Circular measure · 圆的度量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| radians/ˈreɪdɪənz/ | 弧度 | hú dù |
| arc length/ɑːk leŋθ/ | 弧长 | hú zhǎng |
| sector area/ˈsektə ˈeərɪə/ | 扇形面积 | shàn xíng miàn jī |
| segment/ˈseɡmənt/ | 弓形 | gōng xíng |
Why radians 弧度?
- Engineers measure angles in degrees. Mathematicians prefer radians — and for good reason.
- A radian is the natural unit: it's the angle that cuts off an arc equal to the radius.
- With radians, the formulas for arc length 弧长 and sector area 扇形面积 become beautifully simple.
为什么用弧度?
- 工程师用度来测量角。数学家偏好弧度(radians)——而且有充分的理由。
- 一个弧度是自然的单位:它是切出一段等于半径的弧的角。
- 用弧度,弧长和扇形面积的公式变得优美地简单。
What is a radian?
- A radian is the angle that cuts off an arc equal to the radius.
- The link: $\pi \text{ radians} = 180^\circ$.
- degrees → radians: multiply by $\dfrac{\pi}{180}$; radians → degrees: multiply by $\dfrac{180}{\pi}$.
Common conversions. $90^\circ = \dfrac{\pi}{2}$ rad; $\;60^\circ = \dfrac{\pi}{3}$ rad; $\;45^\circ = \dfrac{\pi}{4}$ rad; $\;30^\circ = \dfrac{\pi}{6}$ rad.
什么是弧度?
- 一个弧度是切出一段等于半径的弧的角。
- 联系:$\pi \text{ radians} = 180^\circ$。
- 度 → 弧度:乘以 $\dfrac{\pi}{180}$;弧度 → 度:乘以 $\dfrac{180}{\pi}$。
常见的转换。 $90^\circ = \dfrac{\pi}{2}$ 弧度;$\;60^\circ = \dfrac{\pi}{3}$ 弧度;$\;45^\circ = \dfrac{\pi}{4}$ 弧度;$\;30^\circ = \dfrac{\pi}{6}$ 弧度。
π radians equals how many degrees? · π 弧度等于多少度?
By definition, π radians = 180°. · 根据定义,π 弧度 = 180°。
60° in radians is: · 60° 用弧度是:
60° × π/180 = 60π/180 = π/3. · 60° × π/180 = 60π/180 = π/3。
Arc length and sector area
- For a sector with radius $r$ and angle $\theta$ in radians:
- arc length $s = r\theta$,
- sector area $A = \tfrac12 r^2 \theta$.
The elegance of radians: arc length $= r\theta$ and sector area $= \tfrac{1}{2}r^2\theta$. No messy $\dfrac{\theta}{360}$ fractions needed.
Radians only. The formulas $s = r\theta$ and $A = \tfrac{1}{2}r^2\theta$ work only when $\theta$ is in radians. If your angle is in degrees, convert first.
弧长与扇形面积
- 对一个半径 $r$、角 $\theta$ 以弧度计的扇形:
- 弧长(arc length)$s = r\theta$,
- 扇形面积(sector area)$A = \tfrac12 r^2 \theta$。
弧度的优雅:弧长 $= r\theta$ 而扇形面积 $= \tfrac{1}{2}r^2\theta$。不需要凌乱的 $\dfrac{\theta}{360}$ 分数。
只有弧度。 公式 $s = r\theta$ 和 $A = \tfrac{1}{2}r^2\theta$ 只在 $\theta$ 以弧度计时有效。如果你的角是度,先转换。
The sector · 扇形
s = rθ · A = ½r²θ
Drag the angle · 角度 θ (in radians) and the radius · 半径 r — the arc length and sector area follow. · 拖动角 θ(以弧度计)和半径 r——弧长和扇形面积随之而来。
A sector has radius 5 and angle 2 radians. What is the arc length (s = rθ)? · 一个扇形半径 5、角 2 弧度。弧长(s = rθ)是多少?
s = rθ = 5 × 2 = 10. · s = rθ = 5 × 2 = 10。
A sector has radius 4 and angle 0.5 radians. What is its area (A = ½r²θ)? · 一个扇形半径 4、角 0.5 弧度。它的面积(A = ½r²θ)是多少?
A = ½ × 4² × 0.5 = ½ × 16 × 0.5 = 4. · A = ½ × 4² × 0.5 = ½ × 16 × 0.5 = 4。
The formula s = rθ works when θ is measured in degrees. · 公式 s = rθ 在 θ 以度测量时有效。
s = rθ only works when θ is in radians. Convert degrees to radians first. · s = rθ 只在 θ 以弧度计时有效。先把度转换成弧度。
Segment 弓形 area
- A segment is the region between a chord and the arc.
- Segment area = sector − triangle = $\tfrac12 r^2(\theta - \sin\theta)$.
- The triangle area is $\tfrac{1}{2}r^2\sin\theta$ (using the sine rule for area).
A tangent meets the circle once and is perpendicular to the radius there
弓形面积
- 一个弓形(segment)是一条弦和弧之间的区域。
- 弓形面积 = 扇形 − 三角形 = $\tfrac12 r^2(\theta - \sin\theta)$。
- 三角形面积是 $\tfrac{1}{2}r^2\sin\theta$(用面积的正弦规则)。
一条切线接触圆一次,并在那里垂直于半径
A segment has r = 6 and θ = π/3 radians. The segment area is ½r²(θ − sin θ). Find it (2 dp). · 一个弓形 r = 6、θ = π/3 弧度。弓形面积是 ½r²(θ − sin θ)。求它(2 位小数)。
½ × 36 × (π/3 − sin(π/3)) = 18 × (1.0472 − 0.8660) = 18 × 0.1812 = 3.26. · ½ × 36 × (π/3 − sin(π/3)) = 18 × (1.0472 − 0.8660) = 18 × 0.1812 = 3.26。
Worked example
- Sector: $r = 5$, $\theta = 2$ rad.
- Arc length $= 5 \times 2 = 10$.
- Sector area $= \tfrac{1}{2}(5^2)(2) = 25$.
- Segment area $= \tfrac{1}{2}(25)(2 - \sin 2) = 12.5(2 - 0.909) = 13.6$.
算例
- 扇形:$r = 5$,$\theta = 2$ 弧度。
- 弧长 $= 5 \times 2 = 10$。
- 扇形面积 $= \tfrac{1}{2}(5^2)(2) = 25$。
- 弓形面积 $= \tfrac{1}{2}(25)(2 - \sin 2) = 12.5(2 - 0.909) = 13.6$。
You've got it
- $\pi$ radians $= 180^\circ$; convert by $\times\dfrac{\pi}{180}$ or $\times\dfrac{180}{\pi}$
- arc length $s = r\theta$; sector area $A = \tfrac12 r^2\theta$ ($\theta$ in radians)
- segment $= \tfrac12 r^2(\theta - \sin\theta)$
你掌握了
- $\pi$ 弧度 $= 180^\circ$;通过 $\times\dfrac{\pi}{180}$ 或 $\times\dfrac{180}{\pi}$ 转换
- 弧长 $s = r\theta$;扇形面积 $A = \tfrac12 r^2\theta$($\theta$ 以弧度计)
- 弓形 $= \tfrac12 r^2(\theta - \sin\theta)$