I wrote a comment about the Mighty logo. It has distorted letters, but let's assume you have already got an order to keep the letters undistorted.
Making it in a typographically pleasant way would be easiest if you had as wide and as heavy looking font and you used Indesign's Type on Path tool. In practice it should be the same that was used as the starting point when your letters were made by cutting holes. Otherwise it doesn't offer any fine typography benefits.
After typing the text you simply could place your shapes one by one in the same places and rotated as much with big zoom. I guess Indesign has so much serious typographic knowledge behind it that it's very difficult to beat the result without years of practicing.
If you simply want to rotate and place your shapes on a circle you already have got one suggestion. I have another variant:
Your text (sorry for my bad copies) is here about 150mm wide and 27mm high:
I am not sure how many degrees wide sector of a circle you want to be covered, but I guess you want sector angle S = 90 degrees.
If the bottom line of the text should sit on the circle curve with the same spacing as in your horizontal version you'll need a circle which has diameter D=(W/Pi)(360 degrees/S).
Many letters are narrow in the bottom, so the diameter very likely should be smaller. You must subtract some amount from that theoretical diameter. My first guess is to subtract the height of the letters. So, the formula for the diameter will be
D = (W/Pi)(360 degrees/S)-H
That gives with W=150 mm, H=27 mm and S=90 degrees diameter D=164 millimeters.
Using the first version of diameter without subtraction (=190 mm) makes too sparse looking result. The other extreme is to pack the bottoms of the letters:
The width of the red line is 116 mm. That gives diameter 148 mm.
We'll try both D=164 mm and D=148 mm. Calculated D can be used as shown in the answer of user Kyle, but you can also calculate how much the letters should be rotated based on the average available sector space.
There's six letters, there's room 90 degrees, the average room is 15 degrees. So, the rotations would be 45, 30, 15, 0, -15, -30 and -45 degrees. That sounds idiotic, but I have succeeded to get acceptable results with it.
In the next image there's on the top right the letters rotated one by one and in the bottom they are placed by eye and having Smart Guides ON onto a circle which has diameter 164 mm.
I said: "Sounds idiotic" because it isn't symmetric if the middle A stands straight, as you easily see:
U has clearly higher elevation than D in the right. We can try to fix it by rotating a little counterclockwise, but for me it looks worse because the A is tilted:
Well try also D=148 mm. It has the same symmetry error, but it doesn't look too tight: