Solution:

We can extend line segments $l, m,$ and $n$ as follows.

They are concurrent since $l$ and $m$ intersect and $l$ and $n$ intersect (they are on the same plane). Since $l$ only intersects the plane of $m$ and $n$ once, it must intersect them both at that one point.

The dashed red lines create equilateral triangles on the lateral faces of the bowl, which all have side length $1.$ In the top plane, we know that $m \perp n,$ so the dashed red lines create an isosceles right triangle with leg length $1.$

The octagon looks like the diagram below.

The area of the octagon is the area of the square minus each of the four corner triangles. This is equal to $$3^2 - 4(\dfrac{1}{2} \cdot 1^2) = 7.$$

Thus, B is the correct answer.