In this interactive applet, an informal derivation of the relationship between the circumference and area of a circle is presented.
A vertical slider in the control area guides you through the steps that need to be followed in order to show how the circumference and area of a circle are connected. In each step, a short text shows up which either explains what is happening in the main window or sets questions to initiate discussion in the classroom. The refresh button on the top right corner clears everything up and allows the user to repeat the process.
An outline of the proof:
► A circle with radius r and circumference 2pi × r is given. The circle is initially divided into 3 blue and 3 green sectors all identical. The 6 sectors are rearranged to form a shape that is something like a "curved" and "bending" rectangle.
◍ Students are asked how does the area of this new shape compare to the area of the given circle.
► The next step explains that since this new shape was made of the sectors that the given circle was divided into, its area must be equal to the area of the circle.
◍ Students are then asked what are the lengths of the two curved bases (blue and green) and the two straight sides of the shape.
► The next step shows that since the blue sectors are half the original circle, the corresponding blue curved line must be equal to half the circumference of the circle thus pi × r. Similarly, the green curved line at the bottom of the shape which corresponds to the other half of the circle (green sectors) must also be equal to pi × r. And the segments on both sides are each equal to r. Then, the user increases the number of sectors that the circle is divided into by dragging a relevant slider.
◍ The students are asked what quadrilateral the shape is transformed into as the number of sectors increases.
◍ What are its dimensions?
► The next step shows the answer which is that as the number of sectors increases, the shape is transformed into a rectangle with base pi × r and height r.
◍ Students are then asked what the area of the rectangle is.
◍ They are also asked how the area of the rectangle relates to the area of the circle.
► The next step answers that the area of the rectangle is base × height and thus pi×r×r = pi×r^2 and concludes that since the area of the rectangle is always equal to the area of the circle, the formula of the area of the circle must also be A = pi × r^2.
This description is also included in the product as a printable word document.
With these easy, intriguing and ready-to-use GeoGebra applets, you will enrich your regular classroom practice and enhance student understanding of math concepts. With the help of a smartboard or a projector, you can demonstrate for the entire class or let students explore independently on their computers. No knowledge of how to use GeoGebra is required, you just use an app. No internet connection is needed.
To be able to see and play with the applet you will need to download and install the free software GeoGebra in your computer or tablet if you haven’t done this before. The process is very simple and quick: The link is
from which you can download GeoGebra Classic 5 (this is the version that I also use).