Area of an inclined surface with respect to the original surface

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In summary, The conversation discusses the calculation of stress in an inclined plane of a bar under tension. The formula for finding the area of an inclined plane with a rectangular cross-section is straightforward, but there is confusion about using an elliptical cross-section and whether or not the formula can be applied to any cross-section shape. The concept of "flux" is suggested as a possible explanation and a diagram is provided for reference.
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TL;DR Summary
Relationship of inclined area with respect to original area
Hi, I have a problem with inclined planes. The idea is to calculate the stress in an inclined plane of a bar under tension for which you need the surface. I have no idea how this surface is derived, though. In the attached file, you can see what I mean. For a rectangular cross-section, it's straightforward, just applying the rectangle area with the new inclined length. Now, everywhere I see, everyone uses the same rectangular bar as an example.

However, in one single textbook, the exercise uses an elliptical cross-section to seemingly represent a random surface. They use the same formula for the area, but without any explanation, apparently trivially and immediately deriving, but I don't see why the area of an inclined elliptical surface with respect to the original surface is the same as the rectangular one.

My suspicion is that it has to do with the vector area which, being the same direction as the normal, is somehow projected onto the other's area vector, but I don't see it. Thanks for the help.

If you cut a cylinder you get an ellipse, which is just a stretched circle, so the area of an ellipse is simply pi*(semi-minor axis)*(semi-major axis). The first is the radius of the cylinder, and the second one you can find in the same way as the rectangular case.

Okay, I see that now. It seems to me that for all common cross-sections this is true, at least the ones I can think of, even compound ones such as an H-beam.

But what about any cross-section? By any I mean, an area enclosed by a loop that doesn't cross itself such as a horseshoe, a star/asterisk, sickle, quarter-moon, etc. Could it be proven whether or not ##A _\theta=\frac {A_0} {cos \theta}## is valid for the area of a section resulting from an inclined plane cutting through a bar with cross-section as described previously, where ##\theta## is the angle of inclination?

As long as the bar stays the same along its axis that formula stays true - all you do is stretch the area in one direction.

Alright, looking at it as a scaling factor in one direction does help. This clears it up, thanks.

1. What is the formula for calculating the area of an inclined surface with respect to the original surface?

The formula for calculating the area of an inclined surface with respect to the original surface is A = Aocosθ, where A is the area of the inclined surface, Ao is the area of the original surface, and θ is the angle of inclination.

2. How is the angle of inclination measured for calculating the area of an inclined surface?

The angle of inclination is measured as the angle between the inclined surface and the original surface. It is typically measured in degrees or radians.

3. Can the area of an inclined surface be greater than the area of the original surface?

Yes, the area of an inclined surface can be greater than the area of the original surface. This is because the inclined surface is stretched out and not parallel to the original surface, resulting in a larger surface area.

4. Is the formula for calculating the area of an inclined surface with respect to the original surface applicable to all shapes?

Yes, the formula A = Aocosθ can be applied to any shape as long as the angle of inclination is known. However, the shape of the inclined surface may affect the accuracy of the calculation.

5. How is the area of an inclined surface with respect to the original surface useful in real-world applications?

The area of an inclined surface with respect to the original surface is useful in various fields such as architecture, engineering, and physics. It can be used to calculate the surface area of inclined roofs, ramps, and other structures. It is also used in calculating the force and work done on an inclined plane in physics problems.

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