How to Find the Area of a Truncated Ellipse?

In summary, the conversation discusses how to find the area of intersection between a cylinder and a plane that forms a truncated ellipse. The suggested method involves finding the length of the perpendicular bisector from the center to the chord, the width of the circle based on distance from the center, and the distance from the center based on position in z. The area of the rectangle and the two semi-ellipses can be found using the formula for the area of an ellipse. Another suggestion is to recast the width in terms of theta and use an integral to find the area.
  • #1
AlexK864
2
0
Does anyone know how to find the area of an intersection between a cylinder of height 8 and radius 6 and a plane that passes through the cylinder, forming a chord of 10 units at the top and bottom faces of the cylinder? The area of intersection curves with the cylinder, forming a truncated ellipse, not a rectangle. I'm thinking you could divide the truncated ellipse into a rectangle and two sections of a circle, and find the rectangle by pretending the two chords are opposite edges of a rectangular prism, but I don't know how to find the width of the prism. Can anyone help?
 
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  • #2
First find the length of the perpendicular bisector from the center to the chord using Pythagorean thm.
Then find the width of the circle based on distance from the center (r cos).
Then find the distance from center (height) based on position in z. (linear)
 
  • #3
Ok so I figured out the area of the rectangle, 60 sqrt 3, and I know that the height of each semi-ellipse is 1, and their length is 6 sqrt 3, so how would I find the areas of the two semi-ellipses?
 
  • #4
area of an ellipse is ##\pi a b ## where a is minor semi-axis and b is major semi-axis.
 
  • #5
What if you recast width in terms of theta where theta goes from ## -sin^{-1}\frac{\sqrt 11}{6}## to ##sin^{-1}\frac{\sqrt 11}{6}##?
then you will have an integral
## \int_{-sin^{-1}\frac{\sqrt 11}{6}} ^{sin^{-1}\frac{\sqrt 11}{6}} width(\theta)length(\theta)d\theta##.
 

1. What is an area of a truncated ellipse?

The area of a truncated ellipse refers to the space enclosed by the curved boundary of a truncated ellipse shape.

2. How do you calculate the area of a truncated ellipse?

The formula for calculating the area of a truncated ellipse is A = π * a * b, where a and b are the semi-major and semi-minor axes of the ellipse, respectively.

3. What is the difference between a truncated ellipse and a regular ellipse?

A truncated ellipse is a portion of an ellipse shape that has been cut off or removed, while a regular ellipse is a complete, unaltered shape. The area of a truncated ellipse will be smaller than that of a regular ellipse.

4. Can the area of a truncated ellipse be negative?

No, the area of a truncated ellipse cannot be negative. It is always a positive value, as it represents the physical space enclosed by the shape.

5. What are the real-world applications of calculating the area of a truncated ellipse?

The area of a truncated ellipse is used in various fields such as engineering, architecture, and physics for designing structures and calculating volumes of objects with curved surfaces, such as tanks, tunnels, and domes.

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