Ratios

Similar polygons are similar in all respects.....

If we divide each polygon into 10 equal triangles...the base of each triangle will be formed by a side.....and the side [base] of a triangle in the larger polygon will be  3+1/3  = 10/3   that of the base of a triangle in the smaller one.......and the height of each corresponding triangle in both figures will have the same relationship....[10/3].......so the ratio of the area of a triangle in the larger polygon to a corresponding triangle in the smaller polygon will just be the square of the ratio of the sides = [10/3]^2 = 100/9

And......the ratio of the area of one triangle in the larger polygon to the area of a triangle in the smaller polygon will have the same relationship as the area of the larger polygon to the area of smaller polygon  = [10/3]^2  = 100/9

That's what I thought... thank you for confirming that in my mind!

Hi Shades and CPhill.

Huh, that sounded like a lot of work chris.

If the sides of two similar figures have the ratio of  \(1:\frac{10}{3}\)

Then the area of the 2 figures have the ratio of

    \(1^2:\frac{10^2}{3^2}\\ 1:\frac{100}{9}\)

That is of course the small one to the large one, you asked for it the other way around so the sides need to be swapped.