How do I create a cylinder with a taper at a specific angle?

Question

I'm creating a cylinder in Blender and want to taper the top or angle the sides to a specific 75-degree angle. How can I achieve this by scaling the top horizontal face appropriately in Blender?

Answer 1

Just in case you want a mathematical formula to get a scale factor (although, as I said, that's just maths and not a real Blender question - scroll to the bottom to find out how you can use the maths in Blender):

If you look at the desired cylinder's cross section, there are some things we know: the height $h$ of the cylinder and the radius $r$ (with a height of 2 m and a radius of 1 m for the default primitive object).

Apart from that, the border edges going upwards should be angled by $\beta=75°$ and the height is of course orthogonal, so we can build a right triangle with a remaining angle of $\alpha=15°$ .

What you now need to find is the length $a$ which is how much the lower radius exceeds the desired upper radius to build the triangle. The new radius $b$ is then just the radius $r$ minus $a$ , and to get $b$ you have to scale the original radius by a factor of $\frac{r-a}{r}$ , which in case of a default radius of 1 m comes down to simply $1 - a$ .

And because we have all angles and the value for the height, you can easily calculate the value for $a$ with the following formula:

$$a=h\cdot\frac{\sin{\alpha}}{\sin{\beta}}$$

With the given values this results in:

$$a=2\cdot\frac{\sin{15°}}{\sin{75°}}\approx 0.535898$$

So the upper radius has to be scaled by a factor of

$$\frac{1-0.535898}{1}=1-0.535898=0.464102$$

to get a cylinder (or actually a frustum) with the sides angled at 75°. I hope all the variables did not confuse too much, I thought if you wanted to do this with cylinders of other heights and radii and maybe have different angles as well, it would be better to not only give you formulas with fixed values.

Basically, the whole operation can be written in a single line formula:

$$\text{upper radius }= \text{lower radius}-\text{height}\cdot\frac{\sin{15°}}{\sin{75°}}$$

which can help you since Blender accepts mathematical formulas in value fields. It gets a little bit more complicated since you have to convert the angles in radians:

$$\text{upper radius }= \text{lower radius}-\text{height}\cdot\frac{\sin{\pi\frac{15°}{180°}}}{\sin{\pi\frac{75°}{180°}}}$$

Unfortunately, you cannot enter this complex formula as a factor while you are scaling, but here is the solution:

Now to the part how you can use maths in Blender

Okay, this looks a bit complicated now. But as I mentioned above, this is not a cylinder you are talking about, but a frustum or truncated cone. So what you can actually do instead of creating a cylinder and scaling down the upper face to get the desired angle, just do the following:

Press Shift + A > Add > Cone . Then open the properties at the bottom left of the viewport and just enter

1-2*sin(pi*15/180)/sin(pi*75/180)

into the value for Radius 2 , that is Radius 1 subtracted by the Depth multiplied by the angle calculations. And the result is the angled cylinder.

Answer 2

Under Mesh Edit Mode Overlays you can enable Edge Angle and see the angles when scaling using S to scale and Shift + Z

Answer 3

You could as well, just make a single edge, rotate it by the angle you want(I randomly used 25° in the GIF), move it away from the center(object space origin) by the length of your desired radius and then use Screw modifier to make the cylinder out of it: