1.8 圆圈 - 定义、公式、互动和实例-interactive

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  圆圈 - 定义、公式、互动和示例

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  What is a Circle?
  ::什么是圆圈?

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  A circle is a simple closed curve, with a set of all points at a constant distance from a fixed (center) point , $\begin{array}{r}A,\end{array}$   in the same plane . Since the circle has only one center , you can name the circle by naming its center. In this way, you can name this as circle $\begin{array}{r}A.\end{array}$
  ::圆是一个简单封闭的曲线,与固定(中点)A的固定点(A)保持一定距离,有一组所有点。由于圆只有一个中心,您可以通过命名圆的中心命名圆。这样,您就可以将这个点命名为圆 A。

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  Radius of a Circle
  ::圆圆的半径

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  The distance from the center point to the circle is called the radius   $\begin{array}{r}(r).\end{array}$   In other words, a line segment joining the center of a circle with any point on the circle is called a radius (plural: radii) of that circle. $\begin{array}{r}AB\end{array}$   is a radius of the circle $\begin{array}{r}A.\end{array}$
  ::从中点到圆圈的距离称为半径(r)。换句话说,连接圆圈中心及圆圈任何点的线条段称为圆圈半径(多元:半径)。AB是圆A的半径。

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  Diameter of a Circle
  ::圆形直径

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  The distance from one side of the circle to the other through the center point is called the diameter   $\begin{array}{r}(d).\end{array}$   In other words, a line segment joining any two points on a circle and passing through the center of the circle is called a diameter of that circle. The diameter of a circle is twice its radius. $\begin{array}{r}PQ\end{array}$   is a diameter of the circle $\begin{array}{r}A.\end{array}$
  ::从圆的一面到另一面到中间点的距离称为直径(d)。换句话说,连接圆上任何两个点并穿过圆的中心的线段称为圆的直径。圆的直径是圆的两倍。PQ是圆A的直径。

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  The perimeter of a circle is called its circumference . The ratio between the and diameter of any circle is  $\begin{array}{r}\pi \end{array}$  or “ pi ,” is a Greek letter that stands for an irrational number approximately equal to 3.14. Because  $\begin{array}{r}\pi \end{array}$ is the ratio between the circumference and the diameter, the circumference of a circle is equal to the diameter times $\begin{array}{r}\pi .\end{array}$
  ::圆周的周界称为环绕。任何圆圈与直径之比为 或 “ pi ” , 是希腊字母, 表示非理性数字, 约等于 3. 14。 由于 是环绕与直径之比, 圆圈的环绕等于直径时间 。

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  $\begin{array}{r}C=\pi d\end{array}$
  ::Cd

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  CK-12 Interactive: Circles
  ::CK-12 互动:圆

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  Area of Circle
  ::圆圆区域

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  The formula for the area of a circle can be derived by dissecting a circle into wedges and rearranging them to form a shape that is close to a parallelogram . The can then be formed into a shape close to a rectangle . 
  ::圆区域公式可以通过将圆分解成边缘并重新排列以形成接近平行图的形状来得出。然后可以形成接近矩形的形状。

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  The lengths of the sides of the "parallelogram" are $\begin{array}{r}r\end{array}$ and  $\begin{array}{r}\frac{2\pi r}{2}=\end{array}$ $\begin{array}{r}\pi r\end{array}$ . If you imagine cutting the wedges smaller and smaller, the parallelogram will look closer and closer to a rectangle with dimensions  $\begin{array}{r}\pi r\end{array}$  and $\begin{array}{r}r.\end{array}$  The rectangle is made by cutting up the circle, the area of the circle is equal to the area of the rectangle. 
  ::“ 平行图” 边的长度是 r 和 2 r2 r 。 如果您想象着切开小和小的边框, 平行图将会接近和接近与 °r 和 r 维的矩形。 矩形是通过切开圆形而形成的, 圆圈的面积与矩形的面积相等 。

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  $\begin{array}{rl}& \text{Circle area}\\ & =\text{Rectangle area}\\ & =\text{base}\cdot \text{height}\\ & =\pi r\cdot r\\ & =\pi r^2\end{array}$
  ::圆形区域 = 矩形区域 = 基数 rrr2

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  Circumference of Circle in Terms of Radius
  ::半半半径环环环

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  What is the circumference of a circle in terms of radius , $\begin{array}{r}r?\end{array}$
  ::圆的半径环形是多少, r?

