imc

We extend line segments $\ell ,m,$ and $n$ to their point of concurrency, as shown below: We claim that lines $\ell ,m,$ and $n$ are concurrent: In the lateral faces of the bowl, we know that lines $\ell$ and $m$ must intersect, and lines $\ell$ and $n$ must intersect. In addition, line $\ell$ intersects the top plane of the bowl at exactly one point. Since lines $m$ and $n$ are both in the top plane of the bowl, we conclude that lines $\ell ,m,$ and $n$ are concurrent. In the lateral faces of the bowl, the dashed red line segments create equilateral triangles. So, the dashed red line segments all have length $1.$ In the top plane of the bowl, we know that $\overleftrightarrow{m}\perp \overleftrightarrow{n}.$ So, the dashed red line segments create an isosceles triangle with leg-length $1.$ Note that octagon has four pairs of parallel sides, and the successive side-lengths are $1,\sqrt{2},1,\sqrt{2},1,\sqrt{2},1,\sqrt{2},$ as shown below: The area of the octagon is $$3^2-4\cdot \left(\frac{1}{2}\cdot 1^2\right)=\boxed{7}.$$