Section: 给定面积的圆的半径或直径 | 10.9给定面积的圆的半径或直径

Section outline

Richard and his classmate, Sid, are building a large wooden replica of a clock for their science class project. The clock’s face will be a perfect circle with an area of 530.66 square inches. Richard has to cut the long hand, or dial, which will extend from the center of the clock to the edge , but he isn’t sure how long it should be. How can Richard use what he knows about the area of the clock to determine how long the dial should be?
::理查德和他的同学希德正在为他们的科学课项目建造一个大型的木制时钟复制品。时钟的表情将是一个完美的圆圈,面积为530.66平方英寸。理查德必须切断长手或拨号,从时钟中心延伸到边缘,但他不确定应该花多久时间。理查德如何利用他所知道的时钟区域来确定拨号应该花多久?

In this concept, you will learn to find the radius or diameter of a circle when you have been given the area.
::在此概念中, 当给定区域时, 您将学会找到圆的半径或直径。

Finding Radius or Diameter of a Circle
::查找半径或圆形的直径

The equation for the area of a circle is $\begin{aligned} A = π r^{2} \end{aligned}$ .
::圆区域方程式为 A = r 2。

If you are given the area of a circle, you can use this equation and work backwards to find the unknown radius and diameter of the circle.
::如果给您给定了一个圆的区域,您可以使用此方程式,向后工作以找到圆的未知半径和直径。

Here is an example.
::举一个例子。

The area of a circle is $\begin{aligned} 153.86 i n^{2} \end{aligned}$ . Find the radius and diameter.
::圆的面积为153.86 i n 2。查找半径和直径。

This problem requires you to figure out two different things. Let’s find the radius and then use that measure to figure out the diameter.
::这个问题需要你找出两个不同的问题。让我们找到半径,然后用这个尺度来计算直径。

Start with the formula for finding the area of a circle.
::以查找圆区域公式开始。

$\begin{aligned} A = π r^{2} \end{aligned}$

First, substitute the given information in the equation. You have the area and you know that pi is 3.14.
::首先, 替换方程式中给定的信息。您有区域, 您知道pi是 3. 14 。

$\begin{aligned} 153.86 = (3.14) r^{2} \end{aligned}$

Next, divide the area by pi. This will help to get one step closer to figuring out the radius.
::接下来,将区域除以 pi 。这将帮助更接近了解半径的一步。

$\begin{aligned} 3.14 \bar{) 153.86} \end{aligned}$

Remember, when you divide decimals, to move the decimal two places in the divisor and the dividend.
::记住,当您将小数点除以小数点时,将小数点后两位位移到分数和红利中。

$\begin{aligned} 314 \bar{) 15386} \end{aligned}$

$\begin{aligned} \overset{49}{314 \bar{) 15386}} \\ \underline{- 1256} \\ 2826 \\ \underline{- 2826} \\ 0 \end{aligned}$

So far, the answer is 49, but that is not the radius because $\begin{aligned} r^{2} \end{aligned}$ is on the right side of the equal sign.
::到目前为止,答案是49, 但这不是半径, 因为 r 2 在平等标志的右侧。

$\begin{aligned} 49 = r^{2} \end{aligned}$

Then, you need to figure out which number times itself is equal to 49. You can figure this out by finding the factors of 49.
::然后,你需要弄清楚多少乘数本身等于49, 你可以通过找出49的因数来弄清楚。

$\begin{aligned} 7 \times 7 = 49 \end{aligned}$

Now, you know that the radius is 7 inches because $\begin{aligned} 7 \times 7 = 49 \end{aligned}$ .
::你知道半径是7英寸因为7×7=49

The radius is 7 inches.
::半径是7英寸

Finally, to figure out the diameter, which is twice the radius, multiply the radius by 2.
::最后,要找出直径,也就是半径的两倍, 半径乘以2。

$\begin{aligned} 7 \times 2 = 14 \end{aligned}$

The diameter is 14 inches.
::直径14英寸

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Richard and the clock he is making for his science class project.
::早些时候,你得到一个问题关于理查德和他为科学课项目制作的钟表。

