8.14 球体表面面积

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  环球表面面积

  

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  Erin is decorating a sphere shaped piñata with pink construction paper. Each sphere has a radius of 2.4 feet and Erin made four piñatas. How much construction paper will she need to cover each piñata completely? How much will she need in total?
  ::Erin正在用粉红色建筑纸装饰一个圆球形的皮纳塔。 每个圆球的半径为2.4英尺,Erin做了4皮纳塔。她需要多少建筑纸来完全覆盖每个皮纳塔?她总共需要多少?

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  In this concept, you will learn to calculate the of spheres.
  ::在这个概念中,你会学会计算球体的范围。

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  Surface Area
  ::地表地区

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  A sphere is a that exists in three-dimensional space . Spheres consist of all the points that are equidistant from a center point . Every point on the sphere is the distance of the radius from the center. Spheres are perfectly round.
  ::球体是一个存在于三维空间中的球体。球体由所有点组成,这些点与中心点相近。球体上的每一点都是半径与中心之间的距离。球体是圆形的。

  

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  Spheres do not have any faces because they are round. Still, you can think of its surface as a flat plane that you can unroll. Imagine you could wrap a sphere in wrapping paper, like a present. The amount of wrapping paper needed to cover the figure exactly represents its surface area.
  ::球体没有任何面孔, 因为它们是圆的。 但是, 您仍然可以把它的表面想象成平面, 可以打开它。 想象一下, 您可以用包装纸将球体包起来, 比如一个礼物。 覆盖这个数字所需的包装纸数量代表它的表面面积 。

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  If you could “unroll” the sphere and show it as a rectangle , the rectangle would have a width that is equivalent to the diameter of the sphere. Its length would be the same as the of the sphere (circumference is the distance around a circle). Now this gives you something you can work with, because you can use the area formula for to find the area of the “unrolled” sphere.
  ::如果您可以“翻转”球体, 并将它显示为矩形, 矩形的宽度将相当于球体的直径。 其长度将与球体的长度相同( 频度是圆圈的距离 ) 。 这给了您一些可以工作的东西, 因为您可以使用区域公式查找“ 滚动” 球的面积 。

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  The formula for the area of rectangles is: $\begin{align*}A=lw\end{align*}$ .
  ::矩形区域的公式是:A=lw。

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  The length is the circumference, so the formula becomes:
  ::长度是周长,所以公式变成:

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  $$\begin{align*}\begin{array}{rcl} l &=& C \\ C &=& 2 \pi r \\ \therefore A &=& 2 \pi r \times w \end{array}\end{align*}$$
  ::l=CC=2A=2rxw

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  Now, the width is equal to the diameter of the “roll”. The diameter is twice the radius. Now your formula becomes:
  ::宽度等于“ 滚动” 的直径。 直径是半径的两倍。 现在, 您的公式变成 :

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  $$\begin{align*}\begin{array}{rcl} w &=& d \\ d &=& 2r \\ \therefore A &=& 2 \pi r \times 2r \\ A &=& 4 \pi r^2 \end{array}\end{align*}$$
  ::w=d=2rA=2r×2rA=4r2

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  So, the formula you use for finding the surface area of a sphere is   $\begin{align*}SA = 4 \pi r^2\end{align*}$   .
  ::因此,你用来寻找球体表面面积的公式是 SA=4r2。

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  All you need to do is substitute the measure of the radius for $\begin{align*}r\end{align*}$  in the formula and solve for $\begin{align*}SA\end{align*}$ , the surface area.
  ::您需要做的只是将半径的量度替换为公式中的 r, 并解决 SA 的表面区域 。

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  Let’s look at an example.
  ::让我们举个例子。

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  What is the surface area of the sphere below?
  ::下面球体的表面面积是多少?

