I need help with constructing a "3D" cone out of paper with precise angles. I just can't seem to wrap my head around it. - I need to make an accurate and precise model of a cone out of paper, with it's angle around the cap/spike equal to $90^° $ - I know it seems obvious but getting the Side Area to stick itself into a 3D model is going very wrong so far for me - even though I used this relationship:$$\theta = 2\pi \sin\frac\alpha2$$ image link
Sorry, my mistake. Corrected answer

$ L =\frac{R}{sin(\alpha/2)}$ . Cut a sector of circle with radius $L$ and angle it makes

$ \theta = 360^o \dfrac{R}{L}$
See the figure.. It explains. Take a rectangular paper with one side $2 \pi R$. 

Now from cone figure 

$ L =\frac{R}{sin(\alpha/2)}$ 

Now from paper figure what is $P$?

Simply

$P = \sqrt{L^2-(\pi R)^2}=\sqrt{\frac{R^2}{sin^2(\alpha/2)^2}-(\pi R)^2}$

$P = R \sqrt{cosec^2(\alpha/2)-\pi ^2}$

Draw a perpendicular bisector in paper with length $P$, Complete triangle , cut and glue. :)