Can you solve this efficiently?

Aiden is drawing a pattern on a large piece of paper. He starts by drawing a segment of length 1 cm, then turns 138◦ to the left and draws a segment of length 2 cm, and again turns 138◦ to the left. In this manner, he draws segments of length 1 cm, 2 cm, 1 cm, 2 cm, and so on with each segment drawn following a 138◦ turn. Aiden discovers that the pattern eventually starts to retrace itself exactly. What is the total length, in centimetres, of the pattern before it starts to retrace?

Note: this program is not to be solved using a computer program.