Section: 涉及圆圈的综合数字领域 | 9.19 涉及圆形的综合数字领域

Section outline

Most of the outdoor basketball court was already repainted green. The only part left to redo was the brown key. The painters need to know the total area of the space remaining in order to buy enough paint. Given the dimensions shown below, what area should the workers use to calculate their supplies?
::大部分户外篮球场已经被重新油漆为绿色。剩下的唯一要重做的部分是棕色键。画家们需要知道剩余空间的总面积, 以便购买足够的油漆。鉴于下面的尺寸, 工人应该用哪个区域来计算他们的供给?

In this concept, you will learn how to find the area of composite figures containing circles.
::在这个概念中,你将学会如何找到包含圆形的复合数字区域。

Finding the Area of Composite Figures
::寻找综合数字区域

The formula to find the area of a rectangle is $\begin{aligned} A = l w \end{aligned}$ . The formula to find the area of a circle is $\begin{aligned} A = π r^{2} \end{aligned}$ . To find the area of a composite shape , simply find the area of each individual shape and add them together. The order in which you calculate the areas does not matter, and the commutative property states that it does not matter which order you add them in.
::查找矩形区域的公式是 A=lw。查找圆形区域的公式是 Ar2 。要找到复合形状的面积,只需找到每个形状的面积并将其加在一起。您计算区域的顺序并不重要,而通量属性则指出,命令将其添加到哪个位置并不重要。

Let’s look at an example.
::让我们举个例子。

Find the area of the figure below.
::查找下图的区域。

First, find the area of the rectangle.
::首先,找到矩形区域

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & 6 (8) \\ A & = & 48 \end{array} \end{aligned}$

::A=lwA=6(8)A=48

The area of the rectangle is 48 square inches.
::矩形区域为48平方英寸。

Next, recognize that you have been given a diameter and need to divide that by 2 to get the radius .
::接下来, 承认您被赋予直径, 并且需要将直径除以 2 以获得半径。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ A & = & 3.14 (4)^{2} \\ A & = & 3.14 (16) \\ A & = & 50.24 s q . i n . \end{array} \end{aligned}$

::Ar2A=3.14(4)2A=3.14(16)A=50.24平方米英寸。

The area of a full circle is 50.24 square inches.
::全圆面积为50.24平方英寸。

Then, find the area of one half the circle, a semi-circle, by dividing your final answer by 2.
::然后找到半个圆圈的面积,半圆圈, 将最后答案除以 2 。

$\begin{aligned} A = 25.12 \end{aligned}$

::A=25.12

The area of the semi-circle is 25.12 square inches.
::半圆圈面积为25.12平方英寸。

Finally, add the two areas together.
::最后,将这两个领域加在一起。

$\begin{aligned} \begin{array}{rcl} A & = & A_{r} + A_{s c} \\ A & = & 48 + 25.12 \\ A & = & 73.12 s q . i n . \end{array} \end{aligned}$

::A=Ar+AscA=48+25.12A=73.12平方米英寸。

The answer is the composite figure has an area of 73.12 square inches.
::答案是综合数字的面积为73.12平方英寸。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the outdoor basketball court, which needs some paint.
::早些时候,你被问及户外篮球场的问题, 这需要一些油漆。

The workers had the following court dimensions and need to know the total area to figure out their supplies.
::这些工人具有以下法院层面,需要了解查明其供应品的总面积。

First, find the area of the rectangle.
::首先,找到矩形区域

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & 16 (12) \\ A & = & 192 \end{array} \end{aligned}$

The area of the rectangle is 192 square feet.
::矩形区域为192平方英尺。

Next, recognize that you have been given a diameter and need to divide that by 2 to get the radius. The problem states that the diameter of the circle is the same as the width of the rectangle, 3 feet.
::接下来, 承认您被赋予直径, 并且需要将直径除以 2 以获得半径。问题显示圆的直径与矩形的宽度相同, 即 3 英尺。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ A & = & 3.14 (6)^{2} \\ A & = & 3.14 (36) \\ A & = & 113.04 \end{array} \end{aligned}$

The area for a full circle is approximately 113 square feet.
::全圆面积约为113平方英尺。

Then, remember that you have a semi-circle and divide this area by 2.
::那么,记住你有一个半圆圈将这个区域除以2

