Spheres
Earth is a big blue planet covered mostly with oceans. It is the fifth-largest planet in our solar system and for now, the only one known to have liquid water on its surface.
With this vital water supply, our home planet is the only place to host an estimated 8.7 million species in the known universe.
Is there enough water supply on earth for all of the species including us?
Although our Earth is not perfectly round, it maintains the general shape of a sphere.
Remember, the set of all points equidistant to a certain point on a two-dimensional plane is called a
If we think of the same definition in 3D, then it becomes a sphere.
The word "sphere" is from Greek meaning "globe".
There are many terms related to our world involving the word “sphere”.
For instance, the blanket of gases that surrounds Earth is called the atmosphere. The atmosphere is sliced up into the different zones as the troposphere, stratosphere, mesosphere, and thermosphere.
The Earth has northern and southern hemispheres. Hemisphere means
If you divide the Earth into the hemispheres the resulting flat surface is called the Great Circle.
There
Great circles are used in planning routes for aircraft as the air currents and weather conditions.
The shortest possible air path between two points on a curved surface always lies on the great circle that passes through both of those points.
We know that two-thirds of the earth’s surface is covered with water. But how much is that?
Until now, to be able to find the surface area of the solids, we have always used the
Remember nets are the two-dimensional coats of the 3D Solids.
Let’s try to draw the
If it is not possible to draw the nets of the spheres how do we have the 2D maps of our world?
If a globe were flattened out into a map the result would be wrinkled and torn. The size, shape, and relative location of continents would change. Since drawing an accurate net of a sphere is impossible, we reflect the spherical surface of Earth to a flat piece of paper by using different projections.
Unfortunately, there is no truly correct way of representing the earth as a flat image. All the 2D maps are distorted in some manner!
None of these flat figures can fold up to a sphere.
if we cannot draw its net how can we find the surface area of a sphere?
There are different ways to come up with the surface area and volume formulas of a sphere. We may start using the 3D Solids that we already knew about.
Think about a hemisphere with a radius r fitting inside the smallest possible rectangular prism.
In this case, the dimensions of the rectangular prism in terms of “r” must be
So the volume of the prism is
The volume of the hemisphere is definitely much less than that.
This time let's put the hemisphere in a cylinder.
The smallest possible cylinder that we can fit the hemisphere has the same
The volume of this cylinder is
This time, let’s insert a cone with the maximum possible volume inside of the hemisphere.
The cone shares the same
The volume of this cone is
If we double these boundaries to reach the volume of a whole sphere;
In fact, here we are following the footsteps of the great Mathematician of ancient times, Archimedes. Although he is very famous for the “Eureka” story, he has countless other inventions and contributions to mathematics, science, and engineering.
He studied circles, cylinders, and spheres until he died and made approximations for the number pi and he discovered the volume and the area of the sphere by using the shapes he already knew about like cylinders and cones.
According to a legend, even his last words were “Don’t disturb my circles”.
Think of a cylinder of which its diameter and height are equal to each other.
and a sphere with the same diameter
Archimedes proved that there is exactly the same ratio between their volumes and surface areas.
To find this ratio, let’s use the water experiment we have used before when comparing the pyramids and prisms.
We need to have a cylinder and cone with the same heights as the diameter of their bases (h = 2r) and a sphere with exactly the same diameter.
We already know that the volume of the cone is
Now Let's (add a sphere to the hourglass and) pour the remaining water the sphere:
The Sphere is full and the cylinder is completely emptied now!
You can rotate the hourglass to repeat the experiment.
We have just demonstrated that the volume of the sphere is the
Archimedes has also found that the same ratio exists between the surface areas of the cylinder and sphere as well.
Surface Area of a Cylinder =
Surface Area of the Cylinder with
If we use the same ratio of
This
Thanks to Archimedes, now we can go back to our initial question. We can calculate the surface area and volume of our world to find the true amount of water in it.
The radius of Earth at the equator is 6378 kilometers. Although it is not a perfect sphere, we can still use the sphere’s volume formula to make a fair approximation.
The surface area of the Earth can also be found by using the Archimedes formula
About 70 percent of the Earth’s surface is covered by water. It means the surface area of water is approximately
The average depth of oceans is about 4 km, so we can calculate the volume of the water on Earth as
This means that while the water covers 70% of the Earth’s surface, it has a volume of much less than 1% of Earth’s entire volume.
Not only our world but also other planets and orbits are spherical too.
Let’s compare the size of the moon and other planets with respect to the radius, surface area, and the volume of the Earth.
Remember that, the volume of a sphere is directly proportional to
| Radius | Volume | Surface Area | |
|---|---|---|---|
 Earth | r | V | A |
 Moon | |||
 Mars | |||
 Saturn |
Spheres have unique properties that make them the favorite shape for many sports, droids, fruits even for soap bubbles.
For instance, the spherical shape of a soap bubble minimizes surface tension whereas the round figure of BB-8 gives it the ability to move in any direction
The round shape of the soccer ball makes it bounce evenly and gives uniform responses from each direction.
Round shapes of fruits can be a result of the equally-distributed outward expansion as well as the surface area to volume ratio of the spheres.
Let’s compare the surface area and volume of some of the 3D Solids to see them which one of them encloses the most volume for a given surface area;
Now we have also seen that the
If spheres have so many properties then why not use them in architecture too?
Throughout this unit, we have witnessed how architecture gets its inspiration from the beauty of solid geometry. Just think about the symmetry and the beauty of the spaces created by spheres.
The only problem with creating spherical surfaces is the manufacturing of curved panels of glass or other materials. In the 1940s, the mathematician Buckminster Fuller improved the concept of approximating curved surfaces by using flat triangular panels called geodesic domes and surfaces.
Walt Disney Epcot “Spaceship Earth”
(Orlando, US)
It took more than two years to build Disney’s world-famous spherical attraction. Its name “Spaceship Earth” is also popularized by Buckminster Fuller.
Spaceship Earth is a complete sphere, supported by three pairs of legs with 50 meters of diameter. The volume of the sphere is
11,520 isosceles triangles are planned to be used to create a perfect geodesic sphere of this size.
Like Epcot’s Spaceship Earth, there are many other modern examples of spherical buildings all around the woırld.
Slideshow:
Tianjin Binhai Library (Tianjin, China)
Besides its perfectly round beauty, the properties of the sphere make it a perfect choice for the modern structures.
Amazon Spheres (Seattle, Washington)
Remember, spheres enclose the most volume for a given surface area of any geometric solid.
So, it requires fewer building materials than more conventional buildings based on shapes such as rectangular prisms.
La Géode (Paris, France)
It also means less heat is lost or gained through the exterior, and the structure receives less force from strong winds that could potentially damage it.
Biosphère (Montreal, Canada)
Also, for the same reason of soap bubbles being spherical, the spherical shapes distribute the load of the building throughout the entire structure, which gives the building great strength.