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Area and Circumference of a Polygon Inside a Circle

Area and Circumference of a Polygon Inside a Circle
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2023-03-19 02:24
CirclePolygonCalculusGeometric Construction
In collections geometriadinamica.es
This example demonstrates how to use GeoGebra to calculate the area and circumference of a polygon inside a circle. The key mathematical concepts involved include geometric shape area and circumference calculations, as well as polygon properties. The construction points include using Polygon, Segment, Midpoint, Line, Point, OrthogonalLine, Intersect, Circle, and If GeoGebra commands/objects. Suitable for middle school math, geometry, and computational geometry.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → poly1, a, b, c', d'
  2. Segment(A, C) → c
  3. Midpoint(c) → E
  4. Segment(D, B) → d
  5. Midpoint(d) → C_1
  6. Line(C_1, E) → e
  7. Line(A, e) → g
  8. Line(B, e) → h
  9. Line(C, e) → i
  10. Line(D, g) → f
  11. Point(e) → O1
  12. OrthogonalLine(O1, e) → j
  13. Intersect(h, j) → A_1
  14. Circle(O1, A_1) → k
  15. Intersect(k, e, 2) → D_1
  16. Line(D_1, j) → m
  17. Intersect(g, m) → A1
  18. Circle(O1, A1) → p
  19. Intersect(p, h, 2) → B1_1
  20. Intersect(p, f, 2) → D1_1
  21. If(Distance[A1, B1_1] > Distance[A1, D1_1], B1_1, D1_1) → B1
  22. Segment(A1, B1) → l
  23. OrthogonalLine(B1, l) → n
  24. Intersect(i, n) → C1
  25. OrthogonalLine(C1, n) → q
  26. OrthogonalLine(A1, l) → r
  27. Intersect(r, q) → D1
  28. Polygon(A1, B1, C1, D1) → poly2, a_1, b_1, d_1, f_1
  29. Midpoint(d') → F
  30. Midpoint(b) → G
  31. Line(F, G) → s
  32. Point(s) → O2
  33. Line(D, s) → t
  34. Line(C, s) → c_1
  35. Line(B, s) → e_1
  36. Line(A, s) → g_1
  37. OrthogonalLine(O2, s) → h_1
  38. Intersect(t, h_1) → H
  39. Circle(O2, H) → k_1
  40. Intersect(k_1, s, 2) → I
  41. Line(I, h_1) → i_1
  42. Intersect(e_1, i_1) → B2
  43. Circle(O2, B2) → p_1
  44. Intersect(p_1, t, 2) → D2
  45. Intersect(p_1, c_1, 1) → C2
  46. Intersect(p_1, g_1, 1) → A2
  47. Polygon(B2, D2, C2, A2) → poly3, a2, b2, c2, d2

AI Drawing Prompt Reference

Draw a quadrilateral with its four vertices on a circle, and the circle is inscribed in the quadrilateral. Draw the two diagonals of the quadrilateral, find their midpoints, and connect these midpoints with a line. Choose a point on this line as the center of a new circle that is tangent to one side of the quadrilateral. This circle intersects the lines containing the sides of the quadrilateral, forming a new inscribed quadrilateral. Repeat a similar process using the midpoints of the other pair of opposite sides to construct another circle and a third inscribed quadrilateral. Show all three nested quadrilaterals, each inscribed in a circle, and display their areas and perimeters.
The prompt above was compiled from the original drawing commands and is for reference only.