10.3三角面积

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  三角面积

  

  [Figure 1]

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  Terry has two tickets to his favorite college football team’s next home game, so he is planning to take his younger brother with him to the game. Terry wants to make two pennants for the game, so he buys a piece of material shaped like a  rectangle  with a base of 18 inches and a  height  of 12 inches. Terry plans to divide the material into two triangles. How can Terry figure out the  area  of each of the triangles that will be cut from the material for his pennants?
  ::特里有两张参加他最喜欢的大学足球队下一场家庭比赛的门票,因此他打算带弟弟去参加比赛。 泰瑞想为比赛做两张笔记本,因此他买了一块像长方形一样的材料,底部18英寸高12英寸。 泰瑞计划将材料分为两个三角形。 泰瑞怎样才能找出三角形中每个三角形的面积,这些三角形将与其笔记本上的材料分隔开来?

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  In this concept, you will learn to find the areas of triangles given a base and a height.
  ::在此概念中, 您将学会找到三角形区域的基数和高度 。

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  Finding the Area of a Triangle
  ::查找三角区域

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  Triangles are not   because they only have three sides, but they are related to parallelograms. Take a look the parallelogram below and see if you can figure out the connection.
  ::三角形不是因为它们只有三边,而是它们与平行图相联。看看下面的平行图,看看能否找到连接。

  

  [Figure 2]

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  If you look carefully, you will notice that the parallelogram can be divided into two triangles.
  ::如果您仔细看一看, 您就会注意到平行图可以分为两个三角形 。

  

  [Figure 3]

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  A  rectangle  is a type of parallelogram with four right  angles . You can divide a rectangle into two triangles also.
  ::矩形是一个具有四个右角度的平行图形类型。也可以将矩形分为两个三角形。

  

  [Figure 4]

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  Notice that a rectangle is divided into two  right triangles , triangles containing a  right angle .
  ::注意矩形分为两个右三角形,三角形包含一个右角。

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  A  square  is a type of rectangle with four equal sides and four right angles. A square can also be divided into two right triangles.
  ::方形是一种矩形,有四个等向和四个右角。方形也可以分为两个右三角形。

  

  [Figure 5]

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  If a parallelogram can be divided into two triangles, then the area of a  triangle  is one-half the  .
  ::如果平行图可以分为两个三角形,则三角图的面积为 半。

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  Let’s look at how this works.
  ::让我们看看这如何运作。

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  What is the area of this parallelogram?
  ::这个平行图的方面是什么?

  

  [Figure 6]

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  To find the area of the parallelogram, multiply the base by the height.
  ::要找到平行图的区域,请将基数乘以高度。

  $$\begin{array}{r}\begin{array}{rcl}A& =& bh\\ A& =& 2(5)\\ A& =& 10\text{sq. inches}.\end{array}\end{array}$$

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  A parallelogram can be divided into two triangles.
  ::平行图可以分为两个三角形。

  

  [Figure 7]

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  If you divide the area of the parallelogram in half, that will give you the area of one of the triangles.
  ::如果将平行图的面积除以一半,就会给您一个三角形的面积。

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  $\begin{array}{r}10÷2=5\text{sq. inches}\end{array}$
  ::10 2 = 5 平方英寸

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  Based on this information, you can write the following formula for finding the area of a triangle.
  ::基于此信息,您可以为查找三角形区域写入以下公式。

  $$\begin{array}{r}A=\frac{1}{2}bh\end{array}$$

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  A triangle is one-half of a parallelogram, so the formula for the parallelogram multiplied by one-half is the formula for finding the area of a triangle.
  ::三角形是平行图的1.5倍,平行图的公式乘以1.5倍,是找到三角形区域的公式。

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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 Terry and his sports pennants.
  ::早些时候,你得到一个问题 特里和他的体育公寓。

