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Tiếng ViệtCircles and PiSpheres, Cones and Cylinders
In the previous sections, we studied the properties of circles on a flat surface. But our world is actually three-dimensional, so lets have a look at some 3D solids that are based on circles:
A A cylinder is a three-dimensional solid consisting of two congruent, parallel, circular sides (the bases), joined by a curved surface. You could also think of a cylinder as a “circular prism”.
A A cone is a three-dimensional solid that has a circular base joined to a single point (called the vertex) by a curved side. You could also think of a cone as a “circular pyramid”. A right cone is a cone with its vertex directly above the center of its base.
Every point on the surface of a A sphere is a three-dimensional solid consisting of all points that have the same distance from a given center. This distance is called the radius of the sphere.
Notice how the definition of a sphere is almost the same as the definition of a
Cylinders
Here you can see the cylindrical Gasometer in Oberhausen, Germany. It used to store natural gas which was used as fuel in nearby factories and power plants. The Gasometer is 120m tall, and its base and ceiling are two large circles with radius 35m. There are two important questions that engineers might want to answer:
- How much natural gas can be stored? This is the
??? of the cylinder. - How much steel is needed to build the Gasometer? This is (approximately) the
??? of the cylinder.
Let’s try to find formulas for both these results!
Gasometer Oberhausen
Volume of a Cylinder
The top and bottom of a cylinder are two congruent circles, called bases. The height h of a cylinder is the perpendicular distance between these bases, and the radius r of a cylinder is simply the radius of the circular bases.
We can approximate a cylinder using a A prism is a three-dimensional solid that has two congruent faces that are polygons (called the bases), whose corresponding vertices are joined by parallel segments. The remaining faces of a prism are all rectangles or parallelograms.
Even though a cylinder is technically not a prism, they share many properties. In both cases, we can find the volume by multiplying the area of their base with their height. This means that a cylinder with radius r and height h has volume
Remember that radius and height must use the same units. For example, if r and h are both in cm, then the volume will be in
In the examples above, the two bases of the cylinder were always directly above each other: this is called a right cylinder. If the bases are not directly above each other, we have an oblique cylinder. The bases are still parallel, but the sides seem to “lean over” at an angle that is not 90°.
The Leaning Tower of Pisa in Italy is not quite an oblique cylinder.
The volume of an oblique cylinder turns out to be exactly the same as that of a right cylinder with the same radius and height. This is due to Cavalieri’s Principle states that if two solids have the same height and the same cross-sectional area at every level, then they both have the same volume. We can use this fact to derive that the volume of prisms and cylinders is the area of their cross-section multiplied by their height. Bonaventura Cavalieri (1598 – 1647) was an Italian mathematician and monk. He developed a precursor to infinitesimal calculus, and is remembered for Cavalieri’s principle to find the volume of solids in geometry. Cavalieri also worked in optics and mechanics, introduced logarithms to Italy, and exchanged many letters with Galileo Galilei.
Imagine slicing a cylinder into lots of thin disks. We can then slide these disks horizontal to get an oblique cylinder. The volume of the individual discs does not change as you make it oblique, therefore the total volume also remains constant:
Surface Area of a Cylinder
To find the surface area of a cylinder, we have to “unroll” it into its flat The net of a polyhedron is what you get when you “unfold” its polygonal faces onto a flat surface.
There are two
- The two circles each have area
- The height of the rectangle is
??? of the circles:
This means that the total surface area of a cylinder with radius r and height h is given by
Cylinders can be found everywhere in our world – from soda cans to toilet paper or water pipes. Can you think of any other examples?
The Gasometer above had a radius of 35m and a height of 120m. We can now calculate that its volume is approximately
Cones
A A cone is a three-dimensional solid that has a circular base joined to a single point (called the vertex) by a curved side. You could also think of a cone as a “circular pyramid”. A right cone is a cone with its vertex directly above the center of its base.
The radius of the cone is the radius of the circular base, and the height of the cone is the perpendicular distance from the base to the vertex.
Just like other shapes we met before, cones are everywhere around us: ice cream cones, traffic cones, certain roofs, and even christmas trees. What else can you think of?
Volume of a Cone
We previously found the volume of a cylinder by approximating it using a prism. Similarly, we can find the volume of a cone by approximating it using a A pyramid is a polyhedron that has a polygon as base, and triangular faces around the outside, that taper to a vertex. In a right regular pyramid, the base is a regular polygon and the vertex is directly above the center of the base.
