You asked me in the comments if there is a Geometry Nodes solution. Sure there is one if not more. Here I will first show you quite a simple one which has a flaw that I will erase with a slightly more complex one later.
I start with a 60° pie, the origin is at its bottom vertex.

I add a Geometry Nodes modifier to it where I use a Mesh Circle node with 6 vertices and place the pie on the vertices with an Instance on Points node which I align to the Normal with an Align Rotation to Vector node. For this it is important that the Mesh Circle needs to have the Fill Type set to None (which is the default anyway), not N-Gon or Triangles. As axis which should be aligned I select Y, since the base pie object is oriented in +Y direction. After instancing I use a Rotate Instances node with Local Space disabled to rotate the result 30° so that you have one piece at the top like you show in your reference.

Now this is all not too complicated and you could use the Radius of the Mesh Circle to move the pie pieces closer together or further apart. However, if you wanted to move them together to close without a gap, you would have to set the Radius to 0. But the problem with that is, you lose the Normal information which is needed for the rotation and all six pieces would simply lie on top of each other.
To avoid that, you can only make sure to use a small but never equal to 0 Radius (which would leave a gap, maybe imperceptible though) or you need to have the pie's origin not a the bottom vertex but maybe at for example Y = 0.1, in this case you have closed the gap already at a Radius of 0.1, which is not the worst solution.
But if you do not want to care about staying above a certain radius or placing the origin at a determined location offset the bottom, you can do a workaround in Geometry Nodes.
First of all, I set the Radius of the Mesh Circle to 1 m, which is not explicitly necessary but comes in handy if you want to move the pieces outwards to a specific distance. Then I take a Capture Attribute node to read the Normal of the circle's vertices. This captured attribute goes into the Vector input of the Align Rotation to Vector node.
Now I can use either a Transform Geometry node to scale the circle or in my case I use a Scale Elements node. For the latter the type is set to Edge, not Face, since the Mesh Circle has no faces.
The good thing with this method is you can keep the origin at the bottom vertex, scale all the way down to 0 and do not lose the normal orientation for the pieces. Also with a Radius set to 1 m, a scaling factor of 0.1 means the distance to the center will be 0.1 m for convenience.
