Regular Polygon to Circle (area & perimeter)

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2017-02-14 13:57

This example demonstrates how a regular polygon approaches a circle as the number of sides increases, visually illustrating the change in area and perimeter. Key math concepts include limits, geometric transformations, and analytical geometry. Construction highlights GeoGebra commands like Intersect, Circle, Point, Rotate, Polygon, and Segment, emphasizing parametric drawing and limit concepts. Suitable for high school and above, ideal for understanding geometric limits and parametric constructions.

GeoGebra Commands & Parameters

Intersect(xAxis, yAxis) → center
Circle(center, radius) → c
Point(c) → A
Rotate(A, (360 * ° / sides), center) → B
Polygon(A, B, sides) → poly1, b, n, n_5, p_5, q_5, r_5, s_5, t_5, a_5, b_5, c_5, d_5, p, e_5, f_6, g_6, h_6, i_6, j_6, k_6, l_6, m_6, n_6, q, p_6, q_6, r_6, s_6, t_6, a_6, b_6, c_6, d_6, e_6, r, f_7, g_7, h_7, i_7, j_7, k_7, l_7, m_7, n_7, p_7, s, q_7, r_7, s_7, t_7, a_7, b_7, c_7, d_7, e_7, f_8, t, g_8, h_8, i_8, j_8, k_8, l_8, m_8, n_8, p_8, q_8, f_1, r_8, s_8, t_8, a_8, b_8, c_8, d_8, e_8, f_9, g_9, g_1, h_9, i_9, j_9, k_9, l_9, m_9, n_9, p_9, q_9, r_9, h_1, s_9, t_9, a_9, b_9, c_9, d_9, e_9, f_{10}, g_{10}, h_{10}, i_1, i_{10}, j_{10}, k_{10}, l_{10}, m_{10}, n_{10}, p_{10}, q_{10}, r_{10}, s_{10}, d, j_1, t_{10}, a_{10}, b_{10}, c_{10}, d_{10}, e_{10}, f_{11}, g_{11}, h_{11}, i_{11}, k_1, j_{11}, k_{11}, l_{11}, m_{11}, n_{11}, p_{11}, q_{11}, r_{11}, s_{11}, t_{11}, l_1, a_{11}, b_{11}, c_{11}, d_{11}, e_{11}, f_{12}, g_{12}, h_{12}, i_{12}, j_{12}, m_1, k_{12}, l_{12}, m_{12}, n_{12}, p_{12}, q_{12}, r_{12}, s_{12}, t_{12}, a_{12}, n_1, b_{12}, c_{12}, d_{12}, e_{12}, f_{13}, g_{13}, h_{13}, i_{13}, j_{13}, k_{13}, p_1, l_{13}, m_{13}, n_{13}, p_{13}, q_{13}, r_{13}, s_{13}, t_{13}, a_{13}, b_{13}, q_1, c_{13}, d_{13}, e_{13}, f_{14}, g_{14}, h_{14}, i_{14}, j_{14}, k_{14}, l_{14}, r_1, m_{14}, n_{14}, p_{14}, q_{14}, r_{14}, s_{14}, t_{14}, a_{14}, b_{14}, c_{14}, s_1, d_{14}, e_{14}, f_{15}, g_{15}, h_{15}, i_{15}, j_{15}, k_{15}, l_{15}, m_{15}, t_1, n_{15}, p_{15}, q_{15}, r_{15}, s_{15}, t_{15}, a_{15}, b_{15}, c_{15}, d_{15}, a, a_1, e_{15}, f_{16}, g_{16}, h_{16}, i_{16}, j_{16}, k_{16}, l_{16}, m_{16}, n_{16}, b_1, p_{16}, q_{16}, r_{16}, s_{16}, t_{16}, a_{16}, b_{16}, c_{16}, d_{16}, e_{16}, c_1, f_{17}, g_{17}, h_{17}, i_{17}, j_{17}, k_{17}, l_{17}, m_{17}, n_{17}, p_{17}, d_1, q_{17}, r_{17}, s_{17}, t_{17}, a_{17}, b_{17}, c_{17}, d_{17}, e_{17}, f_{18}, e_1, g_{18}, h_{18}, i_{18}, j_{18}, k_{18}, l_{18}, m_{18}, n_{18}, p_{18}, q_{18}, f_2, r_{18}, s_{18}, t_{18}, a_{18}, b_{18}, c_{18}, d_{18}, e_{18}, f_{19}, g_{19}, g_2, h_{19}, C, D, E, F, G, H, I, J, K, h_2, L, M, N, O, P, Q, R, S, T, U, i_2, V, W, Z, A_1, B_1, C_1, D_1, E_1, F_1, G_1, j_2, H_1, I_1, J_1, K_1, L_1, M_1, N_1, O_1, P_1, Q_1, f, k_2, R_1, S_1, T_1, U_1, V_1, W_1, Z_1, A_2, B_2, C_2, l_2, D_2, E_2, F_2, G_2, H_2, I_2, J_2, K_2, L_2, M_2, m_2, N_2, O_2, P_2, Q_2, R_2, S_2, T_2, U_2, V_2, W_2, n_2, Z_2, A_3, B_3, C_3, D_3, E_3, F_3, G_3, H_3, I_3, p_2, J_3, K_3, L_3, M_3, N_3, O_3, P_3, Q_3, R_3, S_3, q_2, T_3, U_3, V_3, W_3, Z_3, A_4, B_4, C_4, D_4, E_4, r_2, F_4, G_4, H_4, I_4, J_4, K_4, L_4, M_4, N_4, O_4, s_2, P_4, Q_4, R_4, S_4, T_4, U_4, V_4, W_4, Z_4, A_5, t_2, B_5, C_5, D_5, E_5, F_5, G_5, H_5, I_5, J_5, K_5, a_2, L_5, M_5, N_5, O_5, P_5, Q_5, R_5, S_5, T_5, U_5, i, b_2, V_5, W_5, Z_5, A_6, B_6, C_6, D_6, E_6, F_6, G_6, c_2, H_6, I_6, J_6, K_6, L_6, M_6, N_6, O_6, P_6, Q_6, d_2, R_6, S_6, T_6, U_6, V_6, W_6, Z_6, A_7, B_7, C_7, e_2, D_7, E_7, F_7, G_7, H_7, I_7, J_7, K_7, L_7, M_7, f_3, N_7, O_7, P_7, Q_7, R_7, S_7, T_7, U_7, V_7, W_7, g_3, Z_7, A_8, B_8, C_8, D_8, E_8, F_8, G_8, H_8, I_8, h_3, J_8, K_8, L_8, M_8, N_8, O_8, P_8, Q_8, R_8, S_8, i_3, T_8, U_8, V_8, W_8, Z_8, A_9, B_9, C_9, D_9, E_9, j_3, F_9, G_9, H_9, I_9, J_9, K_9, L_9, M_9, N_9, O_9, k_3, P_9, Q_9, R_9, S_9, T_9, U_9, V_9, W_9, Z_9, A_{10}, j, l_3, B_{10}, C_{10}, D_{10}, E_{10}, F_{10}, G_{10}, H_{10}, I_{10}, J_{10}, K_{10}, m_3, L_{10}, M_{10}, N_{10}, O_{10}, P_{10}, Q_{10}, R_{10}, S_{10}, T_{10}, U_{10}, n_3, V_{10}, W_{10}, Z_{10}, A_{11}, B_{11}, C_{11}, D_{11}, E_{11}, F_{11}, G_{11}, p_3, H_{11}, I_{11}, J_{11}, K_{11}, L_{11}, M_{11}, N_{11}, O_{11}, P_{11}, Q_{11}, q_3, R_{11}, S_{11}, T_{11}, U_{11}, V_{11}, W_{11}, Z_{11}, A_{12}, B_{12}, C_{12}, r_3, D_{12}, E_{12}, F_{12}, G_{12}, H_{12}, I_{12}, J_{12}, K_{12}, L_{12}, M_{12}, s_3, N_{12}, O_{12}, P_{12}, Q_{12}, R_{12}, S_{12}, T_{12}, U_{12}, V_{12}, W_{12}, t_3, Z_{12}, A_{13}, B_{13}, C_{13}, D_{13}, E_{13}, F_{13}, G_{13}, H_{13}, I_{13}, a_3, J_{13}, K_{13}, L_{13}, M_{13}, N_{13}, O_{13}, P_{13}, Q_{13}, R_{13}, S_{13}, b_3, T_{13}, U_{13}, V_{13}, W_{13}, Z_{13}, A_{14}, B_{14}, C_{14}, D_{14}, E_{14}, k, c_3, F_{14}, G_{14}, H_{14}, I_{14}, J_{14}, K_{14}, L_{14}, M_{14}, N_{14}, O_{14}, d_3, P_{14}, Q_{14}, R_{14}, S_{14}, T_{14}, U_{14}, V_{14}, W_{14}, Z_{14}, e_3, f_4, g_4, h_4, i_4, j_4, k_4, l_4, l, m_4, n_4, p_4, q_4, r_4, s_4, t_4, a_4, b_4, c_4, m, d_4, e_4, f_5, g_5, h_5, i_5, j_5, k_5, l_5, m_5
Segment(center, A) → g

AI Drawing Prompt Reference

Draw a circle centered at the origin, and then draw a regular polygon around the center, gradually increasing the number of sides of the polygon to see how it approaches a circle. Measure and display the perimeter and area of the polygon, as well as the perimeter and area of the circle.

The prompt above was compiled from the original drawing commands and is for reference only.