1, the answer is 12. The dihedral angle formed by each side surface and the bottom surface is 60, so the distance between the projection point of the apex of the triangular pyramid on the ground and each side of the bottom surface is equal. If X is set, the distance between the apex and the bottom surface on the side surface is also equal, which is 2x, and the side surface area is 0.5*(3+4+5)*2x= 14x. In addition, in the triangle in the lower right corner, the area =0.5*3*4=0.5*(3+4+5)*x=6, so 14x= 12.

The radical sign of 2 and 3. Mainly to find a straight line perpendicular to the plane A. BD- diagonal AC. , calculate the sine value of the angle between this line and BC, and then calculate the cosine value.

The square root of 3.6. Let the length, width and height be x, y and z respectively, then yz=√2, xz=√3 and xy=√6. Calculate x, y and z, which are √3, √2 and 1 respectively, and then the diagonal length can be calculated.

4. The root number 3*a is 2/3 times higher and the surface area ratio is 3/3. If the regular tetrahedron is A.C.BD and the side length of the regular tetrahedron is 2*a, the height of the regular tetrahedron can be calculated. The side is half the square root of 6*a, the surface area of regular tetrahedron is 2√3*a*a, and the surface area of cube is 6 * a * a.

5. 16a is 3/3. If the surface area of the ball is 36∏, the radius of the ball is 3, the center of the ball is at the height of a regular tetrahedron, if the regular tetrahedron is x, the height is one third of the square root of 6*x, and the distance from the center to the four vertices is the same as 3, then the distance from the center to the bottom is one third of the square root of 6*x-3, and the projection of the center on the ground is the center of an equilateral triangle, and the distance from the center to the vertex of the triangle is 3.

This is the general process. I don't know if the result is right. Do the math yourself.