box has dimensions of 30 cm x 20 cm x 8 cm.
A Marketing agency built a 6 meters tall tipped-over cereal box outside the Vancouver Art Gallery in 2012. Can you guess the dimensions of this cereal box? Length:
When you try to send boxes from one place to another via delivery services, the company asks you the dimensions of your package in the form of “l x w x h” to calculate the size and weight of the box to determine the shipping rate.
Try identifying the dimensions of the cuboids in the form of “l x w x h”:
Volume
According to a legend, there was a devastating plague in Greece in 430BC. The Delians (citizens of a Greek Island, Delos) went to the temple of Apollo who told them that the plague will stop when they double the volume of his altar.
IMAGE
Here is a line of length 1.
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A square where all sides have length 1 has area 1.
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A cube where all edges have length 1 has volume 1.
We can find the volume of more complex shapes by determining how many cubes of volume 1 it contains.
The Apollo’s altar was a cubical temple with dimensions of five meters as its length, width, and height.
The Delians doubled the sides of the altar, but the plague did not stop. What went wrong?
One way to find the volume of a cuboid is to fill the cuboid with unit cubes.
Volume is the number of cubic units that fill a 3D region, without any gaps or overlaps.
Let’s look at Apollo’s altar. If the edge length of the original altar was 5 units, then its base is a 5x5 square that has
We continue to fill the altar by adding more cubes. The first layer is a 5 x 5 square made up from
There will be
It means the original cubic altar has a volume of 125 cubic units.
The Delians did not follow the instructions carefully. They
When we double the sides of the cıube as the Delians have done with the incorrect altar, there will be
Doubling the dimensions of a three-dimensional figure will increase its volume by a factor of
The Delians did not double the altar - they made it 8 times bigger.
They should have doubled the volume instead of the side lengths.
To find the volume of the cube, the number of cubes in the first layer is multiplied by the height. This first layer is called the base and usually refers to the top and bottom faces of the cube.
Recall the original altar. We counted
We calculated the volume of the cubic altar by multiplying
Base Area =>
In other words, the volume of a cube is the product of its length,
A cube has the same value as its length, width, and height. The volume of a cube with a side length of “a” units is
Multiplying three edge lengths allows us to determine the volume of cube efficiently.
A cube with an edge length of 3 centimeters has a volume of
A cube with an edge length of 4 inches has a volume of
Volume is always measured in cubic units, such as cubic inches (in3), cubic feet
Greek temples have a very predictable layout. At the center, there is a section called Cella that holds a statue of the God to which the temple is dedicated. The cella is surrounded by a long string of columns. They are not always in cubic forms but mostly in cuboids.
Helping to prevent any future conflicts, we can find a general method to find the volume of all cuboids?
The Model of Aphrodite Temple in Greece
The first layer of the cuboid has
The total number of cubes to be used to create this cuboid is
The volume is 120 cubic units.
We again calculated the volume by multiplying
In other words, the formula for the volume of all cuboids is the
Now it’s your turn – find the volumes of the following cuboids.
Volume=
Volume=
Volume=
Some cuboids may have the same volume although they have different shapes.
Use the unit cubes to create a cuboid so that the volume is 24 cubic units.
It is
How many different cuboids can you create by using 24 cubes?
By finding the missing dimensions of the different cuboids with a volume of 24 cubic units, we can observe how the base area and the height change when we have a certain amount of cubes to create the volume.
| Cuboids | Dimensions | |||
|---|---|---|---|---|
| Length | Width | Height | Volume | |
| A | 6 | 2 | 24 | |
| B | 1 | 8 | 24 | |
| C | 4 | 3 | 24 | |
| D | 2 | 12 | 24 | |
If the base area of a cuboid smaller than the other, then its height has to be
If two cuboids have the same height, then the one with the greater base area has a
We now know about how to find the volume of the cuboids, can we find another measure about them as well?