$\begin{aligned} A_{s c} = 56.5 \end{aligned}$

::Asc=56.5

The area of the semi-circle is 56.5 square feet.
::半圆圈面积为56.5平方英尺。

Add the two areas together.
::将这两个区域加在一起。

$\begin{aligned} \begin{array}{rcl} A & = & A_{r} + A_{s c} \\ A & = & 192 + 56.5 \\ A & = & 248.5 s q . f t \end{array} \end{aligned}$

The answer is the composite figure has an $\begin{aligned} area = 248.5 square feet \end{aligned}$ . The workers will need to by enough paint to cover 248.5 sq. ft.
::答案是综合数字的面积=248.5平方英尺。工人需要有足够的油漆来覆盖248.5平方英尺。

Example 2
::例2

The figure below is two semi-circles with a square in the middle. The square has a side length of 6 inches.
::下图为两半圆形半圆形,中间为方形,侧长6英寸。

First, draw a picture.
::首先,画一张图。

Next, determine the area of the square.
::接下来,确定广场的面积。

$\begin{aligned} \begin{array}{rcl} A & = & s^{2} \\ A & = & 6^{2} \\ A & = & 36 \end{array} \end{aligned}$

::A=S2A=62A=36

The area of the square is 36 square inches.
::广场面积为36平方英寸。

Then, recognize that since the square has four equal sides, the diameter of the circle is also 6 inches.
::然后,承认既然广场有四个平面,圆的直径也是6英寸。

The formula for the area of one semi-circle is the formula for the area of a circle divided by 2.
::一个半圆圈区域的公式是圆区域除以2的公式。

$\begin{aligned} A_{s c} = \frac{π r^{2}}{2} \end{aligned}$

::Ascr22

Since there are two semi-circles the same size, you may simply add them together to end up with the area for the complete circle.
::由于两个半圆的大小相同,你可以简单地将它们加在一起,最后加上整个圆的区域。

Remember that you have been given a diameter, not a radius, so divide the diameter by 2.
::记住给您一个直径, 不是半径, 所以直径除以 2 。

Then, use the formula for the area of a circle.
::然后,使用圆的区域的公式。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ A & = & 3.14 (3)^{2} \\ A & = & 3.14 (9) \\ A & = & 28.26 \end{array} \end{aligned}$

::Ar2A=3.14(3)2A=3.14(9)A=28.26

The area of the two semi-circles that form one full circle is 28.26 square inches.
::构成一整圈的两个半圆形区域为28.26平方英寸。

Finally, add the two areas together.
::最后,将这两个领域加在一起。

$\begin{aligned} \begin{array}{rcl} A & = & A_{r} + A_{s c} \\ A & = & 36 + 28.26 \\ A & = & 64.26 \end{array} \end{aligned}$

::A=Ar+AscA=36+28.26A=64.26

The answer is the composite figure has an $\begin{aligned} area = 64.26 square inches \end{aligned}$ .
::答案是综合图中面积=64.26平方英寸。

Example 3
::例3

Find the composite area of a rectangle with a length of 5 feet and a width of 3 feet connected to a semi-circle with the same diameter as the width.
::寻找长为5英尺和宽度为3英尺的矩形的复合区域,该矩形与宽度直径相同的半圆环相连。

First, find the area of the rectangle.
::首先,找到矩形区域

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & 5 (3) \\ A & = & 15 \end{array} \end{aligned}$

::A=lwA=5(3)A=15

The area of the rectangle is 15 square feet.
::矩形区域为15平方英尺。

Next, recognize that you have been given a diameter and need to divide that by 2 to get the radius. The problem states that the diameter of the circle is the same as the width of the rectangle, 3 feet.
::接下来, 承认您被赋予直径, 并且需要将直径除以 2 以获得半径。问题显示圆的直径与矩形的宽度相同, 即 3 英尺。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ A & = & 3.14 (1.5)^{2} \\ A & = & 3.14 (2.25) \\ A & = & 7.065 \end{array} \end{aligned}$

::Ar2A=3.14(1.5)2A=3.14(2.25)A=7.065

The area for a full circle is 7.065 square feet.
::全圆面积为7.065平方英尺。

Then, remember that you have a semi-circle and divide this area by 2.
::那么,记住你有一个半圆圈将这个区域除以2

