1.9 复合形状(综合图) -- -- 定义、公式、互动和实例-interactive

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• 选择节复合形形(综合图) - 定义、公式、互动和实例

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  复合形形(综合图) - 定义、公式、互动和实例

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  What is a Composite Shape?
  ::什么是复合形状?

  A composite shape or a composite figure is a two-dimensional figure made up of basic two-dimensional shapes such as triangles, , circles, semi-circles, etc.

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  Perimeter of Composite Figures
  ::综合数字周边

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  To find the perimeter of a composite two-dimensional figure, add the lengths of the sides.
  ::要找到一个复合二维图的周界,请加上两侧的长度。

  

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  Area of Composite Figures
  ::综合数字领域

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  There are two general methods for finding the area of a composite shape . 
  ::找到复合形状区域有两种一般方法。

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  Additive Areas Method
  ::适应地区方法

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  Find the individual areas of each piece of the composite shape. The area of the composite shape will be the sum of the individual areas .
  ::查找复合形状每个部分的单个区域。复合形状的区域将是单个区域的和。

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  Subtractive Areas Method
  ::减减区域法

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  Find the area of a shape larger than the composite shape and the areas of the pieces of the larger shape not included in the composite shape. The area of the composite shape will be the difference between the area of the larger shape and the areas of the pieces of the larger shape not included in the composite shape.
  ::查找大于复合形状的形状面积和未包含在复合形状中的较大形状块块的面积。复合形状的面积将是较大形状区域与未包含在复合形状中的较大形状块块区域之间的差别。

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  Regardless of what method you use, you will often have to think carefully in order to find the dimensions necessary for determining the area or perimeter.
  ::无论你使用何种方法,你往往必须仔细思考,以便找到确定面积或周界所需的尺寸。

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  CK-12 Interactive: Composite Shapes
  ::CK-12 互动:复合形状

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  Finding Area of Composite Shape U sing the  A dditive A reas M ethod
  ::使用 " Additive地区方法 " 查找复合形状区域

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  Find the area and perimeter of the composite shape below.
  ::找出下面复合形状的面积和周界。

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  You can find the area, using the additive areas method. The shape has been broken into 1 triangle and 2 rectangles ( although it could have been broken up differently).
  ::您可以使用添加区域方法找到此区域。形状已被折成一个三角形和两个矩形(尽管它本可以被以不同方式分割)。

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  First, find all of the missing side lengths. 
  ::首先,找到所有缺失的侧边长度

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  For the right triangle , you know the hypotenuse is 5 inches because of the tick mark showing it is congruent to the other segment that is 5 inches . The height is 4 inches because the full height of the shape is 10 inches and the height above the triangle is given as 6 inches . Finally, you know the base is 3 inches because of the Pythagorean Theorem (or, because the full base is 15 inches and other portions are 7 inches and 5 inches ).
  ::右三角形的下限为 5 英寸, 因为刻记显示它与其他部位相匹配 5 英寸。 高度为 4 英寸, 因为形状的全高度为 10 英寸, 三角形上方的高度为 6 英寸。 最后, 您知道基座为 3 英寸, 因为有 Pythagorean 定理( 或者, 因为整个底座为 15 英寸, 其他部分为 7 英寸, 5 英寸 ) 。

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  Now you can find the perimeter by finding the sum of all the side lengths.
  ::现在,你可以通过找到所有侧边长度的总和 找到周界。

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  $\begin{array}{r}P=5+7+6+5+10+12+3=48\text{in}\end{array}$
  ::P=5+7+6+5+5+10+12+3=48

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  Find the area of each of the three pieces and then the total area.
  ::找出三块的每个区域 然后找出总面积

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  $\begin{array}{rl}\text{Area of Triangle, }{A}_1& =\frac{bh}{2}=\frac{3\cdot 4}{2}=6{\text{in}}^2\\ \text{Area of Rectangle \#1, }{A}_2& =bh=12\cdot 4=48{\text{in}}^2\\ \text{Area of Rectangle \#2, }{A}_3& =bh=5\cdot 6=30{\text{in}}^2\\ \text{Total Area, }{A}_1+A_2+A_3& =6+48+30=84{\text{in}}^2\end{array}$
  ::三角区域, A1=bh2=342=6, 矩形2区域 #1, A2=bh=124=48, 矩形2区域2, A3=bh=56=30, 共区域2, A1+A2+A3=6+48+30=84

   

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  F inding A rea of Composite Shape U sing the S ubtractive A rea M ethod. 
  ::使用减缩面积方法寻找复合形状区域。

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  Find the area of the composite shape:
  ::查找复合形状的区域 :

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  To  find the area using subtractive areas method:
  ::使用减让区域方法查找区域:

