From area to perimeter

Let $x$ be the spacing of the grid lines.

$A=trapezoid=2\left(\dfrac{1}{2}\right)(x+4x)(3x)=15x^2$

$B=triangle=2\left(\dfrac{1}{2}\right)(x)(2x)=2x^2$

$C=square=2(2x)(3x)=12x^2$

$D=triangle=2\left(\dfrac{1}{2}\right)(2x)(3x)=6x^2$

$E=triangle=\dfrac{1}{2}(8x)(2x)=8x^2$

So we have,

$84=(8x)(8x)-(15x^2+2x^2+12x^2+6x^2+8x^2)$

$84=64x^2-43x^2$

$x=2$

Computation of the perimeter (consider the lower photo):

$2Red=2\sqrt{6^2+6^2}=12\sqrt{2}$

$2Blue=2\sqrt{4^2+2^2}=4\sqrt{5}$

$2Green=2(6)=12$

$2Violet=2(2)=4$

$2Brown=2\sqrt{4^2+6^2}=4\sqrt{13}$

$2Black=2\sqrt{4^2+8^2}=8\sqrt{5}$

Thus, the perimeter is

$P=12\sqrt{2}+4\sqrt{5}+12+4+4\sqrt{13}+8\sqrt{5}=$ $\boxed{12\sqrt{2}+12\sqrt{5}+4\sqrt{13}+16}$

Note:

1. $A,B,C,...etc$ are areas of the colored regions (consider the upper photo only).

2. $2Red$ means the total length of the two lines colored red, and so on..(consider the lower photo only).