In previous chapters, we studied the properties of circles in the plane.But our world is actually 3D, so let's look at some circle-based 3D objects:
A
A
every point on the surface
Note that the definition of a sphere is almost the same as the definition of a
cylinder
Here you can see the cylindricalGas meterin Oberhausen, Germany.It used to store natural gas, which was used as fuel for nearby factories and power plants.The gas tank is 120 meters high, and the base and ceiling are two large circles with a radius of 35 meters.Engineers may want to answer two important questions:
- How much natural gas can be stored?This is
cylinder. - How much steel is needed to build a gas meter?it is (approximately)
cylinder.
Let's try to find formulas for these two results!
Gazomierz Oberhausen
cylinder volume
The upper and lower parts of the cylinder are two congruent circles calledthe base.TentallHThe cylinder distance is the vertical distance between these bases, whileradiusRThe radius of a cylinder is simply the radius of the base of the circle.
We can use an approximate cylinder
Although cylinders are not technically prisms, they share many of the same properties.In both cases, we can find the volume by multiplying their areasAccording toi itall.That is, the radius isRand heightHtom
Note that radius and height must use the same units.For example, ifRIHare in centimeters, then the volume will be
In the example above, the two bases of the cylinder are always theredirectly above each other:it's calledright cylinder.If the bases do not lie directly on top of each other, we have aukośny cylinder.The bases are still parallel, but the sides appear to be "tilted" at an angle other than 90°.
Tenthe leaning tower of pisaNot quite tilted cylinder in Italy.
It turns out that the volume of the oblique cylinder is exactly the same as that of the right cylinder of the same radius and height.It's because
Imagine cutting a cylinder into many thin discs.We can then move these discs horizontally to make an inclined cylinder.When tilted, the volume of the individual discs does not change, so the total volume remains the same:
cylinder surface area
To find the surface area of a cylinder, we need to "spread" it into a plane
there are two
- Two circles have an area
. - The height of the rectangle is
and the width of the rectangle is the same as circles: .
This means the total area of a cylinder with a radiusRand heightHwas given by (who)
Steel cylinders can be found all over the world – from soda cans to toilet paper and water pipes.Can you think of any other examples?
TenGas meterThe upper radius is 35 m and the height is 120 m.We can now calculate that its volume is approximately
cone cells
A
TenradiusThe radius of the cone is the radius of the base of the circle, whiletallThe vertical distance from the base of the cone to the top.
Like other shapes we've encountered before, cones are all around us:Ice cream cones, traffic cones, some roofs and even Christmas trees.What else can you come up with?
cone volume
Previously, we approximated the volume of a cylinder using a prism.We can use the same
Here you can see A
This also means that we can also use the volume equation:
Note the similarity to the equation for the volume of a cylinder.imagine drawing a cylinderOCones with the same base and height - the so-calleddescribed cylinder.The cone will now take exactly
comments:You might think that an infinite number of small sides is a bit "imprecise" as an approximation.Mathematicians spent a lot of time trying to find a more direct way to calculate the volume of a cone.In 1900, a great mathematician
Like a cylinder, a cone is not necessarily "straight".If the top is directly above the center of the bottom, we haveright cone.Otherwise, we call itoblique cone.
Again, we can use Cavalieri's principle to show that all diagonal cones have the same volume, provided they have the same base and height.
cone surface area
Finding the surface area of a cone is a bit tricky.As before, we can expand the cone into its web.Move the slider to see what happens:In this case we get a circle and a
Now we just need to add the fields of these two components.TenAccording tois a circle of radiusRso its area is
radiusdepartmentThe same as the distance from the edge to the vertex of the cone.it's calledSkew heightOtherConical, not the same as normaltallH. We can find the height of the slope with
Tenarc lengthsector is the same
In the end, all we have to do is placeAccording toand areadepartmentto get the total area of the cone:
pole
A
You can think of a sphere as "three dimensional".
wprevious section, you learned how the Greek mathematicians
sphere volume
To find the volume of a sphere, we need to use Cavalieri's principle again.Let's start with the 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" in the center.
By moving the slider below, you can see the cross sections of these two shapes at a certain height above the base:
Let's try to find the cross-sectional area of these two solids at a distancetallHon top of the base.
The cross-section of the hemisphere is always there
TenradiusXPart of the cross section isrectangular triangle,So we can take advantage
Now the cross-sectional area is
| A | = |
The cross section of a cut cylinder is always
The radius of the hole isH.We can find the area of the torus by subtracting the area of the hole from the area of the larger circle:
| A | = | |
| = |
It looks like both units have the same cross-sectional area in each layer.According to Cavalieri's principle, two solids must be the same
| = | ||
| = |
The ball contains
The earth is (approximately) a sphere with a radius of 6,371 kilometers.Therefore, its volume is
| 1 |
The average density of the earth is
This is 6 and 24 zeros!
