如何计算圆扇区的面积？

扇形定义为以两个半径和圆弧为界的圆内的部分。该扇区的面积取决于两个半径之间的相应角度。

特性：

• 一个圆的扇区总是起源于圆的中心。
• 半圆是圆中最常见的部分，半径之间的夹角等于 180°。
• 半圆的面积取决于圆的半径。

部门类型

根据扇区的相应两个半径之间的角度，存在三种类型的扇区。他们是：

• 小扇区：当两个半径之间的夹角小于180°时。包围的区域也小于半圆。
• 半圆：当两个半径之间的夹角等于180°时。
• 大扇区：当两个半径之间的夹角大于 180°时。包围的面积也大于半圆。

扇区面积公式

对于半径等于“r”单位且扇形角度为θ （以度为单位）的圆，面积由下式给出，

半径为r的圆

Area of sector = θ / 360° × πr² 

当θ以弧度给出时，面积由下式给出

Area of sector = 1/2 × r²θ

证明：

For a circle with radius r units, the area is given by πr².

Now the fraction of the area enclosed by the sector will be the same as the fraction of the angle enclosed by the sector in the circle.

Thus, the fraction of area enclosed = θ / 360°

So, the area enclosed by the sector = (θ / 360°) × πr²

示例问题

问题 1. 求一个圆的扇形面积，该圆的内角为 60^度，圆的半径为 5 个单位。是大行业还是小行业？

解决方案：

Give, the angle of the sector = θ = 60°

The radius of the circle = 5 units

Thus, the area of the sector = 60°/360° × π × 5² = 25π/6

Approximating the value of π = 3.14, we get,

Area of sector = 25 × 3.14 / 6 = 13.08 sq. units

Since, angle of sector is less than 180°, it is a minor sector.

问题 2. 求角度为 π/2 弧度且圆的半径为 8cm 的扇形面积。

解决方案：

Since the angle of the sector is given in radian, we can write,

Area of the sector = 1/2 × r² × θ

Given, the radius of the circle is 8cm. Thus,

Area of sector = 1/2 × 8² × π/2 = 16π cm²

Approximating the value of π = 3.14, we get,

Area of sector = 16 × 3.14 = 50.24 cm²

问题 3. 对于给定面积为 50cm ²的圆，有面积为 25cm ² 、45cm ²和 13cm ²的三个扇区。将给定的扇区分为次要扇区、半圆形和主要扇区。

解决方案：

The area of the circle is 50cm².

Thus, half of the area of the circle is 50/2 = 25cm². 

Thus, the sector with an area of 25cm² is a semi-circle.

The sector with an area of 45cm² has a greater area than a semi-circle. Thus, it is a major sector.

Lastly, the sector with an area of 13cm² has a smaller area than a semi-circle. Thus, it is a minor sector.

问题 4. 如果一个半径为 5 英寸的比萨饼被分成 6 等份，求每片比萨饼的围合面积和角度。

解决方案：

Since we divide a pizza into 6 equal pieces, each piece represents a sector with an angle equal to one-sixth of the total angle of pizza, that is 360^o.

So, the angle of each pizza slice = 360°/6 = 60°.

So, the area of each sector is given by,

Area of each slice = (θ / 360°) × πr²,

where,

θ = 60°

r = 5 inches

Thus, we get, area of each slice = 60°/360° × π × 5² = 25π/6 sq. inch 

Putting the value of π = 3.14, we get

Area of each slice =  25 × 3.14 / 6 = 13.08 sq. inch