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  The circumference of a circle in terms of radius  $\begin{array}{r}r\end{array}$  is .
  ::圆圆圆的半径 r 是 。

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  Arc of a Circle
  ::圆圆弧

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  The arc of a circle is a portion of the circumference of a circle. A minor arc is an arc smaller than a semicircle . The larger of the two arcs is called the major arc . Arcs are named by their endpoints. Minor arc can be written as the letters  $\begin{array}{r}AB\end{array}$  with a curving line above them,  $\begin{array}{r}\widehat{AB}\end{array}$ . If you want to indicate the major arc , add an extra point and use three letters in the name.  $\begin{array}{r}\widehat{AKB}\end{array}$  which is the long arc from  $\begin{array}{r}A\end{array}$  to  $\begin{array}{r}B\end{array}$  going around the bottom via  $\begin{array}{r}K.\end{array}$
  ::圆弧是圆环圈的一部分。小弧小弧小于半圈。两个弧中较大的弧称为大弧。弧按其端点命名。小弧可以写成字母ABAB,上面有弯曲线。如果您想要标明大弧,请添加一个额外点,并使用三个字母。AKBZ是A到B通过K绕到底部的长弧。

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  The formula,  $\begin{array}{r}C=\pi d\end{array}$  gives us the arc-length (that is, circumference, or curved line) for the entire circle. Sometimes we need to work with just a portion of a circle's arc, i.e. fractional part of the circumference.
  ::公式, Cd 给了我们整个圆圈的弧长度( 即环形或曲线线 ) 。 有时我们需要使用圆圈弧的一部分, 也就是圆圈的分形部分来工作 。

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  Consider the following proportion : 
  ::考虑以下比例:

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  $\begin{array}{r}\frac{\text{arc measure}}{{360}^{\circ }}=\frac{\text{arc length }}{\text{circumference}}\end{array}$
  ::弧度量度360 弧长度环绕

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  If we solve the proportion for , and replace "arc measure" with its equivalent "central angle", we can establish the formula:
  ::如果我们解决 的比例, 并用等量的“ 中央角度” 替换“ 弧度” , 我们可以确定公式 :

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  $\begin{array}{r}\text{arc length}=\frac{\text{central angle}}{{360}^{\circ }}\cdot \text{circumference}\end{array}$
  ::弧长=中角360 环度

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  For example, an arc measure of  $\begin{array}{r}{60}^{\circ }\end{array}$ is one-sixth of the circle ( $\begin{array}{r}{360}^{\circ }\end{array}$ ), so the length of that arc will be one-sixth of the circumference of the circle. 
  ::例如,圆的弧度为60is 1.6(360),因此弧的长度将是圆环的六分之一。

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  Calculating Arc Length of a Circle
  ::圆圆的计算弧弧长度

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  Play with a circle by dragging one of the dots that define the endpoints of the arc. The arc length is recalculated as you drag.
  ::拖动一个点来决定弧的终点,以圆来玩。弧的长度会随着拖动重新计算。

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  Sector of a Circle
  ::圆圆的扇区

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  If we start with a circle with a marked radius, and turn the circle a bit, the area marked off looks something like a wedge of pie or a slice of pizza; this is called a sector of the circle. Hence, a sector of a circle is a region bounded by an arc of the circle and the two radii to the endpoints of the arc, where the smaller area ( shaded portion) is known as the minor sector and the larger being the major sector .
  ::如果我们先从一个带有标记半径的圆开始,再稍微转转圆圈,标记的区域看上去象一个饼或披萨的边缘,这被称为圆的一个区块。因此,圆的一个区块是圆弧的区域,圆的两条弧形区域与弧的终点交界,在弧的端点,较小的区域(阴色部分)被称为小区块,大区块是主要区块。

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  When finding the area of a sector, you are actually finding a fractional part of the area of the entire circle. Hence, the area of a sector can be expressed using its central angle or its arc length. 
  ::当找到一个扇区的区域时,您实际上正在找到整个圆圈区域的一部分。因此,一个扇区的区域可以用其中心角或弧长度表示。