If you compare the volume equations for cylinders, cones, and spheres, you'll probably notice one of the most satisfying relationships in geometry.Imagine we have a cylinder whose height is the same as the diameter of its base.We can now perfectly match the cone and sphere to it:
+
This cone has a radius
=
This sphere has a radius
This cylinder has a radius
Note that if we
surface area of the sphere
Finding the formula for the surface area of a sphere is very difficult.One reason is that we can't expand and "flatten" the surface of the sphere like we did with the cone and cylinder before.
This is a particular problem when trying to create a map.The earth has a curved three-dimensional surface, but any printed map must be flat and two-dimensional.This means geographers have to cheat:By stretching or compressing specific areas.
Here you can see several different types of maps calledprovide.Try to move the red square to see what area it isActuallyIt looks like a globe:
Mercator
Cylindrical
Robinson
Morverde
Pay attention to the size and shape of the square as you move around the mapactualChanging the area on the 3D Earth.
To find the surface area of a sphere, we can re-approximate it with a different shape—for example, a polyhedron with many faces.As the number of faces increases, the polyhedron begins to resemble a sphere more and more.
Coming soon: Proof of the surface area of a sphere
FAQs
What is the relationship between a cone sphere and cylinder? ›
The formula for the volume of a sphere is 4⁄3πr³. For a cylinder, the formula is πr²h. A cone is ⅓ the volume of a cylinder, or 1⁄3πr²h.
What is the formula for cylinder cone and sphere area? ›- The volume of a cylinder = Area of the base × Height of the cylinder = πr²h.
- Lateral Surface Area = Perimeter of base × height = 2πrh = πdh.
- Total Surface Area = Lateral Surface Area + Area of bases = 2πrh + 2πr² = 2πr (h+r)
So, the ratio of the volumes of the cylinder, the cone and the sphere is 3 : 1 : 2. Q.
What is the difference between a cone and a sphere? ›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. has a circular base that is joined to a single point (called the vertex). A sphere is a three-dimensional solid consisting of all points that have the same distance from a given center.
Is a 3d circle a sphere or cylinder? ›Circle is a 2-dimensional figure. Sphere is a 3-dimensional figure. Circle does not have volume.
How many cones are in a sphere? ›1 sphere has the same volume as 2 cones.
How many edges does a cylinder sphere and cone have? ›Cones have 1 edge. Cylinders have 2 edges. Sphere has no edge.
How many spheres can fit in a cylinder? ›There can be five sphere can be made from cylinder.
What is the ratio of curved surface area of sphere cylinder and cone? ›Since, all the areas are having only r as variables, we can eliminate this by taking ratio. So, the ratio of curved surface area of sphere, cylinder and cone is $4:4:\sqrt{5}$. Note: The key step for solving this question is the concept of curved surface area of various solids.
What is the relationship between the volume of a cone and the volume of a cylinder? ›Thus, the volume of a cone is equal to one-third of the volume of a cylinder having the same base radius and height.
What is the ratio of the volume of a cylinder a cone and a hemisphere? ›
The ratio of their volumes is 1:2:3.
What is the ratio of the volume of a sphere to a cylinder? ›Note : The volume of a sphere is 2/3 of the volume of a cylinder with same radius, and height equal to the diameter.
What is the formula for calculating volume of a sphere? ›The formula for the volume of a sphere is V = 4/3 π r³, where V = volume and r = radius. The radius of a sphere is half its diameter. So, to calculate the surface area of a sphere given the diameter of the sphere, you can first calculate the radius, then the volume.
What is the formula for volume of cylinders? ›A cylinder's volume is π r² h, and its surface area is 2π r h + 2π r². Learn how to use these formulas to solve an example problem.
What is a 3 dimensional circle called? ›2D: circle – 3D: sphere.
Is a ball a circle or sphere? ›A ball is spherical; it's shaped like a sphere — a three-dimensional version of the two-dimensional circle.
What are the 3 dimensions of a sphere? ›Unlike a circle, which is a plane shape or flat shape, defined in XY plane, a sphere is defined in three dimensions, i.e. x-axis, y-axis and z-axis.
What is the center of a sphere called? ›In geometry, a spherical sector, also known as a spherical cone, is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere.
Is a ice cream cone a sphere? ›An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone.
