Section: 圆环环 | 9.15 圆环环

Section outline

The theme of the school dance this year is “The 50’s.” For decorations, the 7 ^th grade class is gathering old records and gluing a sparkly trim around the edges. How much trim do they need for each record if its diameter is 12 inches?
::今年学校舞蹈的主题是“50年代 ” 。对于装饰品来说,七年级班正在收集旧纪录,在边缘周围布满闪闪发光的短片。如果每张记录直径为12英寸,他们需要多少修剪?

In this concept, you will learn how to find the circumference of a circle using a given diameter.
::在此概念中,您将学会如何用给定直径找到圆环环。

Finding the Circumference of a Circle
::寻找圆环环

The perimeter is the total distance around the edges of a two-dimensional shape. To find the perimeter of a shape made from line segments, such as a square or triangle , you simply add the lengths of the sides together.
::周界是二维形状边缘周围的总距离。要找到由线段(如方形或三角形)形成的形状的周界, 您只需将两维形状的长度加在一起即可。

$\begin{aligned} P_{Square} = s + s + s + s or P_{S} = 4 s \end{aligned}$
::PSquare=s+s+s+s+s或PS=4s

$\begin{aligned} P_{Triangle} = s_{1} + s_{2} + s_{3} \end{aligned}$
::PTriangle=s1+s2+s3 组合键=s1+s2+s3

The perimeter of a circle is called the circumference and is equal to the total distance around the edge .
::圆环的周界称为环绕,等于边缘周围的总距离。

The diameter , d, of a circle is the straight-line measurement from one point on the circle to another point on the circle that passes directly through the circle’s center .
::圆的直径, d,是一个圆的直径是从圆上一个点到圆上另一个点的直线测量,该点直接穿过圆的中心。

A circle is a unique shape in that every point along the circumference is exactly the same distance from the circle’s center. This measurement of one half the diameter is called the radius , r.
::圆是一个独特的形状, 环绕的每个点都与圆圈的中心完全相同。直径的一半的测量方法叫做半径, r。

$\begin{aligned} d = 2 r \end{aligned}$
::d=2r d=2r

One of the unique qualities of a circle is that its diameter and circumference have a proportional relationship . This means that no matter what size the circle is, the proportional relationship , or ratio , between its circumference and diameter is always the same.
::圆圈的独特特性之一是其直径和环绕具有比例关系。这意味着无论圆圈大小,其周圆和直径之间的比例关系或比例总是相同的。

The formula for the circumference of a circle is the value of pi times the circle’s diameter: $\begin{aligned} C = π d \end{aligned}$
::圆环的公式是圆的直径比值的倍数: Cd

Since the diameter is twice the radius, the formula can also be written: $\begin{aligned} C = 2 π r \end{aligned}$
::由于直径是半径的两倍,也可以写出公式: C=2r

You already know that pi is an irrational number and goes on forever. The rounded value of pi, 3.14, is usually used for calculations.
::您已经知道 Pi 是一个非理性的数字, 并且会一直持续下去。 pi 的四舍五入值 3. 14 通常用于计算。

Let’s look at an example.
::让我们举个例子。

The circles above have different radii and different circumferences. Use these values and the circumference formula to solve for pi.
::上述圆圈有不同的弧度和环形。使用这些值和环形公式来解决 pi 。

First, substitute the values of the first circle into the circumference equation.
::首先,将第一个圆的值替换为环形方程。

$\begin{aligned} 6.28 = π \times 2 \end{aligned}$

Next, perform the necessary calculations to isolate $\begin{aligned} π \end{aligned}$ .
::接下来,执行必要的计算分离。

$\begin{aligned} \begin{array}{rcl} 6.28 \div 2 & = & π \times 2 \div 2 \\ 3.14 & = & π \end{array} \end{aligned}$

Then, substitute the values of the second circle into the equation.
::然后,将第二个圆的值替换为方程。

$\begin{aligned} 12.56 = π \times 4 \end{aligned}$

Perform the necessary calculations to isolate $\begin{aligned} π \end{aligned}$ .
::12.564 进行必要的计算,以分离 _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} \begin{array}{rcl} 12.56 \div 4 & = & π \times 4 \div 4 \\ 3.14 & = & π \end{array} \end{aligned}$

Pi is what is called a constant. It always stays the same.
::Pi就是所谓的常数。它总是保持不变。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the school dance decorations.
::之前,你在学校的舞蹈装饰上遇到一个问题

The students need to know how much sparkly trim they will need to go around each record if its diameter is 12 inches.
::学生们需要知道,如果每个记录的直径是12英寸,他们需要多少闪烁的修剪才能翻遍每一个记录。