  

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  First, substitute what you know into the surface area formula.
  ::首先,将您所知道的替换为表面积公式。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi r^2 \\ SA &=& 4 \pi (8)^2 \end{array}\end{align*}$$
  ::SA=4r2SA=4(8)2

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  Next, use algebra to calculate the surface area.
  ::接下来,使用代数计算表面积。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi (8)^2 \\ SA &=& 4 \pi(64) \\ SA &=& 256 \pi \\ SA &=& 804.25 \end{array} \end{align*}$$
  ::SA=4(8)2SA=4(64SA=256.__SA=804.25)

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  The answer is 804.25.
  ::答案是804.25。

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  The surface area of the sphere is $\begin{align*}804.25 \ cm^2\end{align*}$   .
  ::球体表面面积为804.25厘米2。

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  If you wanted to make your measurement more accurate, you could say that the surface area is $\begin{align*}256 \pi \ cm^2\end{align*}$   .
  ::如果您想提高测量的准确度, 你可以说表面积是256-cm2。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about  Erin and her piñata s.
  ::早些时候,你得到一个问题 关于艾琳和她的piñatas。

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  Erin wants to know how much pink construction paper she needs to decorate a sphere shaped piñata with a radius of 2.4 feet. She also wants to know the total amount to decorate four piñatas. 
  ::Erin想知道她需要多少粉红色建筑纸来装饰一个圆球形皮纳塔,半径为2.4英尺。她还想知道装饰4皮纳塔的总金额。

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  First, substitute what you know into the surface area formula.
  ::首先,将您所知道的替换为表面积公式。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi r^2 \\ SA &=& 4 \pi (2.4)^2 \end{array}\end{align*}$$
  ::SA=4r2SA=4(2.4)2

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  Next, use algebra to calculate the surface area.
  ::接下来,使用代数计算表面积。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi (2.4)^2 \\ SA &=& 4 \pi(5.76) \\ SA &=& 23.04 \pi \\ SA &=& 72.38 \end{array}\end{align*}$$
  ::SA=4(2.4)2SA=4(5.76)SA=23.04__SA=72.38

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  Then multiply by 4 to find the total amount of construction paper.
  ::然后乘以4 找到建筑造纸的总量。

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  $$\begin{align*}\begin{array}{rcl} SA_{\text{Total}} &=& 4 \times 72.38 \\ SA_{\text{Total}} &=& 289.52 \end{array}\end{align*}$$
  ::SATatal=4×72.38 SATotal=289.52 SATatal=289.52

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  The answer is 289.52.
  ::答案是289.52

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  Erin would need $\begin{align*}72.38 \ ft^2\end{align*}$    of pink construction paper for one piñata and $\begin{align*}289.52 \ ft^2\end{align*}$    for all four.
  ::Erin需要72.38平方英尺的粉红色建筑纸来购买一台皮纳塔和289.52平方英尺的所有四张。

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  Example 2
  ::例2

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  The sphere below has a radius of 8.5 inches. What is the surface area of the sphere?
  ::下面的球体半径为8.5英寸 球体的表面面积是多少?

  

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  First, substitute what you know into the surface area formula.
  ::首先,将您所知道的替换为表面积公式。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi r^2 \\ SA &=& 4 \pi (8.5)^2 \end{array}\end{align*}$$
  ::SA=4r2SA=4(8.5)2

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  Next, use algebra to calculate the surface area.
  ::接下来,使用代数计算表面积。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi (8.5)^2 \\ SA &=& 4 \pi(72.25) \\ SA &=& 289 \pi \\ SA &=& 907.9 \end{array} \end{align*}$$
  ::SA=4(8.5)2SA=4(72.25)SA=289__SA=907.9

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  The answer is 907.9.
  ::答案是907.9。

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  The surface area of the sphere is $\begin{align*}907.9 \ in^2\end{align*}$   .
  ::球体表面面积为907.9英寸2。

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  If you wanted to make your measurement more accurate, you could say that the surface area is   $\begin{align*}289 \ \pi \ in^2\end{align*}$    .
  ::如果您想提高测量的准确度, 你可以说表面积是289 英寸2。

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  Example 3
  ::例3

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  Find the surface area of a sphere with a radius of 5 inches.
  ::找到半径为5英寸的球体表面区域。

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  First, substitute what you know into the surface area formula.
  ::首先,将您所知道的替换为表面积公式。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi r^2 \\ SA &=& 4 \pi (5)^2 \end{array}\end{align*}$$
  ::SA=4r2SA=4(5)2

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  Next, use algebra to calculate the surface area.
  ::接下来,使用代数计算表面积。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi (5)^2 \\ SA &=& 4 \pi(25) \\ SA &=& 100 \pi \\ SA &=& 314.16 \end{array}\end{align*}$$
  ::SA=4(5)2SA=4(25)SA=100SA=314.16。

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  The answer is 314.16.
  ::答案是314.16。

