如何求椭圆的面积？

椭圆是二维形状。它是圆锥曲线的一个组成部分。它是平面上的一条曲线，其中到其两个焦点或焦点的距离之和始终是距指定点的常数。椭圆来自具有两个焦点的圆族。椭圆的偏心率值总是小于一，椭圆的一般方程为

 (x²/a²) + (y²/b²) = 1

其中a代表半长轴的长度， b代表半短轴的长度。

椭圆的一部分

椭圆的一些重要部分是：

• 焦点：椭圆有两个焦点或焦点，坐标分别为 F1(q, 0) 和 F2(-q, 0)，它们之间的距离为 2q
• 中心：它是连接两个焦点的线的中点。
• 长轴：它是椭圆的最长直径。或者我们可以说连接椭圆上两个最远点的线段穿过椭圆的中心。
• 短轴：它是椭圆的最短直径。或者我们可以说它是一条连接椭圆上最近的两个点并穿过椭圆中心的线段。
• Latus Rectum：它是垂直于横轴并穿过焦点的线。

椭圆面积

通过连接与平面中两个固定点保持恒定距离的所有点来创建椭圆。这里的两个固定点称为焦点。因此，椭圆的面积称为其中存在的总面积。用cm ² 、in ² 、m ²等表示。椭圆的面积可以用长半轴的长度和短半轴的长度来计算。椭圆的面积是长半轴和短半轴长度与 π (22/7) 的副产品。公式由下式给出

Area of ellipse A = π × m × n

其中m是半长轴的长度， n是半短轴的长度。

椭圆公式的面积证明

Let us consider A be an ellipse, 2m is the major axis and 2n be the minor axis. It is aligned in the cartesian plane in the reduced form so the equation of the ellipse is

x²/m² + y²/n² = 1

So, 

y = ±n/m√m² – x² ……(1)

Now assume a circle of radius m and whose center lies at the origin. So the equation of the circle is 

x² + y² = m²

So, y = ±√m² – x²  ……(2)

On comparing equations (1) and (2) we conclude that the ellipse is n/m times the circle.

So we can write the area of the two shapes as:

Area of ellipse = n/m x Area of the circle

So, 

Area of ellipse = n/m x (πm²) = πnm

Hence proved

求椭圆面积的步骤

Step 1: Find the length of the semi-major axis or the distance from the farthest point from the center that is m.

Step 2: Find the length of the semi-minor axis or the distance from the nearest point from the center that is n.

Step 3: Now put all these values in the area formula to calculate the area.

Step 4: Now apply the units of area. 

示例：房间的地板被构造成椭圆形，其长轴长度为 15 厘米，短轴长度为 11 厘米。现在找到椭圆的面积。

解决方案：

Given that 

Major axis length(m) = 15 cm

minor axis length(n) = 11 cm

Area of ellipse (A) = π x m x n

= 22/7 x 15 x 11 

= 22/7 x 165 

= 518.57 cm²

使用积分求椭圆的面积

We can also find the area of an ellipse using integration. As we know the general equation of the ellipse is 

x²/m² + y²/n² = 1

y = ±n/m√m² – x² ……(1)

As we know that the ellipse is divided into four quadrants so we find the area or one quadrant then multiple with 4 to get the area of the full ellipse

So, A = 4$\int_{m}^{0} y \,dx$

= 4$\int_{0}^{m} \frac{n}{m}\sqrt{m^2-x^2} \,dx$

= 4n/m$\int_{0}^{m} \sqrt{m^2-x^2} \,dx$

Now put x = m sint, dx = m cost.dt and t = 0 and t = π/2 in the above equation

A = 4n/m$\int_{0}^{\pi/2} \sqrt{m^2-m^2sin^2t} \,.mcost.dt$

= 4n$\int_{0}^{\pi/2} \sqrt{m^2(1-sin^2t)} \,costdt$

= 4mn$\int_{0}^{\pi/2} \sqrt{cos^2t} \,costdt$

= 4mn$\int_{0}^{\pi/2} cos^2t \,dt$

= 4mn$(\frac{t}{2} + \frac{sin2x}{4})^{\pi/2}_0$

= 4mn(π/4)

= πmn

示例问题

问题1：如果长半轴和短半轴的长度分别为6cm和3cm，椭圆的面积是多少？

解决方案：

Given

Length of semi major axis (a) = 6cm

Length of semi minor axis (b) = 3cm

Area of ellipse= π x a x b

                      = (22/7) x 6 x 3

                      = 56.57 cm²

So area of given ellipse is 56.57 cm².

问题2：如果长半轴和短半轴的长度分别为10cm和4.5cm，椭圆的面积是多少？

解决方案：

Given

Length of semi major axis (a) = 10cm

Length of semi minor axis (b) = 4.5cm

Area of ellipse= π x a x b

                      = (22/7) x 10 x 4.5

                      = 141.42 cm²

So area of given ellipse is 141.42 cm².

问题3：如果长轴和短轴的长度分别为10cm和5cm，椭圆的面积是多少？

解决方案：

Given

Length of major axis = 10cm

Length of minor axis = 5cm

So we need to find semi major axis and semi minor axis length. 

Length of semi major axis (a) = Length of major axis/2

                                               = 10/2 = 5cm

Length of semi minor axis (b) = Length of minor axis/2

                                               = 5/2 = 2.5cm

Area of ellipse = π x a x b

                        = (22/7) x 5 x 2.5

                        = 39.25 cm²

So area of given ellipse is 39.25 cm².

问题4：如果长轴和短轴的长度分别为12cm、4cm，求椭圆的面积。

解决方案：

Given

Length of major axis = 12cm

Length of minor axis = 4cm

So we need to find semi major axis and semi minor axis length.

Length of semi major axis (a) = Length of major axis/2

                                               = 12/2 = 6cm

Length of semi minor axis (b) = Length of minor axis/2

                                                = 4/2 = 2cm

Area of ellipse = π x a x b

                        = (22/7) x 6 x 2

                        = 37.71 cm²

So area of given ellipse is 37.71 cm².

问题5：如果长轴和半短轴的长度分别为8cm、2.5cm，求椭圆的面积。

解决方案：

Given

Length of major axis = 8cm

Length of semi minor axis (b) = 2.5cm

So we need to find length of semi major axis 

Length of semi major axis (a) = Length of major axis/2

                                               = 8/2 = 4cm

Area of ellipse = π x a x b

                        = (22/7) x 4 x 2.5

                        = 31.43 cm²

So area of given ellipse is 31.43 cm².