Task 1196

The task

On the 5 by 5 square grid below, each dot is 1 cm from its nearest horizontal and vertical neighbors. What is the product of the value of the area of square $ABCD$ (in cm$^2$) and the value of the perimeter of square $ABCD$ (in cm)? Express your answer in simplest radical form. [asy]unitsize(1cm); defaultpen(linewidth(0.7)); dot((0,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((1,0)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((2,0)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((3,0)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((4,0)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); draw((0,3)--(3,4)--(4,1)--(1,0)--cycle); label("$A$",(3,4),N); label("$B$",(4,1),E); label("$C$",(1,0),S); label("$D$",(0,3),W); [/asy] Note that when we say the grid is 5 by 5 we mean that each row and column contains 5 dots!