$\begin{aligned} A_{s c} = 3.53 \end{aligned}$

::Asc=3.53

The area of the semi-circle is 3.53 square feet.
::半圆圈面积为3.53平方英尺。

Add the two areas together.
::将这两个区域加在一起。

$\begin{aligned} \begin{array}{rcl} A & = & A_{r} + A_{s c} \\ A & = & 15 + 3.53 \\ A & = & 18.53 s q . f t . \end{array} \end{aligned}$

::A=Ar+AscA=15+3.53A=18.53平方米平方英尺。

The answer is the composite figure has an $\begin{aligned} area = 18.53 square feet \end{aligned}$ .
::答案是综合图中面积为18.53平方英尺。

Example 4
::例4

Find the composite area of a square with a side length of 4 mm and a semi-circle with a diameter of the same side length as the square.
::找到方形的复合区域,其侧长为4毫米,半圆圈直径与方形的侧长相同。

First, find the area of the square.
::首先,找到广场的面积

$\begin{aligned} \begin{array}{rcl} A & = & s^{2} \\ A & = & 4^{2} \\ A & = & 16 \end{array} \end{aligned}$

::A=S2A=42A=16

The area of the square is 16 square millimeters.
::广场面积为16毫米。

Next, recognize that you have been given a diameter and need to divide that by 2 to get the radius. The problem states that the diameter of the circle is the same as the side length of the square, 4 mm.
::接下来, 承认您被赋予直径, 并且需要将直径除以 2 以获得半径。问题显示圆的直径与平方的侧长相同, 即 4 毫米。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ A & = & 3.14 (2)^{2} \\ A & = & 3.14 (4) \\ A & = & 12.56 \end{array} \end{aligned}$

::Ar2A=3.14(2)2A=3.14(4)A=12.56

The area for a full circle is 12.56 square millimeters.
::整个圆圈的面积是12.56平方米。

Then, remember that you have a semi-circle and divide this area by 2.
::那么,记住你有一个半圆圈将这个区域除以2

$\begin{aligned} A_{s c} = 6.28 \end{aligned}$

::Asc=6.28

The area of the semi-circle is 6.28 square millimeters.
::半圆圈面积为6.28平方米。

Add the two areas together.
::将这两个区域加在一起。

$\begin{aligned} \begin{array}{rcl} A & = & A_{S} + A_{s c} \\ A & = & 16 + 6.28 \\ A & = & 22.28 s q . m m . \end{array} \end{aligned}$

::A=AS+AscA=16+6.28A=22.28平方米毫米。

The answer is the composite figure has an $\begin{aligned} area = 22.28 s q . m m \end{aligned}$ .
::答案是复合图的面积=22.28平方米毫米。

Example 5
::例5

Find the area of a figure that is made up of a square and a semi-circle. The square has a side length of 8 inches. The circle’s diameter is the same as the side length of the square.
::查找由方形和半圆环组成的数字区域。方形的侧长为8英寸。圆的直径与方形的侧长相同。

First, find the area of the square.
::首先,找到广场的面积

$\begin{aligned} \begin{array}{rcl} A & = & s^{2} \\ A & = & 8^{2} \\ A & = & 64 \end{array} \end{aligned}$

::A=S2A=82A=64

The area of the square is 64 square inches.
::广场面积为64平方英寸。

Next, recognize that you have been given a diameter and need to divide that by 2 to get the radius. The problem states that the diameter of the circle is the same as the side length of the square, 8 inches.
::接下来, 承认您被赋予直径, 并且需要将直径除以 2 以获得半径。问题显示圆的直径与方形的侧长相同, 8 英寸。

$\begin{aligned} \begin{array}{rcl} A & = & π r^{2} \\ A & = & 3.14 (4)^{2} \\ A & = & 3.14 (16) \\ A & = & 50.24 \end{array} \end{aligned}$

::Ar2A=3.14(4)2A=3.14(16)A=50.24

The area for a full circle is 50.24 square inches.
::全圆面积为50.24平方英寸。

Then, remember that you have a semi-circle and divide this area by 2.
::那么,记住你有一个半圆圈将这个区域除以2

$\begin{aligned} A_{s c} = 25.12 \end{aligned}$

::Asc=25.12

The area of the semi-circle is 25.12 square inches.
::半圆圈面积为25.12平方英寸。

Add the two areas together.
::将这两个区域加在一起。

$\begin{aligned} \begin{array}{rcl} A & = & A_{r} + A_{s c} \\ A & = & 64 + 25.12 \\ A & = & 79.12 square inches \end{array} \end{aligned}$

