How to Find Surface Area of Intersecting Cylinders with Different Radii?

[dhtjanda]

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

 

[mathwonk]

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?

 

[M98Ranger]

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!

Best