Composite Figures

Composite Figures: Level 4 Challenges


In the above diagram, each of the grid squares is a unit long. Also, all the arcs are quadrants. If the area of the shape enclosed by arcs \(DB\), \(BJ\), \(JK\), and \(KD\) can be represented as \[\dfrac{a \pi + b}{c},\] where \(a,b,c\) are integers with \(c\) positive, find the maximum value of \(a+b+c\).

Let \(ABCD\) be a square and \(P\) be a point inside the square such that \(PB = 23\) and \(PD = 29\) . Find the area of \(\triangle APC\)

Find the area of \(\color{violet}{Violet}\) region (four square looking region of central circle) if radius of each circle is 4 units.

Inspired by this question.
Image credit goes to Aniket Verma.

As shown, a rectangle (not a square) is partitioned into four triangles: 1 equilateral and 3 right triangles. The areas of two of the right triangles are 30 and 50. Find the area \(x\) of the remaining right triangle.

Bonus: Find the general relationship among the areas of the 3 right triangles.

Suppose we combine a unit circle and a square of the same area, as shown above, such that their centroids lie on the same point. What is the area of the non-overlapping white region?

If your answer is of the following form \[A \arctan\left(\dfrac{\sqrt{\pi\left(B - \pi\right)}}{\pi - C}\right) - D \sqrt{\pi\left(E - \pi\right)},\] where \(A, B, C, D,\) and \(E\) are positive integers, evaluate \(A + B + C + D + E\).


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