::A=Ar+AscA=64+25.12A=79.12平方英寸

The answer is the composite figure has an $\begin{aligned} area = 79.12 s q . i n \end{aligned}$ .
::答案是复合图中有一个区域=79.12平方米。

Review
::回顾

Find the area of each combined figure. Use 3.14 for the value $\begin{aligned} π \end{aligned}$ and round your answer to the nearest hundredths place where needed.
::查找每个合并图的面积。使用 3.14 的值为 __ ,然后在需要的地方将答案绕到最接近的百个位置。

A square and a semi-circle. The square has a side length of 11 mm. The diameter of the circle matches the square ’ s side.
::一个正方形和一个半圆形。正方形的侧长为11毫米。圆形的直径与正方形的侧面吻合。

A square and a semi-circle. The square has a side length of 8.5 inches. The diameter of the circle matches the square’s side.
::方形和半圆形。方形的侧长为8.5英寸。圆形的直径与方形的侧面吻合。

A square and a semi-circle. The square has a side length of 7.25 inches. The diameter of the circle matches the square’s side.
::一个方形和一个半圆圈。方形的侧长为7. 25英寸。圆形的直径与方形的侧面吻合。

A square and a semi-circle. The square has a side length of 13 feet. The diameter of the circle matches the square’s side.
::一个方形和一个半圆圈。方形的侧长为13英尺。圆形的直径与方形的侧面吻合。

A square and a semi-circle. The square has a side length of 15.5 feet. The diameter of the circle matches the square’s side.
::一个正方形和一个半圆形。正方形的侧长为15.5英尺。圆形的直径与正方形的侧面吻合。

A rectangle and a semi-circle. The rectangle has a length of 8 feet and a width of 5 feet. The diameter of the circle matches the width.
::矩形和半圆形。矩形的长度为8英尺,宽度为5英尺。圆形的直径与宽度相符。

A rectangle and a semi-circle. The rectangle has a length of 8.5 feet and a width of 6 feet. The diameter of the circle matches the width.
::矩形和半圆形。矩形的长度为8.5英尺,宽度为6英尺。圆形的直径与宽度相符。

A rectangle and a semi-circle. The rectangle has a length of 9 inches and a width of 4.5 inches. The diameter of the circle matches the length.
::矩形和半圆形。矩形的长度为9英寸,宽度为4.5英寸。圆形的直径与长度相符。

A rectangle and a semi-circle. The rectangle has a length of 7 feet and a width of 4 feet. The diameter of the circle matches the length.
::矩形和半圆形。矩形的长度为7英尺,宽度为4英尺。圆形的直径与长度相符。

A rectangle and a semi-circle. The rectangle has a length of 5.5 feet and a width of 3.5 feet. The diameter of the circle matches the width.
::矩形和半圆形。矩形的长度为5.5英尺,宽度为3.5英尺。圆形的直径与宽度相符。

A triangle and a semi-circle. The triangle has a base of 5 inches and a height of 4 inches. The diameter of the circle matches the base of the triangle.
::三角形和半圆形。三角形的基数为5英寸,高度为4英寸。圆形的直径与三角形的基数相符。

A triangle and a semi-circle. The triangle has a base of 7 inches and a height of 6 inches. The diameter of the circle matches the base of the triangle.
::三角形和半圆形。三角形的基数为7英寸,高度为6英寸。圆形的直径与三角形的基数相符。

A triangle and a semi-circle. The triangle has a base of 5.5 inches and a height of 4 inches. The diameter of the circle matches the base of the triangle.
::三角形和半圆形。三角形的基数为5.5英寸,高度为4英寸。圆形的直径与三角形的基数相符。

Solve each problem.
::解决每一个问题。

Rob is painting large polka dots on a sheet for the backdrop of the school musical. He painted 16 polka dots, each with a radius of 3 feet. What is the total area that the polka dots cover?
::罗布为学校音乐剧的背景画了大圆点。他画了16个圆点,每个圆点的半径为3英尺。波尔卡圆点覆盖的总面积是多少?

The librarian is having the library at her school carpeted. The library is a circular room with a diameter of 420 feet. How many square feet of carpet will she need to order?
::图书馆馆员正在她学校开图书馆,图书馆是一个圆形室,直径为420英尺,她需要订购多少平方英尺的地毯?

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 Area of Composite Figures ::寻找综合数字区域

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 Area of Composite Figures
::寻找综合数字区域

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