Finding distribution of $X^2+Y^2$ where $X,Y\sim N(0,1)$

Finding distribution of $X^2+Y^2$ where $X,Y\sim N(0,1)$

Assume I have two random independent standard normal variables $X,Y\sim
N(0,1)$, How can I find the distribution of $Z=X^2+Y^2$?
I thought integrating the convolution, i.e
$F_Z(z)=\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{X^2}(t)f_{Y^2}(x-t)dt$.
If X is a normal variable, does also $X^2$ is a normal variable (then I
could find the density by simply substituting)?