If you take any point P in the cube abcd-a1b1c1d1 with edge length a, then the probability that the distance between P and point a is not greater than a is: I know everything else, but why is it 1 / 8 of the volume of a sphere centered on a, I think it is 1 / 4!

If you take any point P in the cube abcd-a1b1c1d1 with edge length a, then the probability that the distance between P and point a is not greater than a is: I know everything else, but why is it 1 / 8 of the volume of a sphere centered on a, I think it is 1 / 4!

In a cube, the cut part with a as the center and a as the radius is 1 / 8 of the volume of the sphere with a as the center, and 1 / 4 can only be a hemisphere

As shown in the figure, in the cube abcd-a1b1c1d1, if Ab1, AC and B1C are connected, the shape of △ ab1c is______ ．

Let AB = x, connect Ab1, AC, B1C, we can get that these three lines are the diagonals of the three faces of the cube. According to the Pythagorean theorem, we can get Ab1 = AC = B1C = 2x, so the shape of △ ab1c is equilateral triangle or equilateral triangle. So the answer is equilateral triangle or equilateral triangle

The side length of cube abcd-a1b1c1d1 is a, and the volume of triangular pyramid b-a1c1d is calculated

Do it indirectly
The volume of the whole cube is a ^ 3
After removing the triangles, there are four identical triangles
The bottom area is a ^ 2 and the height is a. if you draw a picture, you can see it clearly
So the volume of b-a1c1d is
a^3-a^2*a/2/3*4=a^3/3

The cube abcd-a1b1c1d1 with edge length of 1 is known, and the volume of b-acb1 is calculated

B-acb1, that is, ABC is the bottom, B1 is the high triangular pyramid
That is, the bottom is the right waist of an equilateral triangle
1/2 * 1*1*1=1/2