Perplexing Pizza Crust

Geometry Level 5

A square box \(ABCD\) of side length \(100\) contains a slice of pizza. The box meets with an inscribed circle at points of tangency \(H\), \(I\), \(J\), \(K\), and there is an inscribed circle quadrant with centre \(D\).

Let \(L\) be the point where arc \(AC\) intersects the line \(DH\), and \(M\) be the point where the same arc intersects the line \(DI\). Find the area of the pizza crust (the shaded region \(HIML\)) to the nearest integer.

Details and Assumptions:

  • Use the approximation \(\tan ^{ -1 }{ (\frac{1}{2}) } =0.464\).

  • Use the approximation \(\pi = 3.142\).

Created by Michael Fuller. Popular geometry problems: "Star Stumper", "Not your average shuriken"

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