Geometry problem: a cone meeting a cylindre.

• Vilestag
In summary, the conversation discusses two equations for a cone and a cylinder, and the question is posed about finding the length of any straight line on the cone inside the cylinder. The conversation also mentions the need for a third equation and the possibility of finding an expression for the average length over all possible angles.
Vilestag
Hi,

I have a cone on the z axis with his summit on height h meeting a cylinder on the x axis. The expressions should be:

cylinder: y2+z2=r2

cone: x2+y2 =(z-h)2tan(phi)2

If we consider any straight line on the cone, what is the length of this line inside the cylinder?

Is it possible to get a theoretical exprssion of this?

I tried the approche of the distance between two points, but i need a third equation to know the three coordinates of each points. Any ideas?

Thanks a lot,

Alx

Welcome to PF!

Hi Alx! Welcome to PF!

(have a phi: φ )
Vilestag said:
cylinder: y2+z2=r2

cone: x2+y2 =(z-h)2tan(phi)2

… i need a third equation to know the three coordinates of each points

the third equation will be a linear one, for a particular line

In fact, I need an expression of the length for ANY line if possible at all.

My problem goes much deeper: I need to find the mean length for all lines for all Φ...

All I have to do is to find an expression only in function of Φ and integrate it on all Φ. As simple as it sounds, I can't figure it out, because it's far from simple. I've done it in 2D (a triangle passing through a circle) and it worked, so i know my approach is good.

Anyway, I'll take any hint I get.

Tanks again,

Alx

Hi Alx!

I know it's complicated, but you'll just have to work through it.

Use y = xtanθ, find the length, and average over θ

Vilestag said:
cone: x2+y2 =(z-h)2tan(phi)2

I think you mean ...

$$x^2 + y^2 = \frac{(h-z)^2}{tan^2\phi}$$

1. How do you find the volume of a cone meeting a cylinder?

The volume of a cone meeting a cylinder can be found by first finding the individual volumes of the cone and cylinder, and then adding them together.

The volume of a cone is calculated using the formula V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone. The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius of the base and h is the height of the cylinder.

2. What is the formula for the surface area of a cone meeting a cylinder?

The formula for the surface area of a cone meeting a cylinder is the sum of the lateral surface area of the cone and the lateral surface area of the cylinder plus the area of the base of the cylinder.

The lateral surface area of a cone is calculated using the formula A = πrl, where r is the radius of the base and l is the slant height of the cone. The lateral surface area of a cylinder is calculated using the formula A = 2πrh, where r is the radius of the base and h is the height of the cylinder. The area of the base of the cylinder is simply πr².

3. What is the relationship between the cone and cylinder in this geometry problem?

The cone and cylinder in this geometry problem are touching at their bases, with the cone's vertex aligned with the center of the cylinder's base.

They also have the same base radius and height, making them similar in shape. The cone can be seen as a "cap" on top of the cylinder, creating a larger, more complex shape.

4. How can this geometry problem be applied in real life?

This geometry problem can have practical applications in fields such as architecture, engineering, and construction. For example, it can be used to design and calculate the volume and surface area of a water tank that has a conical top connected to a cylindrical base.

It can also be applied in manufacturing processes, such as creating a conical mold that fits onto a cylindrical base to create a specific product shape.

5. What are some key properties of a cone meeting a cylinder?

Some key properties of a cone meeting a cylinder include:

• They have the same base radius and height.
• Their volumes can be added together to find the total volume of the shape.
• Their lateral surface areas can be added together to find the total surface area of the shape.
• They are similar in shape, with the cone being a "cap" on top of the cylinder.
• Their intersection point is at the center of the cylinder's base.

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