I need numbers...so yeah...
Check the picture below.
so, a cone inscribed in a square pyramid, like the blue one there, will have the a diameter that is the length of the base of the pyramid, as you can see there, is 6, now, since the radius is half of the diameter, that means r = 3.
we know the slant height is 9, since the slant height and the height segment make a right-triangle with the base's segment, then we use the pythagorean theorem to get the height "h".
Answer: x=29
Work: 5x-67=78 5x=78+67 5x=145
Check the picture below, so the square looks more or less like so, since it has all equal sides, then we can simply get the distance of TE and multiply it times 4 and that's the perimeter.
![~~~~~~~~~~~~\textit{distance between 2 points} \\\\ T(\stackrel{x_1}{-1}~,~\stackrel{y_1}{3})\qquad E(\stackrel{x_2}{-2}~,~\stackrel{y_2}{-3})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ TE=\sqrt{[-2 - (-1)]^2 + [-3 - 3]^2}\implies TE=\sqrt{(-2+1)^2+(-6)^2} \\\\\\ TE=\sqrt{(-1)^2+(-6)^2}\implies TE=\sqrt{37}~\hfill \stackrel{\textit{4 times that much}}{4\sqrt{37}}](https://tex.z-dn.net/?f=~~~~~~~~~~~~%5Ctextit%7Bdistance%20between%202%20points%7D%20%5C%5C%5C%5C%20T%28%5Cstackrel%7Bx_1%7D%7B-1%7D~%2C~%5Cstackrel%7By_1%7D%7B3%7D%29%5Cqquad%20E%28%5Cstackrel%7Bx_2%7D%7B-2%7D~%2C~%5Cstackrel%7By_2%7D%7B-3%7D%29%5Cqquad%20%5Cqquad%20d%20%3D%20%5Csqrt%7B%28%20x_2-%20x_1%29%5E2%20%2B%20%28%20y_2-%20y_1%29%5E2%7D%20%5C%5C%5C%5C%5C%5C%20TE%3D%5Csqrt%7B%5B-2%20-%20%28-1%29%5D%5E2%20%2B%20%5B-3%20-%203%5D%5E2%7D%5Cimplies%20TE%3D%5Csqrt%7B%28-2%2B1%29%5E2%2B%28-6%29%5E2%7D%20%5C%5C%5C%5C%5C%5C%20TE%3D%5Csqrt%7B%28-1%29%5E2%2B%28-6%29%5E2%7D%5Cimplies%20TE%3D%5Csqrt%7B37%7D~%5Chfill%20%5Cstackrel%7B%5Ctextit%7B4%20times%20that%20much%7D%7D%7B4%5Csqrt%7B37%7D%7D)