How to calculate the angle from rotation matrix

Would like to put a contribution here as I was working on the same problem. I add value to the above answers by posting a pure python implementation for converting a 3-D rotation matrix (3x3) to the corresponding roll (Rx) , pitch (Ry) , yaw (Rz) angles.

Reference pseudocode: https://www.gregslabaugh.net/publications/euler.pdf (link is somewhat no longer valid / is broken in 2021... but i include it here for completeness nonetheless)

Reference problem setup: Say we have a 3x3 rotation matrix and we want to extract the Euler angles in degrees. I will make the Python implementation as 'obvious' as possible to make it easy to decipher what is going on in the script. Respective coders can optimize it for their own use.

Assumptions: We rotate the first about the x-axis, followed by the y-axis, and finally the z-axis. This ordering-definition must be respected when you are adapting this code snippet.

"""
Illustration of the rotation matrix / sometimes called 'orientation' matrix
R = [ 
       R11 , R12 , R13, 
       R21 , R22 , R23,
       R31 , R32 , R33  
    ]

REMARKS: 
1. this implementation is meant to make the mathematics easy to be deciphered
from the script, not so much on 'optimized' code. 
You can then optimize it to your own style. 

2. I have utilized naval rigid body terminology here whereby; 
2.1 roll -> rotation about x-axis 
2.2 pitch -> rotation about the y-axis 
2.3 yaw -> rotation about the z-axis (this is pointing 'upwards') 
"""
from math import (
    asin, pi, atan2, cos 
)

if R31 != 1 and R31 != -1: 
     pitch_1 = -1*asin(R31)
     pitch_2 = pi - pitch_1 
     roll_1 = atan2( R32 / cos(pitch_1) , R33 /cos(pitch_1) ) 
     roll_2 = atan2( R32 / cos(pitch_2) , R33 /cos(pitch_2) ) 
     yaw_1 = atan2( R21 / cos(pitch_1) , R11 / cos(pitch_1) )
     yaw_2 = atan2( R21 / cos(pitch_2) , R11 / cos(pitch_2) ) 

     # IMPORTANT NOTE here, there is more than one solution but we choose the first for this case for simplicity !
     # You can insert your own domain logic here on how to handle both solutions appropriately (see the reference publication link for more info). 
     pitch = pitch_1 
     roll = roll_1
     yaw = yaw_1 
else: 
     yaw = 0 # anything (we default this to zero)
     if R31 == -1: 
        pitch = pi/2 
        roll = yaw + atan2(R12,R13) 
     else: 
        pitch = -pi/2 
        roll = -1*yaw + atan2(-1*R12,-1*R13) 

# convert from radians to degrees
roll = roll*180/pi 
pitch = pitch*180/pi
yaw = yaw*180/pi 

rxyz_deg = [roll , pitch , yaw] 

Hope this helps fellow coders out there!

If R is the (3x3) rotation matrix, then the angle of rotation will be acos((tr(R)-1)/2), where tr(R) is the trace of the matrix (i.e. the sum of the diagonal elements).

That is what you asked for; I estimate a 90% chance that it is not what you want.

We can get Euler angles from rotation matrix using following formula.

Given a 3×3 rotation matrix

The 3 Euler angles are

Here atan2 is the same arc tangent function, with quadrant checking, you typically find in C or Matlab.

Note: Care must be taken if the angle around the y-axis is exactly +/-90°. In that case all elements in the first column and last row, except the one in the lower corner, which is either 1 or -1, will be 0 (cos(1)=0). One solution would be to fix the rotation around the x-axis at 180° and compute the angle around the z-axis from: atan2(r_12, -r_22).

See also https://www.geometrictools.com/Documentation/EulerAngles.pdf, which includes implementations for six different orders of Euler angles.