# Four semicircles are packed into a square, as shown. What proportion of the square is filled?

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Four semicircles are packed into a square, as shown. What proportion of the square is filled?

posted Nov 14, 2018

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Nine unit circles are packed into a square, tangent to their neighbors and to the square. What is the length of the longest smooth path connecting two opposite corners of the square?

Assumptions:
- The path must be continuous and follow the lines in the diagram; that is, it must be made up of portions of either the circles or the outside square.
- The path may not change direction suddenly.
- The path may not contain any loops.
- The path may not touch or cross itself at any point.

As shown, a rectangle 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 of the remaining right triangle?

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In the image below there are two tangent semicircles. The side length of the square is 36cm.

What is the length of the radius of the yellow circle?

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A unit circle is divided into 12 congruent regions, as shown. What is the perimeter of one of these regions?