如何证明圆的面积是 pi r 的平方？

圆是一个封闭的二维图形，它有一个中心，平面上的所有点都与它等距。穿过圆的每条线形成反射对称线。除此之外，它在每个角度都围绕中心旋转对称。圆圈的一些例子是轮子、比萨饼、圆形地面等。

圆的性质

圆的特征在于以下一组属性：

• 圆的外线与中心等距。
• 圆的直径将它分成两个相等的部分。
• 具有相等半径的圆彼此全等。
• 不同半径的圆彼此相似。
• 圆的直径被称为最大弦，被认为是半径的两倍。

圆的部分

圆是距圆心固定距离的点的集合。圆的面积被认为是包围在圆内的空间或区域的量度。

• 半径：从中心到边界上一点的距离。它由字母“r”或“R”表示。它用于确定圆的周长。
• 直径：通过圆心和端点的直线。它用字母“d”或“D”表示。

直径公式：圆的直径公式表示为半径的两倍。

Diameter = 2 × Radius

换一种说法，

d = 2r or D = 2R

如果圆的直径已知，则其半径可计算为：

r = d/2 or R = D/2

如何证明圆的面积是 pi r 的平方？

证明：

A circle can be easily segregated into 16 equal sectors which are arranged in the following form. All of the sectors are equal in area. This implies that all the sectors have equal arc length. In case the number of sectors cut from the circle is increased, the parallelogram will eventually look like a rectangle with a length equivalent to πr and breadth equivalent to r.

The area of a rectangle (A) is also considered to be the area of a circle. Therefore, 

• A = πr×r
• A = πr²

示例问题

问题 1. 求半径为 10 厘米的圆的面积？使用 π = 3.14。

解决方案：

Here we need to find the area of the circle,

Given:

Radius of the circle = 10 cm

As we know that

Area of the circle = πr²

Area of the circle = 3.14 × 10 × 10

Area of the circle = 314 cm²

Therefore,

Area of the circle is 314 cm².

问题 2. 如果圆的直径是 24 m，那么求圆的面积？使用 π = 3.14

解决方案：

Here we need to find the area of the circle,

Given:

Diameter of the circle = 24 m

Radius of the circle = 24/2

Radius of the circle = 12 m

As we know that

Area of the circle = πr²

Area of the circle = 3.14 × 12 × 12

Area of the circle = 452.16 m²

Therefore,

Area of the circle is 452.16 m².

问题 3. 如果圆的面积是 3850 cm ²那么求圆的半径？使用 π = 22/7。

解决方案：

Here we have to find the radius of the circle using its area.

Given:

Area of the circle = 3850 cm²

As we know that

Area of the circle = πr²

Area of the circle = 22/7 × r²

3850 = 22/7 × r²

r² = 3850 × 7/22

r² = 1225

r = √1225

r = 35 cm

Therefore,

Radius of the circle is 35 cm when the area of the circle is 3850 cm².

问题 4. 求以350 卢比/m ²的比率为半径为 33 m 的圆形体育馆铺设地毯的成本？使用 π = 3.14。

解决方案：

Here we need to find the cost of carpeting gymnastic hall,

Given:

Radius of the circular gymnastic hall = 33 m

As we know that

Area of the circle = πr²

Area of the circle = 3.14 × 33²

Area of the circle = 3.14 × 33 × 33

Area of the circle = 3419.46 m²

Now,

Cost of carpeting = ₹350 × area of the circular gymnastic hall

Cost of carpeting = ₹350 × 3419.46

Cost of carpeting = ₹1196811

Therefore,

Cost of carpeting circular gymnastic ground is ₹1196811.