# Volume of a slanted cylinder

I have a cylinder of radius 4 and height 10 that is at a 30 degree angle. I need to find the volume.

I have no clue how to do this, I have spent quite a while on it and went through many ideas but I think my best idea was this.

I know that the radius is 4 so if I cut the cylinder in half from corner to corner I will have two side lengths giving me a third side length. So this gives

$$\sqrt{116} = height$$

Or the length of the tall sides.

Now I just plug this into my formula

$$\pi r^2 h$$

$$\pi *16*\sqrt116$$

This is about $34\pi$ which is way off. What did I do wrong?

Picture for reference:

Let's get our terms straight here. $h$ is the height of the cylinder; $\ell$ is the side length, and $r$ is the radius. This cylinder is tilted at $30^\circ$.

The volume of a cylinder like this is given by the formula: $$V = \pi r^2 h$$

For your problem, when you say "height of $10$," I'm assuming you actually mean $\ell=10$. From some trig, we see that: $$h = \ell\sin30^\circ = \frac \ell 2 = 5$$

Thus, our volume is: $$V = \pi (4)^2(5) = 80\pi = 251.3\;\text{cubic units}$$

EDIT:
In the comments, it was mentioned that the vertical length is known. Thus, the solution is much simpler:

$$V=\pi r^2 h = \pi(4)^2(10) = 160\pi = 502.6$$

• No I actually meant that the entire height is 10, the side length is unknown.
– user84004
Jun 29 '13 at 15:20
• @user84004 Ok. The same formulas apply, but you just don't have to do the trig stuff first. (See edit) Jun 29 '13 at 23:07

To expand on the answer above, consider an ordinary cylinder as a stack of thin discs:coins, poker chips, whatever. The stack has a height, given by the total thickness of all the coins, and a certain volume, given by the total volume of all the coins.

Now, sort of srounch the stack over at an angle, so that each coin is offset by a little from the coin below it, each in the same direction. The new cylinder will have the same height (the number and thickness of the coins hasn't changed), and the same volume (the number of coins and the individual volumes haven't changed) . And the same will be true no matter how you change the angle.

The surface area of the cylinder is not, unfortunately, as simple...

• I do believe the surface area is just $2\pi r(\ell + r)$... Jun 28 '13 at 0:49

You should use the standard formula for volume of a cylinder, except that the height is now the projected height of the axis length onto the perpendicular to the base plane. Thus, if the length of the axis of your cylinder is 10, then the projected height is 10*sin(30deg) = 5 (to see this, consider the projection of the cylinder in the plane: you get a right-angled triangle with hypothenusa 10 and angles 30, 90, 60, in which you need the side opposite to 30deg angle). That should give V = pi*r^2 * 5 = 251.328.

EDIT: If you find it difficult to understand why it is this way, recall how one can calculate the area of a parallelogram from that of rectangle with equal base length and height: by cutting off one triangle from the rectangle from the left side and pasting it back on the right side, you obtain a parallelogram of equal area. The slanted and straight cylinders are in fact the same: if you project them in the back plane, you see parallelogram and rectangle, respectively. Oop, when writing this I realized I may have made a mistake: it should not have been multiplied by sin(30deg) after all, because of what I just said. The issue is rather how the radius is measured: it is for the cross-section perpendicular to the axis of the cylinder, or is it the radius of the base, which is at an angle to this axis? In the former case, you will need to find the projected radius first.

• I get more. But did it very casually. Jun 27 '13 at 23:44
• I get more, two, er, too.
– robjohn
Dec 19 '13 at 23:43

assuming that the height giving in the question where to be the slanted height.Therefore the straight height H, must be found before proceeding for further solving using l multiplied the sine of the giving angle 30 degree to give 5, after that you substitute in normal volume formula to give ans. 251.36