Dissection of a Vase
What is this about?
A Mathematical Droodle

The applet is intended to help raise and answer a question about a shape (that might remind one of a vase) whose border consists of six quarter circles drawn with the same radius:

Given the radius, say R, of the arcs, what is the area of the vase?

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This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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The top of the vase if complemented by four circular sectors with central angle of 90° becomes a square of the side equal to the diameter of the arcs and circles, i.e., 2R. One of the sectors (the lowest) is already inside the intended square; the other three find there locations by dragging the scroll bar and come from the remaining three quarters of a circular part of the vase. It follows that - by dissection - the area of the vase equals that of the square.

References