Here you can see a
This also means that we can also use the equation for the volume:
Notice the similarity with the equation for the volume of a cylinder. Imagine drawing a cylinder around the cone, with the same base and height – this is called the circumscribed cylinder. Now, the cone will take up exactly
Note: You might think that infinitely many tiny sides as an approximation is a bit “imprecise”. Mathematicians spent a long time trying to find a more straightforward way to calculate the volume of a cone. In 1900, the great mathematician David Hilbert (1862 – 1943) was one of the most influential mathematicians of the 20th century. He worked on almost every area of mathematics, and was particularly interested in building a formal, logical foundation for maths. Hilbert worked in Göttingen (Germany), where he tutored numerous students who later became famous mathematicians. During the International Congress of Mathematicians in 1900, he presented a list of 23 unsolved problems. These set the course for future research – and four of them are still unsolved today!
Just like a cylinder, a cone doesn’t have to be “straight”. If the vertex is directly over the center of the base, we have a right cone. Otherwise, we call it an oblique cone.
Once again, we can use Cavalieri’s principle to show that all oblique cones have the same volume, as long as they have the same base and height.
Surface Area of a Cone
Finding the surface area of a cone is a bit more tricky. Like before, we can unravel a cone into its net. Move the slider to see what happens: in this case, we get one circle and one
Now we just have to add up the area of both these components. The base is a circle with radius r, so its area is
The radius of the sector is the same as the distance from the rim of a cone to its vertex. This is called the slant height s of the cone, and not the same as the normal height h. We can find the slant height using Pythagoras’ theorem states that in every right-angled triangle,
+ × π | ||
+ × |
The arc length of the sector is the same as the A sector of a circle is a part of its interior, bounded by two radii and an arc. Its area is proportional to the internal angle, as well as the length of the arc. This means that 
+ − × ÷ π |
Finally, we just have to add up the area of the base and the area of the sector, to get the total surface are of the cone:
Spheres
A A sphere is a three-dimensional solid consisting of all points that have the same distance from a given center. This distance is called the radius of the sphere.
You can think of a sphere as a “three-dimensional A circle is the set of all points in two dimensions, at a fixed distance (the radius) from a given point (the center).
In a previous section, you learned how the Greek mathematician Eratosthenes of Cyrene (c. 276 – 195 BCE) was a Greek mathematician, geographer, astronomer, historian, and poet. He spent much of his life in Egypt, as head of the library of Alexandria. Among many other achievements, Eratosthenes calculated the circumference of the Earth, measured the tilt of the Earth’s axis of rotation, estimated the distance to the sun, and created some of the first maps of the world. He also invented the “Sieve of Eratosthenes”, an efficient way to calculate prime numbers.
Volume of a Sphere
To find the volume of a sphere, we once again have to use Cavalieri’s Principle. Let’s start with a hemisphere – a sphere cut in half along the equator. We also need a cylinder with the same radius and height as the hemisphere, but with an inverted cone “cut out” in the middle.
As you move the slider below, you can see the cross-section of both these shapes at a specific height above the base:
Let us try to find the cross-sectional area of both these solids, at a distance height h above the base.
The cross-section of the hemisphere is always a
The radius x of the cross-section is part of a right-angled triangle, so we can use Pythagoras’ theorem states that in every right-angled triangle,
Now, the area of the cross section is
| A | = | + − × ÷ π |
The cross-section of the cut-out cylinder is always a
The radius of the hole is h. We can find the area of the ring by subtracting the area of the hole from the area of the larger circle:
| A | = | |
| = |
It looks like both solids have the same cross-sectional area at every level. By Cavalieri’s Principle, both solids must also have the same The volume of a cylinder is given by the equation where r is the radius of the circular base, and h is the height of the cylinder (the perpendicular distance between the two bases). The volume of a cone is given by the equation where r is the radius of the circular base, and h is the height of the cone (the perpendicular distance from the base to the vertex).
| = | ||
| = | + − × ÷ π |
A sphere consists of
The Earth is (approximately) a sphere with a radius of 6,371 km. Therefore its volume is
+ − × ÷ π | ||
| 1 |
The average density of the Earth is
That’s a 6 followed by 24 zeros!
If you compare the equations for the volume of a cylinder, cone and sphere, you might notice one of the most satisfying relationships in geometry. Imagine we have a cylinder with the same height as the diameter of its base. We can now fit both a cone and a sphere perfectly in its inside:
This cone has radius
This sphere has radius
This cylinder has radius
Notice how, if we
Surface Area of a Sphere
Finding a formula for the surface area of a sphere is very difficult. One reason is that we can’t open and “flatten” the surface of a sphere, like we did for cones and cylinders before.
This is a particular issue when trying to create maps. Earth has a curved, three-dimensional surface, but every printed map has to be flat and two-dimensional. This means that Geographers have to cheat: by stretching or squishing certain areas.
Here you can see few different types of maps, called projections. Try moving the red square, and watch what this area actually looks like on a globe:
As you move the square on the map, notice how the size and shape of the actual area changes on the three-dimensional globe.
To find the surface area of a sphere, we can once again approximate it using a different shape – for example a polyhedron with lots of faces. As the number of faces increases, the polyhedron starts to look more and more like a sphere.
COMING SOON: Sphere Surface Area Proof