Inherited Web API API Controller

I have a couple of classes

Square : Rectangle : Shape (abstract)

and I have a Base Controller inheriting from ApiController that I'd like to use.

public abstract class BaseController<T> : ApiController where T : class
{
    public abstract IEnumerable<T> Get()
    ...
}

and

public class DerivedController : BaseController<Rectangle>
{
    public override IEnumerable<Rectangle> Get()
    ...
}

public class AnotherDerivedController : BaseController<Square>
{
    public new IEnumerable<Square> Get()
    ...
}

/api/rectangle will properly call IEnumerable<Rectangle> Get()

/api/square will give me an error:

Multiple actions were found that match the request: 
System.Linq.IEnumerable`1[Square] Get() on type Web.Api.Controllers.AnotherDerivedController 
System.Linq.IEnumerable`1[Rectangle] Get() on type Web.Api.Controllers.DerivedController

If I change public new IEnumerable<Square> Get() to public override IEnumerable<Square> Get(), I get a compile time error since the return signatures are different

How do I get my code to call the proper method? Is it necessary to explicitly register each class's methods in RegisterRoutes?