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 DBDB, BJBJ, JKJK, and KDKD can be represented as aπ+bc,\dfrac{a \pi + b}{c}, where a,b,ca,b,c are integers with cc positive, find the maximum value of a+b+ca+b+c.

Let ABCDABCD be a square and PP be a point inside the square such that PB=23PB = 23 and PD=29PD = 29 . Find the area of APC\triangle APC

Find the area of Violet\color{#BA33D6}{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 xx 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 Aarctan(π(Bπ)πC)Dπ(Eπ),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,A, B, C, D, and EE are positive integers, evaluate A+B+C+D+EA + B + C + D + E.


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