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  Find the area of the large rectangle , the unshaded triangle , and the unshaded rectangle . Subtract to find the area of the composite shape. Just like with additive areas method, you will first need to find missing side lengths. For the unshaded rectangle , you know the base is 10 inches as the full base is 15 inches and a portion of which is part of the other segment that is 5 inches . The height of the unshaded rectangle is 6 inches . For the triangle, the height is 4 inches because the full height of the shape is 10 inches and the height above the rectangle is given as 6 inches . Finally, you know the base is 3 inches because of the Pythagorean theorem. 
  ::查找大矩形的区域, 未阴影的三角形和未阴影的矩形。 减号以找到复合形状的区域。 就像添加区域方法一样, 您首先需要找到缺失的侧边长度。 对于未阴影的矩形, 您知道底部是10英寸, 因为整根的底部是15英寸, 其中一部分是其它区段的5英寸。 未阴影的矩形的高度是6英寸。 对于三角形, 高度是4英寸, 因为形状的全高度是10英寸, 矩形上方的高度是6英寸。 最后, 您知道底部是3英寸, 因为有Pythagorean 的定理。

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  $\begin{array}{rl}{A}_{\text{large rectangle}}& =bh=15\cdot 10=150{\text{in}}^2\\ A_{\text{unshaded rectangle}}& =bh=10\cdot 6=60{\text{in}}^2\\ A_{\text{unshaded triangle}}& =\frac{bh}{2}=\frac{3\cdot 4}{2}=6{\text{in}}^2\\ \text{Total Area}& =150-60-6=84{\text{in}}^2\end{array}$  
  ::宽矩形= bh= 1510= 150 英寸 2 的未阴影矩形= bh= 106= 60 英寸 2 的未阴影三角形= bh2= 342= 6 英寸 2 Total 区域= 150- 60- 6= 84 英寸 2

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  CK-12 Interactive: Composite Shapes
  ::CK-12 互动:复合形状

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  $$\begin{array}{r}\text{The total area is }12+9=21{\text{ft}}^2\end{array}$$

  
  ::总面积为12+9=21平方英尺

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  CK-12 PLIX: Area and Perimeter of Composite Shapes 
  ::CK-12-PLIX:复合形状面积和周边

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  Composite Shapes - Examples
  ::复合形状 - 示例

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  Example 1
  ::例1

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  A square is inscribed inside a circle . Find the total area of the shaded regions of the circle below. What method for finding the  area makes the most sense in this case? Why?
  ::将一个正方形刻在一个圆内。 查找以下圆的阴影区域的总面积。 在这种情况下, 找到这个区域最合理的方法是什么? 为什么?

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  It makes sense to use subtractive areas method to find the area of the shaded region. Finding the area of the whole circle and subtracting the area of the square is much simpler than trying to calculate the area of each of the four shaded pieces directly.
  ::使用减让区域方法寻找阴影区域的面积是有道理的。 找到整个圆圈的面积并减去广场的面积要比试图直接计算四个阴影块的面积简单得多。

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  $\begin{array}{rl}{A}_{\text{circle}}& =\pi r^2=\pi (3{)}^2=9\pi {\text{in}}^2\\ A_{\text{square}}& =\frac{d_1{d}_2}{2}=\frac{6\cdot 6}{2}=\frac{36}{2}=18{\text{in}}^2\\ A_{\text{shaded region}}& =9\pi -18{\text{in}}^2\end{array}$
  ::在2Asquare=d1d22=662=362=18在2Ashad地区=918英寸

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  Example 2
  ::例2

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  Find the perimeter of the shape.
  ::找到形状的周界

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  The figure below is not drawn to scale. Assume all angles that look like right angles are right angles.
  ::下图不绘制为缩放。 将所有看上去像右角度的角都显示为右角度 。

  

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  First find all of the missing lengths. Note that you will need to use the Pythagorean Theorem to find the side length of 10 cm (the hypotenuse of the right triangle).
  ::首先找到所有缺失的长度。 请注意, 您需要使用 Pythagorean Theorem 来找到 10 cm 的侧长( 右三角形的假设值) 。

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  Now find the perimeter .
  ::现在找到周界

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  $\begin{array}{r}P=17+5+10+11+8+3+17+25=96\text{cm}\end{array}$
  ::P=17+5+10+11+8+3+17+25=96厘米

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  Example 3
  ::例3

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  Choose a method for finding the area of the shape. Justify your method.
  ::选择查找形状区域的方法。 请说明您的方法 。

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  You could choose either method. Subtractive areas method might be slightly easier because you will just need to find the area of the big rectangle and subtract the area of the trapezoid .
  ::您可以选择任意一种方法。 减让区域方法可能比较容易些, 因为您只需要找到大矩形区域, 并减去网格缩放区域 。

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  Example 4
  ::例4

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  Find the area of the shape using the subtractive areas method.
  ::使用减空区域方法查找形状区域 。

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  $\begin{array}{rl}\text{Area}& ={\text{Area}}_{\text{rectangle}}-{\text{Area}}_{\text{trapezoid}}\\ & =bh-\frac{(b_1+b_2)h}{2}\\ & =(25\cdot 17)-\frac{(11+17)(8)}{2}\\ & =(25\cdot 17)-\frac{(28\cdot 8)}{2}\\ & =425-112\\ & =313{\text{cm}}^2\end{array}$
  ::区域 = 区域 = 区域 = 区域 = 区域 = bh- (b1+b2) h2 = (2517) - (11+17) - (8) 2 = (2517) - (288)2 = 425 - 112= 313 cm2

