#### Answer

18

#### Work Step by Step

Let the unknown number be $x$. This means that one-half of the number is $\frac{x}{2}$.
As a result, the square of the unknown number is $x^{2}$ whereas the square of one-half of the number is $(\frac{x}{2})^{2}$.
Since the difference between the square of a number and the square of one-half of the number is 243, we write the following equation and solve it:
$x^{2}-(\frac{x}{2})^{2}=243$
$x^{2}-(\frac{x^{2}}{4})=243$
$\frac{4x^{2}-x^{2}}{4}=243$
$\frac{3x^{2}}{4}=243$
$3x^{2}=972$
$x^{2}=\frac{972}{3}$
$x^{2}=324$
$x=\pm\sqrt {324}$
$x=\pm 18$
Check when x=18:
$18^{2}-(\frac{18}{2})^{2}=243$
$18^{2}-(9)^{2}=243$
$324-81=243$
$243=243$
The problem asks for the square of a positive number, meaning that we should disregard the negative answer, giving us 18 as our answer.