Test analysis: according to the nature of rotation and the inverse theorem of Pythagorean theorem, the judgment and nature of equilateral triangle can get the result;
(1) Referring to the problem-solving ideas given in the title, we can turn △ABP counterclockwise by 90 around point A to get △A DP'. According to the nature of rotation, we can know that △ ABP △ A DP' is an isosceles right triangle, and we can get △ A P' P = 45; Then it is concluded that △DPP' is a right triangle, and the result can be obtained;
(2) The method is the same as (2), and the result can be obtained by combining the properties of regular hexagon.
If △app' is an equilateral triangle, then △app'c = 60.
∵
△ CPP' is a right triangle.
∴∠cp′p=90
∴∠ap′c= 150
∴∠apb= 150；
(1) rotates △ ABP 90 counterclockwise around point A to get △A DP'.
It is concluded that △ ABP △ A DP' and △APP' are isosceles right triangles.
∴∠ap′p=45
∵
△ DPP' is a right triangle,
∴∠dp′p=90
∴∠dp′a= 135
∴∠ APB = 135, the side length of the square is;
(2) The method is the same as (2), the degree of ∠APB is equal to 120, and the side length of a regular hexagon is
Comments: The key to solve this problem is to master the essence of rotation: the two figures are congruent before and after rotation, the included angle of the connecting line between the corresponding point and the rotation center is equal to the rotation angle, and the distance from the corresponding point to the rotation center is equal.