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  The following propositions are true regarding the sector:
  ::关于该部门,以下主张属实:

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  $\begin{array}{r}\frac{\text{area of sector}}{\text{area of whole circle}}=\frac{\text{central angle}\theta }{{360}^{\circ }}=\frac{\text{arc length}}{\text{circumference}}\end{array}$
  ::整个圆形区域面积=中角 360 arc 长长

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  $\begin{array}{r}\text{area of sector}=\frac{\text{central angle}}{{360}^{\circ }}\cdot \text{area of whole circle}\end{array}$
  ::区域面积 = 中角360 圆圆整圈区域

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  Calculating Sector Area of a Circle
  ::计算圆圆的区区

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  Play with a circle by dragging one of the dots that define the endpoints of the sector. The sector area is recalculated as you drag.  
  ::拖曳一个点来决定扇区端点的端点,与圆一起玩。扇区区域会随着拖动重新计算。

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  Let's take a look at some example problems. 
  ::让我们来看看一些例子问题。

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  1. Find the with radius 5 inches.
  ::1. 找到半径为5英寸的半径。

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  $\begin{array}{r}\text{Area}=\pi r^2=\pi (5{)}^2=25\pi {\text{in}}^2\end{array}$
  ::区域@r2(5)2=252英寸

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  $\begin{array}{r}\text{Circumference}=\pi d=\pi (2\cdot 5)=10\pi \text{in}\end{array}$
  ::环境环境(25)=10

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  Leaving your answer with a  $\begin{array}{r}\pi \end{array}$ symbol in it is called leaving your answer “in terms of $\begin{array}{r}\pi \end{array}$ .” This is often preferable because it is the exact answer. As soon as you approximate a value for  $\begin{array}{r}\pi \end{array}$ your answer is not exact. Keep in mind that  $\begin{array}{r}\pi \end{array}$ is not a unit. You should still put the appropriate units in your answers.
  ::在回答中留下一个 符号被称为 符号。 这通常更可取, 因为它是准确的答案。 一旦您对 的回答接近一个数值, 您的回答并不准确。 请记住 不是一个单位。 您仍然应该将合适的单位放在答案中 。

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  2. Find the area and circumference of a circle with diameter 16 cm.
  ::2. 寻找直径16厘米的圆形面积和环形。

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  If the diameter is 16 cm, then the radius is 8 cm.
  ::如果直径为16厘米,则半径为8厘米。

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  $\begin{array}{r}\text{Area}=\pi r^2=\pi (8{)}^2=64\pi {\text{cm}}^2\end{array}$
  ::区域r2(8)2=64cm2

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  $\begin{array}{r}\text{Circumference}=\pi d=16\pi \text{cm}\end{array}$
  ::环形=16厘米

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  CK-12 PLIX: Radius or Diameter of a Circle with Given Area
  ::CK-12 PLIX: 带给定区域的圆形半径或直径

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  3. The shape below is a portion of a circle called a sector. This sector is  $\begin{array}{r}\frac{1}{4}\end{array}$ of the circle. Find the area and perimeter of the sector.
  ::3. 下面的形状是称为一个区块的圆圈的一部分,这个区块是圆圈的14,找出该区块的面积和周界。

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  Since the sector is  $\begin{array}{r}\frac{1}{4}\end{array}$ of the circle, its area will be  $\begin{array}{r}\frac{1}{4}\end{array}$ the area of the circle.
  ::由于该区为圆圈的14个,其面积为圆圈的14个。

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  The perimeter of the sector is the sum of the lengths of the two radii and the arc. Each radius is 4 in. The arc is  $\begin{array}{r}\frac{1}{4}\end{array}$ of the circumference of the full circle. 
  ::区域周边是两个半径和弧的长度之和,每个半径为4英寸。弧是整个圆环的14弧。

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  Circles - Examples
  ::圆圈 - 示例

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  Example 1
  ::例1

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  Find the area of a circle with a circumference of $\begin{array}{r}12\pi \text{cm}\end{array}$ .
  ::查找圆的圆区域,环度为12厘米。

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  Let  $\begin{array}{r}r\end{array}$  be the radius of the given circle.
  ::让 r 成为给定圆的半径。

