anonymous 3 years ago Place four points on the surface of the sphere with radius 1m such that all four points are separated by equal distance from each other.... Find the distance between any two points.

1. hba

@sauravshakya Do you have a diag ?

2. Kainui

My intuition tells me that they will be separated such that they will form a regular tetrahedron by connecting the dots. When you say distance between any two points, do you mean in terms of traversing on the surface of the sphere or going straight between two points?

3. anonymous

traversing through the surface

4. anonymous

I mean on the surface

5. Kainui

It might be difficult, but I suggest drawing a picture and finding out for yourself. The angles should be something like 109.5 degrees. At this point it's just finding the arc length by knowing that this ratio is always true $\frac{ \theta }{ 2 \pi }=\frac{ arc length }{ 2 \pi r }$

6. anonymous

But how did u get angle between any two point is 109.5 degree

7. anonymous

I got this question in the chemistry class

8. Kainui

It's not very easy to draw out, but basically you have 4 arms coming outwards from the center and the angle between each one in 3 dimensions is 109.5 degrees. Only two are ever flat at one point. |dw:1355919400604:dw|

9. anonymous

I also know it is 109.5 degree but how to calculate it geometrically.

10. anonymous

I mean how to show the angle must be 109.5 degree

11. Kainui

Alright let me work on it real quick.

12. Kainui

|dw:1355920398348:dw| cut diagonally on the cube with the tetrahedral angle in the center. |dw:1355920471025:dw|

13. Kainui

Keep in mind, this is all pythagorean theorem up to this point, the sides of the cube all being 1 to keep things simple. The reason the diagonal through the center of the cube is sqrt(3) is because 1^2+(sqrt2)^2=3 because one edge of it is the hypotenuse of a triangle with legs of 1. Now let's look at what we can do here: |dw:1355921319840:dw|

14. Kainui

I hope this explains it well, it's fairly difficult to draw out accurately without being a total piece of garbage.

15. UnkleRhaukus

i always wondered where that number 109.5... came from

16. Kainui