With the first one, you are given that 3 of the sides are perpendicular, so obviously the solid is a section of a rectangular prism. Now each face of these 3 perpendicular sides is a triangle, i.e. half of a rectangle, meaning you effectively have (1/2)*(1/3)=1/6of a rectangular prism. Also, each of these faces is obviously a right angled triangle The other triangular face has the side lengths as the hypotenuses of the 3 right angles triangles, and these are known. So anyway to find the volume of the full cube, you would find each unknown side length of these 3 perpendicular faces, as this effectively finds the dimensions. To do this, label each unknown side x, y, z, and use the pythagoras theorem relations on each known right angled triangle side to get a system of equations in x^2, y^2, z^2. Solving this you should get one side as 9, one as 2sqrt(10) and the other as 6sqrt(10). Multiplying these all together and dividing by 6, you will get the volume of the solid, which turns out to be 180.

Imo you are better at doing other types of maths for problem solving, the australian maths competition problems are all really silly convoluted problems.