# Describe in words the surface whose equation is given. r = 6

The aim of this question is to infer/visualize the shapes/surfaces constructed from a given mathematical function using prior knowledge of standard functions.

The standard equation of a circle in two-dimensional plane is given by:

$x^2 \ + \ y^2 \ = \ r^2 \ … \ … \ … \ ( 1 )$

The standard equation of a sphere in three-dimensional space is given by:

$x^2 \ + \ y^2 \ + \ z^2 = \ r^2 \ … \ … \ … \ ( 2 )$

We will use both of these equations for solving the given question.

Given:

$x^2 \ + \ y^2 \ = \ r^2$

Substituting $r \ = \ 6$:

$x^2 \ + \ y^2 \ = \ ( 6 )^2$

$\Rightarrow x^2 \ + \ y^2 \ = \ 36$

Part (a): Describing the given equation in a two-dimensional plane.

Compared with equation no. (1), we can see that the given equation represents a circle located at the origin with a radius of 6.

Part (b): Describing the given equation in a three-dimensional space.

Compared with equation no. (2), we can see that the given equation is not a sphere since the third axis $z$ is missing.

Using information from part (a), we can see that the given equation represents a circle located in the xy-plane with a radius of 6 for a given fixed value of $z$.

Since $z$ can vary from $– \infty$ to $+ \infty$, we can stack such circles along z-axis.

Hence, we can conclude that the given equation represents a cylinder with radius $6$ extending from from $– \infty$ to $+ \infty$ along $z-axis$.

## Numerical Result

The given equation represents a cylinder with radius $6$ extending from $– \infty$ to $+ \infty$ along $z-axis$.

## Example

Describe the following equation in words (assume $r \ = \ 1$ ):

$\boldsymbol{ x^2 \ + \ z^2 \ = \ r^2 }$

Substituting $r \ = \ 1$:

$x^2 \ + \ z^2 \ = \ ( 1 )^2$

$\Rightarrow x^2 \ + \ z^2 \ = \ 1$

Compared with equation (1), we can see that the given equation represents a circle located in the xz-plane with a radius of 1 for a given fixed value of $y$.

Since, $y$ can vary from $– \infty$ to $+ \infty$, we can stack such circles along y-axis.

Hence, we can conclude that the given equation represents a cylinder with radius $6$ extending from from $– \infty$ to $+ \infty$ along $y-axis$.