Lesson
Challenge
Standards

Estimating the Area of a Disk

What is it?

Theorem: The area of a circle lies between the area of an internal regular polygon and an external regular polygon, each with the same number of sides and all vertices touching the edge of the circle.

You can estimate the area inside the circle by inscribing a polygon where each corner touches the circle and dividing it into triangles.
hexagon.gif
Image from www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/archimedes.html"

Notice that the sides AC and AB are of length r , or the radius of the circle. The angle ABC will be 60 degrees (i.e. one sixth of a revolution, 360/6 =60 degrees). This implies that the sum of the other angles, CAB plus CBA is 120 degrees (since the sum of all the angles in a triangle is 180 degrees). The triangles in this hexagon are all equilateral: all three sides are of length r. The height of each of these triangles is √3 *r/2, so that the area of the hexagon, it follows, is:
archimedes_10.gif



Now you could divide a unit circle into 12 of the 30-60-90 triangles. The area of a right triangle is
h * s/2
where h is the height and s is the base of the right triangle.

Archimedes proved that the
Area of a circle is equal to the area of a right triangle that has a height equal to the radius and base equal to the circumference:.
Archimedes_circle_area_proof_-_inscribed_polygons.png
Image from wikipedia

So, the total area of a unit circle is:
Area = 12*.5 *.8660




Similarly you could compute the area of the 12 triangles by drawing a hexagon outside the circle, with each edge tangent to the circle. Then create a similar area. If you find the mean of the inside and outside areas, you'll have a better estimate of the circle's area.



You can use a 3D printer to print a circle that includes the inner and outer polygons, as well as an inner triangle division.

To create the base circle:
  1. Open SketchUp.

  2. Delete the figure.

  3. Click the Circle Tool (C).

  4. Starting at the origin, draw a circle with a radius of 50.

  5. Press Spacebar, CTRL+click on the circle and select Zoom Extents from the Context Menu

  6. Navigate to the Camera>Standard Views menu and select Bottom

  7. Click the Push/Pull tool (P) and pull the circle down. Type 10 and press ENTER/RETURN. You can orbit to check your results.

  8. Navigate to the Camera>Standard Views menu and select Top

  9. You want to divide the circle into triangles so that you can estimate the volume. Click the Polygon Tool. Type 6s and press ENTER/RETURN. The default for the Polygon Tool is a hexagon, a six sided polygon, but you may have changed the preferences.

  10. Create a hexagon starting from the center of the circle with vertices touching the circumference of the circle.

  11. Click the Line tool (L) and draw six lines connecting all the vertices through the center.

  12. Select the three lines that outline one of the resulting triangle.

  13. Select the Offset Tool (F) and drag inward and type .5, then press ENTER/RETURN

  14. Repeat for all the triangles.


  15. With the Erase Tool (E) delete the center lines:


  16. Click the Push/Pull tool (P) and push each of the six triangles down. Type 8 and press ENTER/RETURN


    This illustrates the basic breakup of a circle.

  17. Select the interior of one of the triangles and copy the selection


  18. Paste the copied triangle


  19. Select the Line Tool (L) and bisect the triangle to create 2 right triangles


  20. Select the Offset Tool (F) and drag inward and type .5, then press ENTER/RETURN


  21. Select the extra lines and delete them:


  22. Click the Push/Pull tool (P) and pull the triangle up. Type 8 and press ENTER/RETURN. Triple click on the entity and select Make Group from the Context Menu


  23. Export the entities as stl files and print.