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  Terry has always been a Gator football fan, so he is excited about attending one of the team’s home games. He has purchased material to make two pennants for the game. The material is shaped like a rectangle, and Terry plans to divide it into two triangles for the pennants. The material has a 38-inch base and a height of 30 inches. Terry wants to know the size of the two triangles after he cuts the material. How can Terry figure out the area of each of the triangles that will form his pennants?
  ::Terry一直是Gator足球球迷,因此他对于参加球队的一场家庭比赛很兴奋。他购买了两张笔记本,为比赛制作两张笔记本。材料形状像矩形,Terry计划将其分为两张三角形。材料有38英寸的底部和30英寸的高度。Terry想了解两张三角形在切开材料之后的大小。Terry如何找出构成他的笔记本的三角形的面积?

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  Here is a look at the piece of material.
  ::这是对一块材料的一目了然。

  

  [Figure 8]

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  First, recall that the area of a rectangle is base times height and a triangle is one half the area of a rectangle.
  ::首先,请记住矩形区域是基时高度,三角形是矩形区域的一半。

  $$\begin{array}{r}\begin{array}{rcl}\text{Base}\times \text{Height}& =& \text{Area of a Rectangle}\\ 18\times 12& =& 216\\ & =& 216\text{ square inches}\end{array}\end{array}$$

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  Next, substitute the area value for the rectangle (or the  $\begin{array}{r}\text{base}\times \text{height}\end{array}$ ) in the triangle area formula.
  ::接下来,将区域值替换为三角形区域公式中的矩形(或基数 x 高度) 。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}bh\\ A& =& \frac{1}{2}(216)\end{array}\end{array}$$

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  Then, solve the equation.
  ::那就解开方程式

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}(216)\\ A& =& 108\text{square inches}\end{array}\end{array}$$

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  The answer is each of the triangles will have an area of 108 square inches.
  ::答案是每个三角形 都有108平方英寸的面积

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  Example 2
  ::例2

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  Find the area of the triangle with a base of 6 feet and a height of 5 feet.
  ::找到三角形的区域,基数为6英尺,高度为5英尺。

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  First, substitute the values for the base and height into the triangle area formula.
  ::首先,将三角区域公式中的基数和高度值替换为基数和高度。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}bh\\ A& =& \frac{1}{2}(6)(5)\end{array}\end{array}$$

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  Next, solve the equation.
  ::下一个,解决方程式。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}(30)\\ A& =& 15\text{square feet}\end{array}\end{array}$$

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  The answer is 15 square feet.
  ::答案是15平方英尺

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  Example 3
  ::例3

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  Using the area of the following parallelogram, find the area of one of the triangles inside the parallelogram.
  ::使用以下平行图的区域,在平行图中找到一个三角形的区域。

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  Area of a rectangle is 12 sq. inches
  ::矩形区域为 12 平方英寸

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  First, recall that the area of a rectangle is base times height and a triangle is one half the area of a rectangle.
  ::首先,请记住矩形区域是基时高度,三角形是矩形区域的一半。

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  Next, substitute the area value for the base and height in the triangle area formula.
  ::下一步,将区域值替换为三角形区域公式中的基值和高度。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}bh\\ A& =& \frac{1}{2}(12)\end{array}\end{array}$$

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  Then, solve the equation.
  ::那就解开方程式

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}(12)\\ A& =& 6\text{square inches}\end{array}\end{array}$$

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  The answer is 6 square inches.
  ::答案是6平方英寸

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  Example 4
  ::例4

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  Using the area of the following parallelogram, find the area of one of the triangles inside the parallelogram.
  ::使用以下平行图的区域,在平行图中找到一个三角形的区域。

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  Area of a parallelogram is 24 sq. feet
  ::平行平面面积为24平方英尺

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  First, recall that the area of a parallelogram is base times height and a triangle is one half the area of a parallelogram.
  ::首先,请记住,平行图的面积是基时高度,三角形是平行图的面积的一半。