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  $\begin{array}{rl}\text{Circumference}& =2\pi r\\ 12\pi & =2\pi r\\ r& =\frac{12\pi }{2\pi }=6\text{cm}\\ \therefore \text{Area}& =\pi r^2\\ \text{Area}& =\pi (6{)}^2=36\pi {\text{cm}}^2\end{array}$
  ::弧度=2r122226厘米

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  Example 2
  ::例2

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  Find the area and perimeter of the sector below. The sector is  $\begin{array}{r}\frac{1}{3}\end{array}$ of the circle.
  ::查找下区段的面积和周界。 区段为圆圈的13。

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  $\begin{array}{rl}& {\text{Area}}_{\text{sector}}=\frac{1}{3}\cdot {\text{Area}}_{\text{circle}}\\ & {\text{Area}}_{\text{sector}}=\frac{1}{3}\cdot \pi r^2\\ & {\text{Area}}_{\text{sector}}=\frac{1}{3}\cdot \pi (9{)}^2\\ & {\text{Area}}_{\text{sector}}=\frac{1}{3}\cdot \pi (81)=27\pi {\text{cm}}^2\\ & {\text{Perimeter}}_{\text{sector}}=\text{length of the arc}+2\cdot \text{radius}\\ & {\text{Perimeter}}_{\text{sector}}=\frac{2\pi r}{3}+2r\\ & {\text{Perimeter}}_{\text{sector}}=\frac{2\pi (9)}{3}+2(9)\\ & {\text{Perimeter}}_{\text{sector}}=6\pi +18\text{cm}\end{array}$   .
  ::区域=13AreacircleAreaa部门=13r2Area部门=13(9)2Area部门=13(81)=27cm2Perime部门=弧+2radiousPerime部门=2r3+2rPerime部门=2(93+2(9)Perime部门=618厘米。

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  CK-12 PLIX: Area of Sectors and Segments of a Circle
  ::CK-12-PLIX:圆形区段和圆形段面积

  Summary

  播放段落 A circle is a simple closed curve, with a set of all points at a constant distance from a fixed (center) point. ::一个圆是一个简单的封闭曲线,所有点与固定(中点)点保持恒定距离。 播放段落 The radius (r) of a circle is the distance from the center point to any point on the circle. ::圆的半径(r)是圆中点到圆中任何一点的距离。 播放段落 The diameter (d) of a circle is the distance across the circle through the center point, and it is twice the radius. ::圆的直径(d)是圆圆穿过中点的距离,是半径的两倍。 播放段落 The circumference (C) of a circle is its perimeter, and it is equal to the diameter times $\begin{array}{r}\pi .\end{array}$   ::圆环环(C)是其周界,等于直径乘以 。 播放段落 The area of a circle can be calculated using the formula $\begin{array}{r}A=\pi r^2,\end{array}$  where $\begin{array}{r}A\end{array}$  is the area and $\begin{array}{r}r\end{array}$  is the radius. ::圆的面积可以使用公式Ar2计算,其中A为区域,r为半径。 播放段落 The arc of a circle is a portion of the circumference of a circle ::圆弧是圆环圈的一部分 播放段落 The arc length is equal to the central angle divided by 360° times the circumference. ::弧长度等于以圆周度360°的中央角除以360°。 播放段落 The sector of a circle is a wedge shaped region bounded by an arc of the circle and the two radii to the endpoints of the arc. ::圆形区域是圆形形区域,圆形区域与圆形弧和圆形弧两端的圆形区域相邻,以圆形弧的终点为界。 播放段落 The area of a sector is equal to the central angle divided by 360° times the area of the whole circle. ::一个扇区面积等于以整个圆的360度面积除以360度的中央角。

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  Circles - Review Questions
  ::圆圈 - 审查问题

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  1. Find the area and circumference of a circle with radius 3   in .
  ::1. 找出圆的面积和圆环,半径3英寸。

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  2. Find the area and circumference of a circle with radius 6   in .
  ::2. 找出半径为6的圆圈的面积和环形。

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  3. Find the area and circumference of a circle with diameter 15   cm .
  ::3. 寻找直径为15厘米的圆形面积和环形。

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  4. Find the area and circumference of a circle with diameter 22   in .
  ::4. 查找直径22英寸的圆圈的区域和环形。