Can 3 cones make a cylinder? ›Each cone fills the cylinder to one-third quantity. Hence, such three cones will fill the cylinder. Thus, the volume of a cone is one-third of the volume of the cylinder.
Which shape is a sphere? ›
The sphere is a three-dimensional shape, also called the second cousin of a circle. A sphere is round, has no edges, and is a solid shape.
Is A cone A prism? ›Pyramids and cones are different from prisms and cylinders in that they have just one base and an apex, or a single point at which the other faces of the solid meet.
Which two shapes make a cylinder? ›A cylinder has two flat ends in the shape of circles. These two faces are connected by a curved face that looks like a tube. If you make a flat net for a cylinder, it looks like a rectangle with a circle attached at each end.
Does a cylinder have 2 edges? ›A cylinder has 2 edges. A cylinder has 2 faces and 1 curved surface.
How many edges a sphere has? ›Sphere has only 1 curved surface. So, it does not have any edges.
Does a cone have 3 edges? ›A cone has 1 edge. A cone has 1 face and 1 curved surface.
Does a cylinder have more volume than a sphere? ›What is the relationship between the volume of the sphere and the volume of the cylinder? (Answer: The sphere takes up two-thirds of the volume of the cylinder.)
Does a sphere have the same volume as a cylinder? ›The volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere.
What is the formula ratio of surface area of a sphere? ›So, what's the surface area of a sphere formula? Surface area of the sphere = 4 π r 2 , where “r” is the sphere's radius.
When a cylinder and a cone are of the same radius and of the same height? ›If a cylinder and a cone are of the same base radius and of the same height, then the ratio of the volume of the cylinder to that of the cone is 3 : 1. Hence, the given statement is true.
Do a cone and a sphere have the same radius? ›
A cone and a sphere have equal radii and equal volume. Hence, the ratio d : h = 1 : 2 .
What happens to the volume of a cone if the dimensions are doubled? ›So, when height is doubled, the volume is also doubled.
What is the ratio of a cone a hemisphere and a cylinder stand on equal bases and have the same height ›Thus, the ratio of their volumes is 1 : 2 : 3. Q. A cone, hemisphere and a cylinder stand on the same base and have equal height.
What is the ratio of volume of sphere to hemisphere? ›So the ratio is 2:1.
What is the ratio of a solid sphere and solid hemisphere have same volume? ›would then be 2:1.
What is the ratio of the radius of the sphere and the cylinder? ›Ratio=4πr24πr2=1:1.
What is the ratio of the surface area of a sphere to a cylinder? ›The surface area of sphere and the curved surface area of a cylinder are in the ratio 2 : 1.
What is the ratio of the volume of a cylinder to the volume of a cylinder having twice the height but the same radius? ›THE RATIO IS 1: 2.
How do you convert sphere area to volume? ›For a sphere, surface area is S= 4*Pi*R*R, where R is the radius of the sphere and Pi is 3.1415... The volume of a sphere is V= 4*Pi*R*R*R/3. So for a sphere, the ratio of surface area to volume is given by: S/V = 3/R.
What is the volume of pi? ›The volume of a right circular cylinder in terms of pi = πr2h cubic units where r is the radius of the cylinder and h is the height of the cylinder.
What is the volume of a circular cylinder? ›
Answer: The volume of a cylinder can be calculated using the formula V=πr2(h). When determining the volume of a cylinder, you are simply finding the area of the circular base shape and then multiplying this by the height.
What does a cylinder and cone have the same? ›The correct option is C
A cylinder and a cone have the same height and same radius of the base.
All have the same surface area. c.
What is the difference between cone and cylinder sphere? ›1. The cone has only one base, while the cylinder has two bases lower and upper bases. 2. The area of the base is π r 2 in the cone while the area of the base is 2 π r 2 in the cylinder.
How many edges does a cylinder a sphere and a cone have? ›Cones have 1 edge. Cylinders have 2 edges. Sphere has no edge.
What is the difference between sphere and cylinder? ›SPH, or sphere, is the amount of lens power needed to correct nearsightedness or farsightedness, whereas CYL, or cylinder, is the amount of lens power needed to correct astigmatism. When people talk about their eye degree or “power,” they are usually referring to their SPH value.
Is a sphere a prism or pyramid? ›Pyramids are polyhedra because they are made up of plane faces. Spheres are not polyhedra because they are curved.
What is the relationship between a circle and a cone? ›A cone is a solid composed of a circle and its interior ( base ), a given point not on the plane of the circle ( vertex ) and all the segments from the point to the circle. The radius of the cone is the radius of the base. The altitude of the cone is the perpendicular segment from the vertex to the plane of the base.