First, write down the formula.
::首先,写下公式。

$\begin{aligned} C = π d \end{aligned}$

::Cd

Next, substitute in what you know.
::下一位,你所知道的替代

$\begin{aligned} C = 3.14 \times 12 \end{aligned}$

::C=3.14x12

Then, multiply to solve the equation.
::然后,乘以解答方程式。

$\begin{aligned} C = 37.68 inches \end{aligned}$

::C=37.68英寸 C=37.68英寸

The answer is the circumference is 21.98 inches. The students will probably round this up to 38 inches.
::答案是周长为21.98英寸学生们可能会把这里围起来38英寸

Example 2
::例2

Find the circumference of the circle below.
::查找下方圆的环形。

First write down the formula.
::先写下公式

$\begin{aligned} C = π d \end{aligned}$
::Cd

Next, substitute in what you know.
::下一位,你所知道的替代

$\begin{aligned} \begin{array}{rcl} C & = & π \times 8 \\ C & = & 3.14 \times 8 \end{array} \end{aligned}$

::C8C=3.14×8

Then, multiply to solve the equation.
::然后,乘以解答方程式。

$\begin{aligned} C = 25.12 \end{aligned}$
::C=25.12 C=25.12

If you were to unroll the circle into a flat line, it would be 25.12 inches long. The answer is the circumference, $\begin{aligned} C = 25.12 i n \end{aligned}$ .
::如果将圆向平面线上拉开,则长度为25.12英寸。答案是环形,C=25.12英寸。

Example 3
::例3

What is the circumference of the circle below?
::下面圆圈的环度是多少?

First, write down the formula.
::首先,写下公式。

$\begin{aligned} C = π d \end{aligned}$

::Cd

Next, substitute in what you know.
::下一位,你所知道的替代

$\begin{aligned} \begin{array}{rcl} C & = & π \times 12.7 \\ C & = & 3.14 \times 12.7 \end{array} \end{aligned}$

::C12.7C=3.14×12.7

Then, multiply to solve the equation, and round.
::然后,乘以解答方程,然后圆。

$\begin{aligned} C = 39.878 \end{aligned}$

::C=39.878 C=39.878

The answer is the circumference, $\begin{aligned} C = 39.88 m \end{aligned}$ .
::答案是环形,C=39.88米

Example 4
::例4

Find the circumference of the circle given the radius.

::查找圆周的环形, 以给定圆周圆的半径。

First, remember that the diameter is twice the radius, and use the formula that includes that variable.
::首先,记住直径是半径的两倍, 并使用包含该变量的公式。

$\begin{aligned} C = 2 π r \end{aligned}$

::C=2r

Next, substitute in the values that you know.
::接下来,取代你所知道的价值观。

$\begin{aligned} C = 2 (3.14) (3) \end{aligned}$

::C=2(3.14)(3)

Then, multiply.
::然后,乘数。

$\begin{aligned} C = 18.84 \end{aligned}$

::C=18.84 C=18.84

The answer is 18.84 inches.
::答案是18.84英寸

Example 5
::例5

Find the circumference of a circle with a diameter of 5 inches.
::找到一个直径为5英寸的圆圈环绕。

First, write down the formula.
::首先,写下公式。

$\begin{aligned} C = π d \end{aligned}$

::Cd

Next, substitute in what you know.
::下一位,你所知道的替代

$\begin{aligned} C = 3.14 \times 5 \end{aligned}$

::C=3.14×5

Then, multiply to solve the equation.
::然后,乘以解答方程式。

$\begin{aligned} C = 15.7 \end{aligned}$

::C=15.7 C=15.7

The answer is the circumference, $\begin{aligned} C = 15.7 inches \end{aligned}$ .
::答案是环形 C=15.7英寸

Review
::回顾

Find the circumference of each circle given the radius or the diameter. Use 3.14 for the value $\begin{aligned} π \end{aligned}$ and round your answer to the nearest hundredths place where needed.
::根据半径或直径,查找每个圆的环形。值为 ° 时使用 3.14 ,然后按需要将您的回答绕到最接近的百位位置。

radius = 2 in
::半半径= 2 英寸

diameter = 4 ft
::直径=4英尺

radius = 4.5 in
::半半径=4.5英寸

diameter = 8 meters
::直径=8米

radius = 12 inches
::半径 = 12 英寸

diameter = 12 mm
::直径=12毫米

radius = 14 mm
::半径=14毫米

diameter = 13 feet
::直径=13英尺

radius = 10 inches
::半径 = 10 英寸

diameter = 7.5 feet
::直径=7.5英尺

radius = 2.5 inches
::半径=2.5英寸

radius = 5.6 feet
::半径=5.6英尺

diameter = 3.75 feet
::直径=3.75英尺

diameter = 4.5 feet
::直径=4.5英尺

diameter = 10.75 meters
::直径=10.75米

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Resources
::资源

Section outline

Finding the Circumference of a Circle ::寻找圆环环

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Finding the Circumference of a Circle
::寻找圆环环

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源