Cone – maybe the first thing that comes to your mind Sport bowling a. Yes one mathematical cone and playing skittles have nothing to do with each other, the geometric bodies even look very different from skittles in sports. Nevertheless, you often encounter them in everyday life and are indispensable in many areas.
cone – Definition, mesh and properties
The cone is a three-dimensional, geometric body.
definition
Before we get into how to calculate the volume or surface area of a cone, let’s take a look at the cone itself.
The cone is one pointed, three-dimensional body with a Circle as a base.
It is made up of the Base G, the tip S, the lateral surface M, the generatrix s and the height h.
This is how you can imagine a cone.
Figure 1: Cone
If you cut open a cone, you get the mesh of a cone, which looks like this:
Figure 2: Mesh of a cone
Draw a cone oblique image
drawing
Instructions
Figure 3: horizontal line with radius left and right and center
draw one horizontal line with the length of diameter. Mark the middle of this line with a point and label it with (center of the base).
Figure 4: horizontal line with radius right and left and center
Draw one now vertical line through the Focus with the length of radius (Half the length of the radius on each side).
Figure 5: Ellipse around endpoints
connect the four endpoints to an ellipse.
Figure 6: vertical height on center
draw one vertical line from the Focus from upwards with the length of the Height.
Figure 7: Connect the endpoints of the horizontal line to the endpoint of the height
Connect the right and left point the horizontal line with the end of elevation lines.
Figure 8: Erase auxiliary lines
Finally you can all lines, the you only for that To draw needed, offerase.
cone figure – Characteristics
- The cone owns
- two surfaces: base and lateral surface G and M;
- a peak
- and a page: the circle/perimeter of the base U.
- the height h of the cone stretches from the top vertically to the footprint G
- The cone is axisymmetric to the cone height hwhich passes through the apex and the midpoint of the base.
types of skittles
There are basically three types of skittles:
- Straight cones
- Leaning Cone
- truncated cones
1. Straight cones
Straight cones are the «normal» cones we’ve dealt with so far. It’s up to them Apex perpendicular to the center of the base.
When we talk about a cone in this article, we mean the right cone.
Figure 9: straight cone
2. Oblique cones
Oblique cones are cones where the apex not vertical on the center of the base stands, but is shifted. As the name suggests, it looks crooked.
Figure 10: oblique cone
The only difference between a straight cone and an oblique cone is how the height is determined. Here is the height the perpendicular which the Top with the Floor space connects.
the length the vertical route in between Top and the extended footprint equals to Height.
Figure 11: Height h of an oblique cone
3. Truncated cones
Truncated cones are cones where, in principle cut off the tip became. So sometimes they will too truncated cones or blunt cones called.
Figure 12: Truncated cone
In contrast to straight and oblique cones, truncated cones have an additional area. Therefore, their surface area and volume are calculated differently.
The following applies to the surface area O of a truncated cone:
The following applies to the volume V of a truncated cone:
R is always the larger radius, while r is always the smaller radius.
If you want a more detailed explanation of the formulas and the whole topic of truncated cones, then read our article!
Kegel in everyday life
You can find cones in many places in everyday life.
Calculating the volume of a cone
The formula for calculating the volume of a cone consists of a formula that general for all pointed bodiesi.e. also pyramids, for example, the following applies:
Then all you have to do is plug in the appropriate numbers for the area of the cone to get a concrete formula for the volume of a cone.
The formula for calculating the volume V of a cone is:
V is the volume, while r is the radius, h is the height, and π is the circle number.
If you want to learn more about the volume of a cone, then read our article on it. There you will find more detailed information and application examples.
The lateral surface of a cone
As already mentioned, one obtains, among other things, the surface area when a cone is broken down into its parts. This lateral surface can be calculated. But let’s take a closer look at what the lateral surface actually is.
the lateral surface M of a cone is a circle section (also called circle segment).
Of the radius this sector corresponds to the surface line s of the cone while the arc length b to the Scope U of the circle of the cone base is equivalent to.
In a figure, the lateral surface then looks like this:
Figure 13: Lateral surface M
The following applies to the lateral surface M of a cone:
If you want to see example problems on the lateral surface, then read our article on the surface area of a cone.
Calculate the surface area of a cone
The formula for the surface area is made up of the sum of the individual areas of the cone. As you saw above, there is a cone out two surfaces: the lateral surface M and the circular base G. If you add the area of the two areas, you get the formula for the surface area.
The surface area O of a cone is:
This formula can still be summarized.
O is the surface area, while r is the radius, s is the generatrix, and π is the circle number. The result of this calculation is given in m².
cone – formulas
In the table below you will find all the formulas you need to calculate a cone.
For reference, here’s another labeled cone so you can see what each size is again.
Figure 14: Cone
parameter
formula
diameter d
surface line s
Scope U
footprint G
lateral surface M
surface area O
Volume V
Calculate the length of the surface line s
In order to calculate the length of the surface line s, we first look at an image of a cone:
Figure 15: labeled cone
From this cone you can see that the surface line s, the height h and the radius r together one triangle form. This triangle forms between r and h a right one anglesince the height is perpendicular to the baseline.
Figure 16: Right angle cone
And what do we know about right triangles? You can Pythagorean theorem use to calculate the length of a side.
In our case corresponds s the hypotenusesince the generatrix is opposite the right angle. h and r correspond then the two catheti.
Figure 17: Pythagorean theorem
If we rewrite the formula appropriately, it will look like this:
But now we want to know the length of side s and not the length squared. That’s why we take the root.
And now we have the formula for calculating the length of the shell side s.
The following applies to the shell side s:
Danger! This procedure only works for straight cones.
Cone – The most important thing
- A cone is one pointed three-dimensional body with a Circle as a base.
- the height h of the cone extends from the top perpendicular to the footprint G
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There are straight cones, oblique cones, and truncated cones.
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Because truncated cones have an additional area, their volume and surface area are calculated differently
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the lateral surface of a cone is a circle section.
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The following applies to the lateral surface M: .
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The volume V of a cone is: .
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The surface area O of a cone is: .