A medial meniscus tear is a tear of the substance of the shock-absorbing medial meniscus.
Looking down at the top of the menisci one can see that they are different in shape, and the C-shaped medical meniscus is better tethered to the outer capsule.
C-shaped medial meniscus
The two menisci are different shapes, with the medial meniscus being more C-shaped than the O-shaped lateral meniscus.
Characteristics of Radial Tears in the Posterior Horn of the Medial Meniscus Compared to Horizontal Tears. Choi C-J, Choi Y-J, Song I-B and Cho C-Hi. Clin Orthop Surg. 2011 Jun; 3(2): 128–132.