# Thread: Find 3D cylinder equations

1. ## Find 3D cylinder equations

This is a question from my calculus 3 class.

"Find the equations for the following right circular cylinders. Each cylinder has a radius a and is tangent to two coordinate planes."

Basically I am given three 3D graphs of cylinders tangent to the 3 different planes. The center points for the different graphs are (a, 0, a), (a, a, 0), and (0, a, a). Basically here is what I have for my cylinder equations:

(a, 0, a) ... ax^2 + az^2 = a
(a, a, 0) ... ax^2 + ay^2 = a
(0, a, a) ... ay^2 + az^2 = a

I am however a little confused as to whether I square the radius or not? Either way, am I doing it right? I feel something is wrong maybe.

I've attached a picture of the graphs in my textbook.

2. A cylinder parallel to an axis just means that the value does not depend on the coordinate along that axis.
For example, the first cylinder is parallel to the y axis, so you don't need to consider y in the equation. All you need is to write the equation of the circle in the xz plane (this automatically becomes a cylinder in 3 dimensions - think about it).

So the equation of the circle for the first one is:
(x-a)^2 + (z - a)^2 = a^2

Do you understand now?

3. Start by finding the equation of each circle in the plane...

The general equation of a circle (in the $\displaystyle x, y$ plane) is $\displaystyle (x - h)^2 + (y - k)^2 = r^2$, where $\displaystyle (h, k)$ is the centre and $\displaystyle r$ is the radius. Though this translates to any of the three coordinate planes.

I'll do Question 1 and leave 2 and 3 for you...

In Q.1, the circle is in the $\displaystyle x-z$ plane, so your equation will involve $\displaystyle x$ and $\displaystyle z$.

The centre is $\displaystyle (a, a)$ and its radius is $\displaystyle a$.

So the equation of the circle is $\displaystyle (x - a)^2 + (z - a)^2 = a^2$.

4. Ohh, okay I see what to do. Thanks guys! I was looking at the cylinder so hard that I forgot about the circle part. I guess I was looking at a cylinder example at the origin, hence my weird equations.

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# cylinder equation in 3d geometry

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