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椭圆的面积公式S=πab的证明,要用到微积分的知识,在这里,给出一种初等证法。高中《平面解析几何》上有这样的题(P126第23题):底面直径为12cm的圆柱被与底面成30°角的平面所截,截面为一椭圆。求该椭圆的方程。其图如右(图1),不难发现:椭圆的长半轴a、短半轴b与底面圆的半径r有如下关系: a·cosa=r b=r (a为椭圆面与底面成的角) “—一·—L_/ 由此,我们以椭圆x~2/a~2+y~2/b~2=1的短半轴b为底面圆半径,构造一个圆柱(高h足够大),然后用一个平面去截圆柱,当截面与底面成a角时,得到椭圆截面x~2/a~2
The proof of the area formula for the ellipse, S=πab, uses knowledge of calculus. Here, an elementary proof is given. High school “Planar Analytic Geometry” has such a question (P126 Question 23): The cylinder with the bottom diameter of 12cm is cut by a plane with an angle of 30° to the bottom, and the cross section is an ellipse. Find the elliptic equation. As shown in the right figure (Fig. 1), it is not difficult to find that the major axis a of the ellipse, the minor axis b, and the radius r of the bottom circle have the following relationship: a·cosa=rb=r (a is the elliptic surface and the bottom surface Angular) "-a.-L_/ From this, we construct a cylinder (height h is large enough) using the short side b of the ellipse x~2/a~2+y~2/b~2=1 as the radius of the bottom circle. ), and then use a plane to cut the cylinder, when the cross-section and the bottom into a angle, get the elliptical cross-section x ~ 2 / a ~ 2