求解重叠区域面积问题
几何画板及绘图内容加载中,请稍等……
①本范例演示两个正方形重叠区域的面积计算;②涉及核心数学概念包括面积、旋转、正方形、周长、最大公约数与最小公倍数;③作图要点运用了Polygon(多边形)、Segment(线段)、Intersect(交点)、Circle(圆)、Point(点)、OrthogonalLine(垂线)、Angle(角度)、PointIn(点在对象内)等GeoGebra命令;④适合高中数学教学,用于几何计算与问题解决训练。
GeoGebra 命令及参数
Polygon(B, A, 4) → poly1, b, a_1, c, d, C, DSegment(A, D) → eSegment(B, C) → fIntersect(e, f) → ECircle(E, b) → gPoint(g) → FPolygon(E, F, 4) → poly2, e_1, f_1, h, i, G, HIntersect(b, e_1) → IIntersect(a_1, i) → JPolygon(I, A, J, E) → poly3, i_1, a_2, j, e_2Intersect(b, i) → KIntersect(d, e_1) → LPolygon(L, B, K, E) → poly4, l, b_1, k, e_3Intersect(d, i) → MIntersect(c, e_1) → NPolygon(M, E, N, D) → poly5, m, e_4, n, d_1Intersect(c, i) → OIntersect(a_1, e_1) → PPolygon(E, O, C, P) → poly6, e_5, o, c_1, pOrthogonalLine(E, b) → aPoint(a) → QSegment(Q, E) → qAngle(H, E, Q) → αOrthogonalLine(E, a_1) → rIntersect(r, d) → RSegment(E, R) → sSegment(R, L) → tSegment(E, L) → g_1Circle(E, s) → h_1Point(h_1) → SCircle(S, t) → k_1Circle(E, g_1) → p_1Intersect(k_1, p_1, 1) → TPolygon(S, T, E) → poly7, e_6, s_1, t_1Circle(E, s) → q_1Segment(M, R) → j_1Segment(M, E) → l_1Point(q_1) → UCircle(E, l_1) → c_2Circle(U, j_1) → d_2Intersect(c_2, d_2, 1) → VPolygon(E, U, V) → poly9, v, e_8, uIntersect(a, c) → WSegment(E, N) → m_1Segment(W, N) → n_1Segment(E, W) → f_2Circle(E, m_1) → g_2Circle(E, f_2) → p_2Point(q_1) → XCircle(X, n_1) → h_2Circle(E, m_1) → q_2Intersect(h_2, q_2, 1) → YPolygon(E, X, Y) → poly10, Z, e_9, A_1Intersect(r, a_1) → B_1Segment(E, B_1) → i_2Segment(B_1, P) → j_2Segment(P, E) → l_2Circle(E, l_2) → r_2Circle(E, i_2) → s_2Point(s_2) → C_1Circle(C_1, j_2) → t_2Intersect(r_2, t_2, 1) → D_1Polygon(E, C_1, D_1) → poly11, d_3, e_{10}, c_3Circle(E, i_2) → f_3Point(f_3) → E_1Segment(P, B_1) → m_2Circle(E_1, j_2) → g_3Circle(E, l_2) → h_3Intersect(g_3, h_3, 1) → F_1Polygon(E, F_1, E_1) → poly12, e_{11}, e_{12}, f_4Circle(E, i_2) → k_3Segment(E, J) → n_2Segment(B_1, J) → a_3Circle(E, n_2) → p_3Point(k_3) → G_1Circle(G_1, a_3) → q_3Intersect(p_3, q_3, 1) → H_1Polygon(E, G_1, H_1) → poly13, h_4, e_{13}, g_4Intersect(a, b) → I_1Segment(I_1, E) → b_3Segment(I_1, I) → i_3Segment(I, E) → j_3Circle(E, b_3) → r_3Point(r_3) → J_1Circle(E, j_3) → s_3Circle(J_1, i_3) → t_3Intersect(s_3, t_3, 2) → K_1Polygon(J_1, K_1, E) → poly14, e_{14}, j_4, k_4Segment(K, I_1) → l_3Segment(E, I_1) → m_3Segment(E, K) → n_3Circle(E, m_3) → c_4Circle(E, n_3) → d_4Point(c_4) → L_1Circle(L_1, l_3) → p_4Intersect(d_4, p_4, 1) → M_1Polygon(L_1, E, M_1) → poly15, m_4, l_4, e_{15}Segment(R, E) → a_4Polygon(B, L, E, I_1) → poly8, r_1, b_2, k_2, b_4Polygon(K, B, R, E) → poly16, k_5, b_5, r_4, e_7Polygon(N, D, R, E) → poly17, n_4, d_5, r_5, e_{16}Polygon(W, D, M, E) → poly18, w, d_6, m_5, e_{17}Polygon(W, C, P, E) → poly19, w_1, c_5, p_5, e_{18}Polygon(O, C, B_1, E) → poly20, o_1, c_6, b_6, e_{19}Polygon(I, A, B_1, E) → poly21, i_4, a_5, b_7, e_{20}Polygon(J, A, I_1, E) → poly22, j_5, a_6, i_5, e_{21}PointIn(poly2) → N_1
AI 绘图提示语参考
画两个正方形,其中一个旋转一定角度与另一个重叠,求重叠部分的面积。先画出正方形的边和顶点,然后找到边的中点并连接,形成新的正方形。接着,利用圆和交点构造重叠区域,并细分重叠部分为多个小多边形,最后计算每个小多边形的面积并求和得到总面积。
以上提示语为根据原始绘图命令整理而来,仅供参考。