I first noticed that the products of equations C and D are both square numbers. Since the products of both are different to the factors, this rules out 1 and so there are only two possible square numbers between 0 and 12: 4 and 9. Therefore the oval and the star each equal 4 or 9, and the rectangle and the square each equal 2 or 3.
Equation B involves the two different integers from equation D. The two possible answers must be [2 x 4 = 8] or [3 x 9 = 27]. 27 isn't a number between 0 and 12, so this can't be the correct equation. Instead, we know that the purple square is equal to 2, the orange oval is equal to 4, and the yellow semicircle is equal to 8. This also means that equation C is [3 x 3 = 9]. We can substitute these numbers into other equations A,E,F,G.
Equations F and H have the same product as one of its factors. This means that one of the factors in each equation has to be either 1 or 0. In equation F, we know that [a x 8 = a]. The red triangles (a) have to be 0 because anything multiplied by 0 is 0. I also know that it can't be 1 as [1 x 8 = 8]. These numbers are the wrong way round for this equation to work. 1 must therefore fit in equation H as the yellow diamond.
As I have substituted some shapes in the equations for numbers I already know, equation A is left as [3 x 4 = a]. a (or the red circle) is easy to find to be 12. 12 is also the solution to equation E: [a x 2 = 12]. By making a the subject, we know that [a = 12/2 = 6].
We are now left with unknown numbers in only equations G and H. G says [2 x a = c]. This means that c is an even number. The numbers remaining to choose from are 5,7,10 and 11, of which 10 is the only even number. This is the blue hexagon. By changing equation G to make the purple star the subject, we know that [a = 10/2 = 5].