Richard knows that the clock’s face is a circle and that the area of the circle is 530.66 square inches. Now, he needs to figure out the length of a dial that will extend from the center of the circle to the edge.
::理查德知道时钟的表情是一个圆圈,圆圈的面积是530.66平方英寸。现在,他需要弄清楚一个从圆圈中心延伸到边缘的拨号长度。

First, substitute the given information into the equation. You have the area and you know that pi is 3.14.
::首先, 将给定的信息替换为方程。您有区域, 您知道pi是 3. 14 。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ 530.66 & = & (3.14) r^{2} \end{array} \end{aligned}$

Next, divide 530.66 by 3.14 to get 169 is equal to the radius squared.
::接下来,530.66除以3.14 获得169等于半径平方。

$\begin{aligned} 169 = r^{2} \end{aligned}$

Then, they asked themselves what number times itself is equal to 169. They found the answer by listing the factors of 169.
::然后,他们问自己,自己多少倍等于169倍,他们通过列出169个因素找到了答案。

$\begin{aligned} 13 \times 13 = 169 \end{aligned}$

The radius of the circle is 13 inches.
::圆的半径是13英寸

So, the dial should be 13 inches long.
::所以拨号应该长13英寸

Example 2
::例2

Use the area of a circle formula to answer the following question.
::使用圆形公式的区域回答下列问题。

If the area of a circle is 314 sq. cm, what is the radius of the circle?
::如果圆的面积是314平方米,圆的半径是多少?

First, substitute the given information in the equation. You have the area and you know that pi is 3.14.
::首先, 替换方程式中给定的信息。您有区域, 您知道pi是 3. 14 。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ 314 & = & (3.14) r^{2} \end{array} \end{aligned}$

Next, divide 314 by 3.14, which will be 100.
::其次,314除以3.14,即100。

$\begin{aligned} 100 = r^{2} \end{aligned}$

Then, ask yourself what number times itself is equal to 100. You can also find the factors of 100.
::那么,问问你自己,多少乘数本身等于100,你也可以找到100的因数。

$\begin{aligned} 10 \times 10 = 100 \end{aligned}$

The radius of the circle is 10 cm.
::圆的半径为10厘米。

Example 3
::例3

The area of a circle is $\begin{aligned} 12.56 c m^{2} \end{aligned}$ . What is the radius? What is the diameter?
::圆的面积是12.56厘米2。圆的半径是多少?圆的直径是多少?

First, substitute the given information in the equation. You have the area and you know that pi is 3.14.
::首先, 替换方程式中给定的信息。您有区域, 您知道pi是 3. 14 。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ 12.56 & = & (3.14) r^{2} \end{array} \end{aligned}$

Next, divide 12.56 by 3.14. You can move the decimal over two places and divide 1256 by 314, which is 4.
::接下来,12.56除以3.14。您可以将小数点移到两个位子上,将1256除以314,即4。

$\begin{aligned} 4 = r^{2} \end{aligned}$

Then, ask yourself what number times itself is equal to 4. You can also find the factors of 4.
::那么,问问你自己,多少倍本身等于4倍,你也可以找到4的因数。

$\begin{aligned} 2 \times 2 = 4 \end{aligned}$

The radius of the circle is 2 cm.
::圆的半径为2厘米。

Finally, to get the diameter, multiply the radius by 2.
::最后,为了获得直径,将半径乘以2。

$\begin{aligned} 2 \times 2 = 4 \end{aligned}$

The diameter is 4 cm.
::直径为4厘米。

The answer is the radius is 2 cm. and the diameter is 4 cm.
::答案是半径为2厘米,直径为4厘米。

Example 4
::例4

The area of a circle is $\begin{aligned} 200.96 m^{2} \end{aligned}$ . What is the radius? What is the diameter?
::圆的面积是200.96米2。圆的半径是多少?圆的直径是多少?

