# 圆形计算器：面积、周长与半径

> 由面积、周长、半径或直径中的任一值求其余值，并根据圆心角计算弧长、弦长、扇形面积和弓形面积。

交互版本：https://www.calcopenly.com/zh/geometry/circle-calculator
主题：几何计算器

Give any one of the radius, diameter, circumference or area and the other three follow from d = 2r, C = 2πr and A = πr². For a central angle θ the calculator also returns the arc length s = rθ (θ in radians), the chord c = 2r sin(θ/2), the sector area πr² × θ/360° and the segment area r²/2 × (θ − sin θ).

The default radius of 5 cm gives a diameter of 10 cm, a circumference of 31.4159 cm and an area of 78.5398 cm². Its 60° arc is 5.236 cm long and the chord across it is exactly 5 cm, because two radii and that chord form an equilateral triangle. Starting from the circumference suits trees, pipes and columns whose diameter cannot be reached: a girth of 100 cm means a diameter of 31.831 cm.

Inputs can be in any length or area unit, including hectares and acres, and every result appears in the unit chosen under "Show results in". The central angle must be above 0° and at most 360°; at 360° the segment is the whole disc.

## 输入

- **已知量** (选项：半径, 直径, 圆周长, 面积)
- **半径**
- **直径**
- **圆周长**
- **面积**
- **弧与弦对应的圆心角**: 介于0°和360°之间
- **结果单位** (选项：毫米 (mm), 厘米 (cm), 米 (m), 千米 (km), 英寸 (in), 英尺 (ft), 码 (yd), 英里 (mi))

## 结果

- 面积 — 主要结果
- 半径
- 直径
- 圆周长
- 弧长
- 弦长
- 扇形面积
- 弓形面积

## 公式

$$
\begin{gathered} A = \pi r^2,\quad C = 2\pi r,\quad s = r\theta \\ c = 2r\sin\tfrac{\theta}{2},\quad A_\text{segment} = \tfrac{r^2}{2}(\theta - \sin\theta) \end{gathered}
$$

## 计算示例

### 半径 5 cm，圆弧 60°

- 已知量: 半径
- 半径: 5 cm
- 弧与弦对应的圆心角: 60 °
- 结果单位: 厘米 (cm)
- **面积: 78.539816 cm²**
- **圆周长: 31.415927 cm**
- **直径: 10 cm**
- **弧长: 5.235988 cm**
- **弦长: 5 cm**
- **扇形面积: 13.089969 cm²**
- 核验来源：Python 3.8 math: pi*5**2, 2*pi*5, 5*radians(60), 2*5*sin(radians(30)) = 5 (equilateral triangle), pi*25*60/360

### 周长 100 cm

- 已知量: 圆周长
- 圆周长: 100 cm
- 弧与弦对应的圆心角: 60 °
- 结果单位: 厘米 (cm)
- **半径: 15.915494 cm**
- **面积: 795.774715 cm²**
- **直径: 31.830989 cm**
- 核验来源：Python 3.8 math: 100/(2*pi), 100**2/(4*pi)

### 面积 1 m²，结果以米为单位

- 已知量: 面积
- 面积: 1 m²
- 弧与弦对应的圆心角: 60 °
- 结果单位: 米 (m)
- **半径: 0.56419 m**
- **圆周长: 3.544908 m**
- 核验来源：Python 3.8 math: sqrt(1/pi), 2*sqrt(pi)

### 直径 12 in，四分之一圆弧

- 已知量: 直径
- 直径: 12 in
- 弧与弦对应的圆心角: 90 °
- 结果单位: 英寸 (in)
- **弦长: 8.485281 in**
- **扇形面积: 28.274334 in²**
- **弓形面积: 10.274334 in²**
- **弧长: 9.424778 in**
- 核验来源：Python 3.8 math: 12*sin(pi/4), 36*pi/4, 36*pi/4 − 18, 6*pi/2

### 完整一周（边界情况）：弦长为 0，弓形为整个圆盘

- 已知量: 半径
- 半径: 2 m
- 弧与弦对应的圆心角: 360 °
- 结果单位: 米 (m)
- **弦长: 0 m**
- **弧长: 12.566371 m**
- **弓形面积: 12.566371 m²**
- **面积: 12.566371 m²**
- 核验来源：Definition: a 360° arc is the whole circumference; Python 3.8 math: 4*pi

### 以弧度输入角度（π/3）

- 已知量: 半径
- 半径: 5 cm
- 弧与弦对应的圆心角: pi/3
- 结果单位: 厘米 (cm)
- **弦长: 5 cm**
- **弧长: 5.235988 cm**
- 核验来源：A 60° chord equals the radius (equilateral triangle); Python 3.8 math: 5*pi/3

## 常见问题

### How do you find the area of a circle from the diameter?

Use A = πd²/4, which is πr² with the radius written as half the diameter. A 10 cm diameter gives 25π ≈ 78.54 cm², and a 12 in pizza covers 36π ≈ 113.10 in². Area grows with the square of the diameter, so one 16 in pizza (201.06 in²) has more area than two 11 in pizzas together (190.07 in²).

### How do you find the radius from the circumference?

Divide the circumference by 2π: r = C/(2π). A tree with a girth of 100 cm has a radius of 15.915 cm and a diameter of 31.831 cm (C/π). Foresters' diameter tapes do this division for you: their scale is graduated in units of π, so wrapping the tape round the trunk reads the diameter directly.

### What is the difference between a sector and a segment of a circle?

A sector is the pie slice between two radii and the arc; a segment is the region between the arc and the chord joining its ends. The segment is the sector minus the triangle formed by the two radii and the chord. For a 90° slice of a circle with a 6 in radius, the sector is 9π ≈ 28.274 in², the triangle 18 in², and the segment 10.274 in².

### How do you calculate arc length?

Multiply the radius by the central angle in radians: s = rθ. With the angle in degrees, use s = 2πr × θ/360. A 60° arc on a 5 cm radius is 5 × π/3 ≈ 5.236 cm, one sixth of the 31.416 cm circumference. Putting degrees straight into s = rθ gives an answer 57.3 times too large, the number of degrees in one radian.

### Is 3.14 accurate enough for pi?

For estimates, yes. 3.14 is 0.05% below π, so a 5 cm radius gives an area of 78.5 cm² instead of 78.54 cm²; the fraction 22/7 is 0.04% too high. The absolute error grows with size: for a radius of 100 m, 3.14 gives 31,400 m², which is 15.9 m² short of the true 31,415.9 m².

### “圆形计算器：面积、周长与半径”有多准确？

准确性取决于输入值和方法的假设。十进制运算使用50位有效数字，但估算、数值方法和源数据的精度可能较低；显示时的舍入并不能消除这些限制。 已按独立来源核验的计算示例：6。 例如，“半径 5 cm，圆弧 60°”根据Python 3.8 math: pi*5**2, 2*pi*5, 5*radians(60), 2*5*sin(radians(30)) = 5 (equilateral triangle), pi*25*60/360进行核验。

### 这种方法出自哪里？

Weisstein, E. W. “Circle”, “Circular Segment” — MathWorld; NIST Digital Library of Mathematical Functions §3.12 — mathematical constant π.

## 来源

- [Weisstein, E. W. “Circle”, “Circular Segment” — MathWorld](https://mathworld.wolfram.com/CircularSegment.html)
- [NIST Digital Library of Mathematical Functions §3.12 — mathematical constant π](https://dlmf.nist.gov/3.12)