Perimeter and surface area of skewed end semi cylinder

[smstoankur]

I want to calculate the surface area of semi cylinder with one or both ends skew at a given angle. To calculate the area/perimeter i need to find out the end profile of the semi cylinder.
If the skew plane is perpendicular to the cylinder axis, the cut surface would be half ellipse with major axis skewed at the angle and minor axis equal to the radius of the semi cylinder. In this case the surface area can be easily calculated since the end part can be easily identified.
But if the skew plane is not perpendicular to the cylinder axis, then the end part would be skew along with bevel. In this case, the end part is again ellipse. I want to calculate the surface area of the semi cylinder for the second part. Please assume the radius to be 1m and length as 100m. The skew angle is 45 degrees at the end and inclined at 45 degrees to the perpendicular axis. Please help me solve this problem.

 

[tiny-tim]

welcome to pf!

hi smstoankur! welcome to pf!

show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[smstoankur]

Hi tiny-tim,

I am only interested in the approximate lateral area of the top part of semi cylinder so I tried solving it using engineering drawing by diving it into number of smaller strips and adding the total. I have attached the two problems which I explained before. The problem I discussed is with the semi circular ends so it is easy to work with. But I have to work with multi radius arches which can be difficult to work with. I am only interested in the lateral surface area. If I can calculate the boundary then I can work out the area by approximation. I was looking for a generalized solution for calculation of the boundary in different cases.

 

[smstoankur]

How to delete this thread?

 

[blue_raver22]

To calculate the surface area of a skewed end semi cylinder, we need to first determine the end profile of the semi cylinder. As mentioned, if the skew plane is perpendicular to the cylinder axis, the end profile would be a half ellipse with a major axis skewed at the given angle and a minor axis equal to the radius of the semi cylinder.

In this case, the surface area can be calculated using the formula:
Surface Area = π * r * (r + √(h^2 + r^2))

Where r is the radius of the semi cylinder and h is the height of the semi cylinder, which can be calculated as the length of the cylinder divided by 2.

However, if the skew plane is not perpendicular to the cylinder axis, the end profile would also be skewed along with a bevel. In this case, the end profile would still be an ellipse, but with a different major axis and minor axis.

To calculate the surface area in this scenario, we can use a similar formula, but we need to first calculate the major and minor axes of the ellipse.

The major axis (a) can be calculated as:
a = r * cos(θ)

Where r is the radius of the semi cylinder and θ is the angle of skew.

The minor axis (b) can be calculated as:
b = r * sin(θ)

Once we have determined the major and minor axes, we can use the same formula as before to calculate the surface area:
Surface Area = π * a * b

In your specific case, with a radius of 1m and length of 100m, the surface area of the semi cylinder with a 45 degree skew and inclined at 45 degrees to the perpendicular axis would be:

Surface Area = π * (1 * cos(45)) * (1 * sin(45)) = π * (0.707 * 0.707) = 0.497π square meters.

I hope this helps you solve your problem. Keep in mind that this is just one way to calculate the surface area of a skewed end semi cylinder and there may be other methods or formulas that could be used depending on the specific scenario. As a scientist, it is always important to carefully consider the variables and equations involved in a problem to determine the most accurate and appropriate solution.