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  5. Find the area of a circle with a circumference of   $\begin{array}{r}32\pi cm\end{array}$ .
  ::5. 寻找圆圈区域,环度为32厘米。

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  6. Find the circumference of a circle with area   $\begin{array}{r}32\pi c{m}^2\end{array}$ .
  ::6. 找到一个圆圈的环形,区域为32°cm2。

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  7. The sector below is  $\begin{array}{r}\frac{1}{2}\end{array}$   of a circle. Find the area of the sector.
  ::7. 以下部门为12个圆圈,寻找部门领域。

  

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  8. Find the perimeter of the sector in #7.
  ::8. 寻找7号区的周边。

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  9. The sector below is  $\begin{array}{r}\frac{3}{4}\end{array}$   of a circle. Find the area of the sector.
  ::9. 以下部门为34个圆圈,寻找部门领域。

  

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  10. Find the perimeter of the sector in #9.
  ::10. 找出9号区的周边。

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  11. The sector below is  $\begin{array}{r}\frac{1}{8}\end{array}$   of a circle. Find the area of the sector.
  ::11. 以下部门为18个圆圈,寻找部门领域。

  

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  12. Find the perimeter of a $\begin{array}{r}\frac{1}{8}\end{array}$  circle sector with a radius of 1.1 inches.
  ::12. 找出18个圆周段的周界,半径1.1英寸。

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  13. The sector below is  $\begin{array}{r}\frac{2}{3}\end{array}$   of a circle. Find the area of the sector.
  ::13. 下面是圆圈的23个部门,找找这个部门的地区。

  

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  14. Find the perimeter of the sector in #13.
  ::14. 查找13号区的周边。

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  15. Explain the formula  $\begin{array}{r}A=\frac{Cr}{2}\end{array}$   where  $\begin{array}{r}A\end{array}$   is the area of a circle,  $\begin{array}{r}C\end{array}$   is the circumference of the circle, and  $\begin{array}{r}r\end{array}$   is the radius of the circle.
  ::15. 解释公式A=Cr2,其中A是圆的区域,C是圆的环绕,r是圆的半径。

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  16. The triangles shown are equilateral with a side length of 1. Find the area of the green sector.
  ::16. 所显示的三角形为对应形,侧长为1. 找出绿区面积。

  

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  17. The right triangle below has a hypotenuse of 5. Find the area of the green region. (This is called a segment   of the circle.)
  ::17. 下面的右三角形的下限为5。 找到绿区域的区域。 (这称为圆的一个部分。 )

  

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  18. Why is a complete circle assigned an angle measure of   $\begin{array}{r}{360}^{\circ }\end{array}$  instead of   $\begin{array}{r}{100}^{\circ }\end{array}$  or some other number? Make an argument for using   $\begin{array}{r}{360}^{\circ }\end{array}$ . Make an argument against and suggest an alternative.
  ::18. 为什么整个圆圈的角度为360,而不是100或其它数字?为使用360提出论据。提出反对意见并提出替代意见。

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  19. Given a circle with radius 1. For each of the following central angles, find the corresponding arc length. Leave the answers in terms of  $\begin{array}{r}\pi \end{array}$ .
  ::19. 给以半径为圆形的圆。 对于以下每个中心角度,请找到相应的弧长度。请将答案保留为 。

  $\begin{array}{r}\begin{array}{}{0}^{\circ }\\ {90}^{\circ }\\ {180}^{\circ }\\ {270}^{\circ }\\ {360}^{\circ }\end{array}\end{array}$

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  A new type of angle measurement unit can be derived from the arc length of a circle with radius 1. S ince  $\begin{array}{r}{360}^{\circ }\end{array}$  corresponds to an arc length of  $\begin{array}{r}2\pi \end{array}$ ,   the new  angle corresponding to a complete circular rotation is  $\begin{array}{r}2\pi \end{array}$  radians.  In addition to measuring  an angle in degrees, we can also measure it in   radians.   Do you prefer this new unit, or degrees? Why?
  ::一种新型角度测量单位可以从半径 1 的圆弧长度中得出。 由于 360 对应弧长度为 2, 与整个圆环旋转相应的新角度是 2 弧度。 除了以度度测量角外, 我们还可以用弧度测量它。 您是否更喜欢这个新单位, 或度 ? 为什么 ?

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