The circle could be as small as that of an atom or as large as a circle in space with a diameter of one light year. In all such cases the ratio of the circumference to the diameter is constant. The ratio of the circumference to the diameter in the red circle is the same for the yellow and black circles.
Good math teachers will have students measure the circumference of several circular objects (like circular lids) as accurately as possible to establish the belief that the ratio of the circumference to the diameter of any circle is Pi.
I remember being taught the sine, cosine, and tangent trigonometry functions from a single textbook diagram of a right triangle. No effort was made to show that the ratios were constant for any given angle no matter the size of the right triangle containing the given angle.
This project has the mouse draw a right triangle with angles equal to 35º, 90º, and 55º. The mouse first draws the hypotenuse of the right triangle a random number of steps. The mouse turns to the right and draws the perpendicular to the base of the triangle. The mouse then turns right through 90º and draws the side adjacent to the 35º angle. The program then computes and displays the sine(35º) using the known lengths of the hypotenuse and opposite side. The program repeats’ this process 9 more times, producing a series of right triangles that demonstrate the sine(35º) is a constant 0.57735… for all of the ten right triangles.
The project can be viewed and downloaded at the following link.
This is a screen shot of the output of the project.
Teachers and students might be interested in a free PDF tutorial on the project that can be obtained by emailing firstname.lastname@example.org.