First, substitute the given information in the equation. You have the area and you know that pi is 3.14.
::首先, 替换方程式中给定的信息。您有区域, 您知道pi是 3. 14 。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ 200.96 & = & (3.14) r^{2} \end{array} \end{aligned}$

Next, divide 200.96 by 3.14. You can move the decimal over two places and divide 20096 by 314, which is 64.
::接下来,将200.96除以3.14。您可以在两个位子上移动小数点,并将20096除以314,即64。

$\begin{aligned} 64 = r^{2} \end{aligned}$

Then, ask yourself what number times itself is equal to 64. You can also find the factors of 64.
::然后,问问你自己,多少倍本身等于64,你也可以找到64的因素。

$\begin{aligned} 8 \times 8 = 64 \end{aligned}$

The radius of the circle is 8 m.
::圆的半径为8米。

Finally, to get the diameter, multiply the radius by 2.
::最后,为了获得直径,将半径乘以2。

$\begin{aligned} 8 \times 2 = 16 \end{aligned}$

The diameter is 16 m.
::直径为16米。

The answer is the radius is 8 m. and the diameter is 16 m.
::答案是半径为8米,直径为16米。

Example 5
::例5

If the area of a circle is $\begin{aligned} 379.94 m^{2} \end{aligned}$ , what is the radius? What is the diameter?
::如果圆的面积是379.94米2,半径是多少?直径是多少?

First, substitute the given information in the equation. You have the area and you know that pi is 3.14.
::首先, 替换方程式中给定的信息。您有区域, 您知道pi是 3. 14 。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ 379.94 & = & (3.14) r^{2} \end{array} \end{aligned}$

Next, divide 379.94 by 3.14. You can move the decimal over two places and divide 37994 by 314, which is 121.
::接下来,将379.94除以3.14。您可以将小数点移到两个位子上,并将37994除以314,即121。

$\begin{aligned} 121 = r^{2} \end{aligned}$

Then, ask yourself what number times itself is equal to 121. You can also find the factors of 121.
::那么,问问你自己,多少次数本身等于121倍,你也可以找到121因素。

$\begin{aligned} 11 \times 11 = 121 \end{aligned}$

The radius of the circle is 11 m.
::圆的半径为11米。

Finally, to get the diameter, multiply the radius by 2.
::最后,为了获得直径,将半径乘以2。

$\begin{aligned} 11 \times 2 = 22 \end{aligned}$

The diameter is 22 m.
::直径为22米。

The answer is the radius is 11 m. and the diameter is 22 m.

::答案是半径为11米,直径为22米。

Review
::回顾

Use the given area to find the radius of each circle. Use the value approximated value 3.14 for the number $\begin{aligned} π . \end{aligned}$
::使用给定区域查找每个圆的半径。使用数值大约为 3.14 的数值来查找 __ 。

$\begin{aligned} A = 12.56 s q . c m \end{aligned}$ .
::A=12.56 sq. cm.

$\begin{aligned} A = 28.26 s q . m \end{aligned}$ .
::A=28.26平方米/米。

$\begin{aligned} A = 50.24 s q . c m \end{aligned}$ .
::A = 50.24 s q. c m.

$\begin{aligned} A = 78.5 s q . f t \end{aligned}$ .
::A=78.5平方米q.f t.

$\begin{aligned} A = 153.86 s q . m \end{aligned}$ .
::A=153.86平方米/米。

$\begin{aligned} A = 200.96 s q . i n \end{aligned}$ .
::A = 200.96 sq. i n.

$\begin{aligned} A = 254.34 s q . f t \end{aligned}$ .
::A=254.34 q.f t.

$\begin{aligned} A = 113.04 s q . m i \end{aligned}$ .
::A=113.04 s q.m i.

$\begin{aligned} A = 452.16 s q . m \end{aligned}$ .
::A=452.16平方米/米。

$\begin{aligned} A = 615.44 s q . c m \end{aligned}$ .
::A = 615.44 sq. c m.

$\begin{aligned} A = 803.84 s q . i n \end{aligned}$ .
::A = 803.84 s q. i n 。

$\begin{aligned} A = 1017.36 s q . f t \end{aligned}$ .
::A=1017.36 q.f t.

$\begin{aligned} A = 1256 s q . f t \end{aligned}$ .
::A = 1256 s q. f t.

$\begin{aligned} A = 1384.74 s q . f t \end{aligned}$ .
::A=1384.74平方米 q.f t.

$\begin{aligned} A = 1962.5 s q . f t \end{aligned}$ .
::A = 1962年5月 q.f t = 1962年5月 q.f t

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 Radius or Diameter of a Circle ::查找半径或圆形的直径

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 Radius or Diameter of a Circle
::查找半径或圆形的直径

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