# Surface area of a circular cylinder cut by a slanted plane

• LaplacianHarmonic
In summary, the surface area of a circular cylinder cut by a slanted plane can be calculated using the formula SA = 2πrh + πr², and the slant height can be determined using the Pythagorean theorem. The angle of the slanted plane does not affect the surface area, and it is possible for the surface area to be greater than that of a complete cylinder if the slanted plane cuts at a sharp angle.
LaplacianHarmonic

## Homework Statement

Cylinder : x^2 + y^2 = 1
Plane that intersects above cylinder: y + z = 2

What is the surface area of the sides of this cylinder?

## Homework Equations

dS= R1 d@ dz
@ is from 0 to 2 pi
z is from 0 to 2 - y

dS=(Zx^2 + Zy^2 + 1)^.5 dA
Where Z = 2 - y

## The Attempt at a Solution

I used
dS=(Zx^2 + Zy^2 + 1)^.5 dA
Where Z = 2 - y

To get
2^.5 r dr d@
r is from 0 to R1
@ is from 0 to 2pi

(2)^.5*(pi)(R1)^2

Is this correct??

LaplacianHarmonic said:

## Homework Statement

Cylinder : x^2 + y^2 = 1
Plane that intersects above cylinder: y + z = 2

What is the surface area of the sides of this cylinder?

## Homework Equations

dS= R1 d@ dz
@ is from 0 to 2 pi
z is from 0 to 2 - y

dS=(Zx^2 + Zy^2 + 1)^.5 dA
Where Z = 2 - y
Please don't write Z when you mean z. Also your first relevant equation is ##dS = R1 d\theta dz##. What is ##R1##? Where does the equation I colored red come from and why do you need it since you already have ##dS##? Anyway, you should set your integral up in terms of ##\theta## and ##z## as your first three lines suggest. And no, your answer is not correct.

R1 is the constant radius of the cylinder. In this case it equals to the number 1.

Where does the equation I colored red come from and why do you need it since you already have ##dS##?

The other equation for dS comes from the other derivation of dS in which I rotate a curve around the path of a circle in 3 dimensions.

Anyway, you should set your integral up in terms of ##\theta## and ##z## as your first three lines suggest.

Okay. I am just learning how to write correctly here. And no, your answer is not correct.

Can you help me? What did I do wrong?

dS= R1 d@ dz
@ is from 0 to 2 pi
z is from 0 to 2 - y
Where R1 = 1, so using proper notation you are saying:
##dS = d\theta dz##
##0\le \theta \le 2\pi##
##0 \le z \le 2-y##.
All you have to do is express ##y## in terms of ##\theta## and set up the double ##dzd\theta## integral you describe.

LCKurtz said:
dS= R1 d@ dz
@ is from 0 to 2 pi
z is from 0 to 2 - y
Where R1 = 1, so using proper notation you are saying:
##dS = d\theta dz##
##0\le \theta \le 2\pi##
##0 \le z \le 2-y##.
All you have to do is express ##y## in terms of ##\theta## and set up the double ##dzd\theta## integral you describe.
So, 2 - sin@?

That gives me the value of 4pi as the final answer.

Does not come out from the

vector V= < 1cos@, 1sin@, z>
dS = <1cos@, 1sin@, 0> for the above vector eqn start.
dS=d@dz

Does not appear to have a vector equation...?

Last edited:
Yes, that is correct. By the way, it is easy to type special characters like ##\theta##. Just type ##\theta##.

Thread reopened after a post has been deleted.

berkeman
And @LaplacianHarmonic -- please read the LaTeX tutorial in the Help/How-To section of the PF under INFO at the top of the page. It is very difficult trying to follow what you are writing when you try to type out equations as text. Thank you.

And after a PM discussion with the OP, this thread will remain closed.

## 1. What is the formula for finding the surface area of a circular cylinder cut by a slanted plane?

The formula for finding the surface area of a circular cylinder cut by a slanted plane is SA = 2πrh + πr², where SA represents the surface area, r is the radius of the circular base, and h is the height of the cylinder.

## 2. How do you determine the slant height of a cylinder cut by a slanted plane?

The slant height of a cylinder cut by a slanted plane can be determined by using the Pythagorean theorem. The slant height, denoted by l, can be found by taking the square root of the sum of the square of the height of the cylinder (h) and the square of the radius (r).

## 3. Can the surface area of a circular cylinder cut by a slanted plane be calculated without knowing the slant height?

Yes, the surface area of a circular cylinder cut by a slanted plane can be calculated without knowing the slant height. This can be done by using the formula SA = 2πrh + πr², where h is the height of the cylinder and r is the radius of the circular base.

## 4. How does the angle of the slanted plane affect the surface area of a circular cylinder?

The angle of the slanted plane does not affect the surface area of a circular cylinder. The surface area is only affected by the height of the cylinder and the radius of the circular base, as shown in the formula SA = 2πrh + πr². The angle of the slanted plane only affects the shape and size of the cross-section of the cylinder.

## 5. Can the surface area of a circular cylinder cut by a slanted plane be greater than the surface area of a complete cylinder?

Yes, it is possible for the surface area of a circular cylinder cut by a slanted plane to be greater than the surface area of a complete cylinder. This can happen if the slanted plane cuts the cylinder at a sharp angle, resulting in a larger cross-sectional area compared to the complete cylinder.

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