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  The surface area of the sphere is $\begin{align*}314.16 \ in^2\end{align*}$    or $\begin{align*}100 \ \pi \ in^2\end{align*}$   .
  ::球体表面面积为314.16英寸2或100英寸2。

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  Example 4
  ::例4

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  Find the surface area of a sphere with a radius of 8 meters.
  ::找到圆球的表面区域 半径为8米

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  First, substitute what you know into the surface area formula.
  ::首先,将您所知道的替换为表面积公式。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi r^2 \\ SA &=& 4 \pi (8)^2 \end{array}\end{align*}$$
  ::SA=4r2SA=4(8)2

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  Next, use algebra to calculate the surface area.
  ::接下来,使用代数计算表面积。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi (8)^2 \\ SA &=& 4 \pi(64) \\ SA &=& 256 \pi \\ SA &=& 804.25 \end{array} \end{align*}$$
  ::SA=4(8)2SA=4(64SA=256.__SA=804.25)

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  The answer is 804.25.
  ::答案是804.25。

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  The surface area of the sphere is $\begin{align*}804.25 \ m^2\end{align*}$  or  $\begin{align*}256 \pi \ m^2\end{align*}$ .
  ::球体表面面积为804.25平方米或256.%2。

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  Example 5
  ::例5

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  Find the surface area of a sphere with a radius of 12 feet.
  ::找到一个圆球的表面区域,半径12英尺。

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  First, substitute what you know into the surface area formula.
  ::首先,将您所知道的替换为表面积公式。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi r^2 \\ SA &=& 4 \pi (12)^2 \end{array}\end{align*}$$
  ::SA=4r2SA=4(12)2

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  Next, use algebra to calculate the surface area.
  ::接下来,使用代数计算表面积。

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  $$\begin{align*}\begin{array}{rcl} SA &=& 4 \pi (12)^2 \\ SA &=& 4 \pi(144) \\ SA &=& 576 \pi \\ SA &=& 1809.6 \end{array} \end{align*}$$
  ::SA=4(12)2SA=4(144)SA=576SA=1809.6

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  The answer is 1809.6.
  ::答案是1809.6

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  The surface area of the sphere is $\begin{align*}1809.6 \ ft^2\end{align*}$  or   $\begin{align*}576 \pi \ ft^2\end{align*}$ .
  ::球体表面面积为1809.6平方英尺或576平方英尺。

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  Review
  ::回顾

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  Find the surface area of each sphere. Use 3.14 to approximate pi.
  ::查找每个球的表面区域。 使用 3. 14 到 大约 pi 。

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  1. A sphere with a radius of 4 in .
     ::一个半径为4的球体。
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  2. A sphere with a radius of 2 in.
     ::一个半径为 2 英寸的球体。
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  3. A sphere with a radius of 3.5 ft.
     ::半径为3.5英尺的球体
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  4. A sphere with a radius of 6.7 in.
     ::一个半径为6.7的球体。
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  5. A sphere with a radius of 12 cm.
     ::半径为12厘米的球体。
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  6. A sphere with a radius of 1.6 ft.
     ::半径1.6英尺的球体
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  7. A sphere with a radius of 9 m.
     ::半径为9米的球体。
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  8. A sphere with a diameter of 9 m.
     ::直径为9米的球体。
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  9. A sphere with a diameter of 18 in.
     ::一个直径为18英寸的球体。
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  10. A sphere with a diameter of 10 cm.
      ::直径为10厘米的球体。
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  11. A sphere with a diameter of 12 m.
      ::直径为12米的球体。
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  12. A sphere with a diameter of 13 ft.
      ::直径13英尺的球体
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  13. A sphere with a diameter of 15 m.
      ::直径为15米的球体。
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  14. What is the surface area of a sphere whose diameter is 22 centimeters?
      ::直径为22厘米的球体的表面面积是多少?
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  15. Bruce is making a sculpture in his art class that is made of 3 spheres. Each sphere has a radius of 2.3 feet. He will paint them all with blue poster paint. If each bottle of paint covers 20 square feet, how many bottles will Bruce need to buy?
      ::布鲁斯正在他的艺术课上雕塑一个由3个球体组成的雕塑。 每个球体的半径为2.3英尺。 他将用蓝色海报涂料画画它们。 如果每瓶涂料覆盖20平方英尺,布鲁斯需要买多少瓶?

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  Review (Answers)
  ::回顾(答复)

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  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 。

   

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  Resources
  ::资源