Intersecting cylinder

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In summary, intersecting cylinders are cylindrical shapes that intersect or cross each other, creating a shared region. They share some properties with regular cylinders, such as a curved surface and circular base, but also have unique properties like a shared base and common axis of rotation. To calculate their volume, you can use the formula V = πr²h for each individual cylinder. Intersecting cylinders have various real-world applications in objects and structures like pipes and bridges. They can also add stability to structures, but proper reinforcement is necessary to ensure optimal stability.
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dhtjanda
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Dear all,

Can anybody help me the formula for finding the surface area of two intersecting cylinders with different radii and perpendicular to each other (similar to Steinmetz Solid but with different radii)?

Thanks for your help
David
 
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have you tried the same idea as when the radii are the same, i.e. slice it by a family of moving planes so that each cylinder is sliced in a rectangular slice?
 
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Hi David,

The formula for finding the surface area of two intersecting cylinders with different radii and perpendicular to each other is a bit complex, but I will try my best to explain it to you. First, let's define some variables:

r1 = radius of the first cylinder
r2 = radius of the second cylinder
h1 = height of the first cylinder
h2 = height of the second cylinder

To find the surface area, we need to consider three different surfaces: the curved surface of the first cylinder, the curved surface of the second cylinder, and the curved surface of the intersecting part. Let's break it down into these three parts:

1. Curved surface of the first cylinder:
The surface area of a cylinder is given by the formula 2πrh, where r is the radius and h is the height. In this case, the radius is r1 and the height is h1. So the surface area of the curved surface of the first cylinder is 2πr1h1.

2. Curved surface of the second cylinder:
Similarly, the surface area of the curved surface of the second cylinder is 2πr2h2.

3. Curved surface of the intersecting part:
To find the surface area of the intersecting part, we need to first find the circumference of the circle formed by the intersecting part. This can be done by using the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the radius of the intersecting part, which we can find by using the formula r = √(r1^2 + r2^2). Once we have the radius, we can find the circumference using the formula 2πr. Once we have the circumference, we can find the surface area of the intersecting part by multiplying it by the height of the intersecting part, which is the smaller of h1 and h2.

So the total surface area of the intersecting cylinders is the sum of these three parts:

Surface area = 2πr1h1 + 2πr2h2 + 2πr(h1 or h2)

I hope this helps. Let me know if you have any further questions or need clarification. Good luck!

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1. What is an intersecting cylinder?

An intersecting cylinder refers to two or more cylindrical shapes that intersect or cross each other at some point. This creates a shared region or intersection between the cylinders.

2. What are the properties of intersecting cylinders?

Intersecting cylinders share some common properties with regular cylinders, such as having a curved surface, a circular base, and a constant radius. However, they also have unique properties, such as a shared base and a common axis of rotation.

3. How do you calculate the volume of intersecting cylinders?

To calculate the volume of intersecting cylinders, you first need to find the volume of each individual cylinder by using the formula V = πr²h, where r is the radius and h is the height. Then, you can add the volumes of all intersecting cylinders to get the total volume.

4. What are some real-world applications of intersecting cylinders?

Intersecting cylinders can be seen in various real-world objects and structures, such as pipes, tunnels, and tunnels. They are also used in engineering and architecture, such as in the design of bridges and tunnels.

5. How do intersecting cylinders affect the stability of structures?

The intersection of cylinders can add strength and stability to structures, as the shared region between the cylinders creates a stronger support system. However, it is important to consider the forces acting on the intersecting cylinders and ensure they are properly reinforced for optimal stability.

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