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Area of a Circle II

Area of a Circle II
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2015-10-19 05:35
CircleTransformationGeometric Construction
In collections Area, Surface Area, Volume, Cross Section, 3D Perimeter & Area
This applet provides an intuitive method for students to conceptually discover the formula for the area of a circle. Key mathematical concepts include the properties of circles, rotation, and area calculation. Using GeoGebra commands such as Circle, CircleArc, Segment, Point, and Intersect, students can observe and explore the changes in the area of a circle. Suitable for students from middle school and above.

GeoGebra Commands & Parameters

  1. Circle((0, 0), 1 / (1 - m)) → d
  2. Circle(L, 1 / (1 - m)) → e
  3. CircleArc((0, 1 / (1 - m)), C, L) → f
  4. Segment(C, A) → a
  5. Point(j) → F
  6. OrthogonalLine(F, i) → b
  7. Intersect(i, b) → G
  8. Segment(F, G) → l
  9. Circle(E, (0.95 * 2 / (2 - m))) → d_2
  10. Segment(H, D) → n
  11. Circle(H, (0.95 * 2 / (2 - m))) → c_2
  12. Segment(I, K) → a_1
  13. CircleArc((0, 1 / (3 - m)), I, J) → f_2
  14. CircleArc((0, 1 / (2 - m)), (0, 0.05), E) → e_2
  15. Segment(N, O) → k
  16. CircleArc((0, 1 / (4 - m)), N, M) → g_2
  17. Segment(P, Q) → b_1
  18. CircleArc((0, 1 / (5 - m)), P, R) → h_2
  19. Segment(S, T) → i_1
  20. CircleArc((0, 1 / (6 - m)), S, U) → k_2
  21. Segment(V, W) → j_1
  22. CircleArc((0, 1 / (7 - m)), V, Z) → p_2
  23. Segment(A_1, B_1) → l_1
  24. CircleArc((0, 1 / (8 - m)), A_1, C_1) → q_2
  25. Segment(D_1, E_1) → m_1
  26. CircleArc((0, 1 / (9 - m)), D_1, F_1) → r_2
  27. Segment(G_1, H_1) → n_1
  28. CircleArc((0, 1 / (10 - m)), G_1, I_1) → s_2
  29. Segment(J_1, K_1) → a_2
  30. CircleArc((0, 1 / (11 - m)), J_1, L_1) → t_2
  31. Segment(M_1, N_1) → b_2
  32. CircleArc((0, 1 / (12 - m)), M_1, O_1) → c_3
  33. Segment(P_1, Q_1) → i_2
  34. CircleArc((0, 1 / (13 - m)), P_1, R_1) → d_3
  35. Segment(S_1, T_1) → j_2
  36. CircleArc((0, 1 / (14 - m)), S_1, U_1) → e_3
  37. Segment(V_1, W_1) → l_2
  38. Segment(A_2, B_2) → m_2
  39. CircleArc((0, 1 / (16 - m)), A_2, C_2) → g_3
  40. CircleArc((0, 1 / (15 - m)), V_1, Z_1) → f_3
  41. Segment(D_2, E_2) → n_2
  42. CircleArc((0, 1 / (17 - m)), D_2, F_2) → h_3
  43. Segment(G_2, H_2) → a_3
  44. CircleArc((0, 1 / (18 - m)), G_2, I_2) → k_3
  45. Segment(J_2, K_2) → b_3
  46. CircleArc((0, 1 / (19 - m)), J_2, L_2) → p_3
  47. Segment(M_2, N_2) → i_3
  48. CircleArc((0, 1 / (20 - m)), M_2, O_2) → q_3
  49. Segment((0, 0), (0, 1)) → j_3
  50. Segment((0, 1), (6.283185307179586, 0)) → l_3
  51. Segment((0, 0), (6.283185307179586, 0)) → m_3
  52. Angle(A, C, P_2) → α
  53. Angle(A, G, F) → β
  54. Segment((-2, 0), (-2, 1)) → s_3
  55. Segment((0, 0), (6.283185307179586, 0)) → t_3

AI Drawing Prompt Reference

Draw a circle centered at the origin with radius 1/(1-m). Then draw a circular arc starting from (0,1/(1-m)) to point L. Next, draw a line segment from the center to point A. Draw a line perpendicular to a given line at point F. The intersection of these lines is point G, and draw a line segment FG. Draw a new circle with radius 0.95*2/(2-m) centered at point E. Draw a line segment from point H to point D. Draw a new circle with radius 0.95*2/(2-m) centered at point H. Draw a line segment from point I to point K. Draw a circular arc with center at (0,1/(2-m)), starting from I and ending at J. Draw a circular arc with center at the origin, starting from (0,0.05) and ending at E. Draw a line segment from point N to point O. Draw a circular arc with center at (0,1/(3-m)), starting from N and ending at M. Draw a line segment from point P to point Q.
The prompt above was compiled from the original drawing commands and is for reference only.