A medial meniscus tear is a tear of the substance of the shock-absorbing medial meniscus.

Both medial and lateral menisci rest on the upper surface of the tibia, tucked between the rounded condyle of the femur and the flattened plateau of the top of the tibia.

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.

A longitudinal meniscal tear, cutting through the fibre-filled matrix of the body of the meniscus.
C-shaped medial meniscus
The two menisci are different shapes, with the medial meniscus being more C-shaped than the O-shaped lateral meniscus.
CLINICAL PUBLICATIONS
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.