The area of the figure is half the product of the diagonals:
A = (1/2)*(5 cm)*(10 cm) = 25 cm²
If both diagonals are doubled, each of the dimensions in the above equation is multiplied by 2, so two factors of 2 are thrown into the product. The area of the larger figure will be 2*2 = 4 times that of the figure shown:
A = (1/2)*(10 cm)*(20 cm) = 100 cm²
100 cm² = 4*(25 cm²)
If both diagonals are doubled, the area changes by a factor of 4.
_____
When every linear dimension of a geometric figure is multiplied by the same scale factor (2, in this case), the new figure is "similar" to the old one. Any area measure of the new figure will be the square of that scale factor times the original area. Any volume of the new figure will be the cube of that scale factor times the original volume.