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  Next, substitute the area value for the base and height in the triangle area formula.
  ::下一步,将区域值替换为三角形区域公式中的基值和高度。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}bh\\ A& =& \frac{1}{2}(24)\end{array}\end{array}$$

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  Then, solve the equation.
  ::那就解开方程式

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}(24)\\ A& =& 12\text{square feet}\end{array}\end{array}$$

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  The answer is 12 square feet.
  ::答案是12平方英尺

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  Example 5
  ::例5

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  Using the area of the following parallelogram, find the area of one of the triangles inside the parallelogram.
  ::使用以下平行图的区域,在平行图中找到一个三角形的区域。

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  Area of a parallelogram is 18 sq. feet
  ::平行平面面积为18平方英尺

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  First, substitute the area value for the base and height in the triangle area formula.
  ::首先,以三角区域公式中的基值和高度取代区域值。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}bh\\ A& =& \frac{1}{2}(18)\end{array}\end{array}$$

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  Then, solve the equation.
  ::那就解开方程式

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}(18)\\ A& =& 9\text{square feet}\end{array}\end{array}$$

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  The answer is 9 square feet.
  
  ::答案是9平方英尺

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  Review
  ::回顾

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  Find the area of each triangle given the following measurements.
  ::根据以下测量,查找每个三角形的区域。

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  1.  
     

     [Figure 9]
     
     ::[图9]

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   Base = 10 in, Height = 4 in
  ::基数 = 10英寸,高度 = 4英寸

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  2. 

     [Figure 10]
     
     ::[图10]

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   Base = 16 meters, Height = 10 meters
  ::基数=16米,高度=10米

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  3. 

     [Figure 11]
     
     ::[表11]

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   Base = 8 in, Height = 6.5 in
  ::基数=8英寸,高度=6.5英寸

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  4. 

     [Figure 12]
       
     ::[图12]

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   Base = 10 cm, Height = 7 cm
  ::基数=10厘米,高度=7厘米

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  5. 

     [Figure 13]
     
     ::[图13]

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   Base = 5 ft, Height = 8.5 feet
  ::基数=5英尺,高度=8.5英尺

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  Find the area of each triangle given the base and height. Round your answers to the nearest hundredths if needed.
  ::查找每个三角形的面积, 给出基数和高度。 如果需要的话, 将您的答案四舍五入到最接近的百位 。

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  6. Base = 4 in, Height = 5 in
     ::基数 = 4 英寸,高度 = 5 英寸
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  7. Base = 6 in, Height = 4 in
     ::基数=6英寸,高度=4英寸
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  8. Base = 8 ft, Height = 7 ft
     ::基数 = 8 英尺,高度 = 7 英尺
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  9. Base = 10 meters, Height = 8 meters
     ::基数 = 10米,高度 = 8米
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  10. Base = 10 meters, Height = 5 meters
      ::基数 = 10米,高度 = 5米
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  11. Base = 12 feet, Height = 14 feet
      ::基地=12英尺,高度=14英尺
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  12. Base = 11 feet, Height = 6 feet
      ::基地=11英尺,高度=6英尺
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  13. Base = 14 inches, Height = 8 inches
      ::基数=14英寸,高度=8英寸
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  14. Base = 22 feet, Height = 19 feet
      ::基地=22英尺,高度=19英尺
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  15. Base = 30 cm, Height = 28 cm
      ::基数=30厘米,高度=28厘米
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  16. Base = 18 inches, Height = 16 inches
      ::基数 = 18 英寸,高度 = 16 英寸
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  17. Base = 13 meters, Height = 10 meters
      ::基地=13米,高度=10米
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  18. Base = 18 meters, Height = 5.5 meters
      ::基数=18米,高度=5.5米
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  19. Base = 12.5 feet, Height = 2.5 feet
      ::基地=12.5英尺,高度=2.5英尺
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  20. Base = 13.75 inches, Height = 1.5 inches
      ::基数=13.75英寸,高度=1.5英寸

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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