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  CK-12 PLIX: Area and Perimeter of Composite Shapes
  ::CK-12-PLIX:复合形状面积和周边

  Summary

  播放段落 A composite shape is a two-dimensional figure made up of basic two-dimensional shapes such as triangles, rectangles, circles, semi-circles, etc. ::复合形状是由三角形、矩形、圆形、半圆形等基本二维形状组成的二维图。 播放段落 To find the perimeter of a composite figure, add the lengths of the sides. ::为了找到一个综合数字的周界,加上两边的长度。 播放段落 The Additive Areas Method for finding the area of a composite shape involves finding the individual areas of each piece of the composite shape and summing them up. ::寻找复合形状区域的 " 添加区域方法 " 涉及寻找复合形状每一部分的单个区域并进行总结。 播放段落 The Subtractive Areas Method involves finding the area of a larger shape and subtracting the areas of the pieces not included in the composite shape. ::减法涉及寻找较大形状的区域,减去复合形状未包括的碎片区域。

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  Composite Shapes - Review Questions
  ::复合形状 - 审查问题

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  The figure below is not drawn to scale. Assume all angles that look like right angles are right angles. Point  $\begin{array}{r}O\end{array}$   is the center of the partial circle.
  ::下图不绘制为缩放。 将所有看上去像右角度的角都显示为右角度。 O点是部分圆的中心 。

  

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  1. Find all missing side lengths for the shape.
  ::1. 找出形状所有缺失的侧边长度。

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  2. Find the perimeter of the shape.
  ::2. 寻找形状的周界。

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  3. Choose a method for finding the area of the shape. Justify your method.
  ::3. 选择查找形状区域的方法,说明方法的合理性。

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  4. Find the area of the shape using the method you chose in #2.
  ::4. 使用您在#2中选择的方法查找形状区域。

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  5. Try to find the area using the other method. If not possible, explain why not.
  ::5. 尝试使用其他方法查找区域,如果不可能,请解释原因。

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  The figure below is not drawn to scale. Assume all angles that look like right angles are right angles.
  ::下图不绘制为缩放。 将所有看上去像右角度的角都显示为右角度 。

  

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  6. Find all missing side lengths for the shape.
  ::6. 找出形状所有缺失的侧边长度。

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  7. Find the perimeter of the shape.
  ::7. 寻找形状的周界。

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  8. Choose a method for finding the area of the shape. Justify your method.
  ::8. 选择查找形状区域的方法,说明方法的合理性。

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  9. Find the area of the shape using the method you chose in #8.
  ::9. 使用您在#8中选择的方法查找形状区域。

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  10. Try to find the area using the other method. If not possible, explain why not.
  ::10. 尝试使用其他方法找到区域,如果不可能,请解释为何不能。

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  The figure below is not drawn to scale. Assume all angles that look like right angles are right angles.
  ::下图不绘制为缩放。 将所有看上去像右角度的角都显示为右角度 。

  

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  11. Find all missing side lengths for the shape.
  ::11. 找出形状所有缺失的侧边长度。

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  12. Find the perimeter of the shape.
  ::12. 寻找形状的周界。

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  13. Choose a method for finding the area of the shape. Justify your method.
  ::13. 选择查找形状区域的方法,说明方法的合理性。

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  14. Find the area of the shape using the method you chose in #13.
  ::14. 使用您在 #13 中选择的方法查找形状区域。

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  15. Try to find the area using the other method. If not possible, explain why not.
  ::15. 尝试使用其他方法找到区域,如果不可能,请解释为何不能。

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  16. Given the rectangle below, find the scalene triangle area using two different methods. (Diagram is not to scale.)
  ::16. 鉴于下面的矩形,用两种不同的方法查找缩放三角区域。 (数字不是缩放。 )

  

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  17. In the coordinate plane, sketch a quadrilateral that is not a parallelogram, trapezoid, or kite, such that the vertex coordinates are integers. (Polygons like this are called   lattice polygons.)  Is it possible to decompose this shape and find its area? If so, find its area. Do you think every quadrilateral whose vertex coordinates are integers can be decomposed to triangles and its area calculated? Experiment with quadrilaterals and decompositions and explain. Extend your exploration to polygons with more sides - it is not necessary to calculate the area, only to decompose the figure in such a way that the area can be calculated. Explain your conclusions. 
  ::17. 在坐标平面上,绘制一个不是平行图、陷阱类或风筝的四边形,使顶点坐标为整数。 (像这样的多边形称为 lattice 多边形。 ) 是否有可能分解这个形状并找到它的区域? 如果是这样的话, 找到它的区域。 您认为每个顶点坐标为整数的四边形都可以分解成三角形, 和计算出的区域? 实验四方和分解并解释。 将您的勘探扩展至多边多边形, 以更多侧面来计算区域 - 不必计算区域, 只能将数字分解成可以计算出的区域。 解释您的结论 。

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  Review (Answers)
  ::审查(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 。