节: 使用公式查找方形的方形和矩形的周边和面积 | 1.13 使用公式查找周边和广场和矩形面积

章节大纲

Raj has a small patch of land that he wants to turn into a vegetable garden. However, he wants to build a small fence around it before he plants anything. He knows that the area of the land is 240 square feet and that the length of one side is 15 feet. In order to build a fence, he needs to know the width of the land too. How can Raj write and solve an equation to determine the width of his future garden?
::Raj拥有一小块土地,他想变成一个蔬菜园,但他想在种植任何植物之前围绕它建造一道小围栏。他知道这片土地面积为240平方英尺,一面长度为15英尺。为了建造一道栅栏,他需要知道土地的宽度。Raj如何写和解决一个方程式来确定他未来花园的宽度?

In this concept, you will learn to find the perimeter and using formulas.
::在这个概念中,你将学会找到周边,并使用公式。

Perimeter and Area of Squares and Rectangles
::周边和广场和矩形区域

A formula is a method that has been proven to work when solving specific types of problems.
::一种公式是在解决特定类型的问题时已证明行之有效的方法。

Let’s explore some of those familiar formulas by looking at , squares, area and perimeter.
::让我们通过观察方块、面积和周界来探索这些熟悉的公式。

The perimeter of a figure is the distance around the figure. Perimeter is the sum of all of the sides in a square or rectangle . Since a rectangle has two sets of parallel sides, the formula for determining perimeter $\begin{aligned} (P) \end{aligned}$ of a rectangle is:
::数字的周界是图周围的距离。周界是方形或矩形内所有侧面的总和。由于矩形有两组平行边, 确定矩形周界( P) 的公式是:

$\begin{aligned} P = 2 L + 2 W \end{aligned}$

$\begin{aligned} L = length \end{aligned}$ and $\begin{aligned} W = width \end{aligned}$
::P=2L+2W L=长度和W=宽度

Let’s look at an example.
::让我们举个例子。

The rectangle above shows its dimensions . Find the perimeter.
::上面的矩形显示它的尺寸找到周界

First, substitute the values for the width $\begin{aligned} (W) \end{aligned}$ and the length $\begin{aligned} (L) \end{aligned}$ into the perimeter formula.
::首先,以周边公式中的宽度(W)和长度(L)值替代值。

$\begin{aligned} P = 2 (12) + 2 (9) \end{aligned}$

::P=2(12)+2(9)

Next, complete the multiplication and addition to find the perimeter.
::下一步完成乘数和加法以找到周界

$\begin{aligned} \begin{array}{rcl} P & = & 2 (12) + 2 (9) \\ P & = & 24 + 18 \\ P & = & 42 \end{array} \end{aligned}$

::P=2(12)+2(9)P=24+18P=42

The answer is 42.
::答案是42岁

The perimeter of the rectangle is 42 inches.
::矩形周围42英寸

Area is the amount of square units inside the figure. Area is found by multiplying the $\begin{aligned} length \times width \end{aligned}$ . The formula for finding the area of a rectangle is:
::区域是图内平方单位的数值。区域通过乘xxwidth 来发现。查找矩形区域的公式是:

$\begin{aligned} A = L \times W \end{aligned}$

::A=LxW

You can use the dimensions from the rectangle above to find the area of this rectangle.
::您可以使用上面矩形的维度来找到这个矩形的区域。

First, fill the values for $\begin{aligned} L \end{aligned}$ and $\begin{aligned} W \end{aligned}$ into the formula for area.
::首先,将L和W的值填入区域公式。

$\begin{aligned} \begin{array}{rcl} A & = & L \times W \\ A & = & 12 \times 9 \end{array} \end{aligned}$

::A=LxWA=12x9

Next, solve for the area by multiplying.
::接下来,通过乘法解决区域问题。

$\begin{aligned} \begin{array}{rcl} A & = & 12 \times 9 \\ A & = & 108 \end{array} \end{aligned}$

::A=12x9A=108 A=12x9A=108

The answer is 108.
::答案是108

The area of the rectangle is $\begin{aligned} 108 i n^{2} \end{aligned}$ .
::矩形区域为108英寸2。

Notice that the unit of measurement for area is squared. That is because you multiplied a unit measure times itself: $\begin{aligned} i n \times i n = i n^{2} \end{aligned}$ . Area is always written in square units.
::注意区域测量单位是正方形。这是因为您乘以单位测量时间本身: inxin=in2。区域总是以正方形单位写出。

You can also find the perimeter and area of a square. Remember that a square has four equal sides given the symbol $\begin{aligned} s \end{aligned}$ . You can use the following formula for finding the perimeter of a square:
::您也可以找到一个广场的周界和地区。记住一个广场的符号 s 具有四个相等的边。您可以使用以下公式来找到一个广场的周界 :

$\begin{aligned} P = 4 s \end{aligned}$

::P=4s

Let’s look at an example.
::让我们举个例子。

A rectangle has a length of 12 feet and a perimeter of 72 feet. Write and solve an equation to determine the width of the rectangle.
::矩形的长度为 12 英尺, 周长为 72 英尺。写和解析一个方程以确定矩形的宽度。

First, substitute the values for the perimeter $\begin{aligned} (P) \end{aligned}$ and the length $\begin{aligned} (L) \end{aligned}$ into the perimeter formula.
::首先,将周边(P)和长度(L)的值替换为周边公式。

$\begin{aligned} 72 = 2 (12) + 2 W \end{aligned}$

::72=2(12)+2W

Next, complete the multiplication.
::下一步,完成乘法。

$\begin{aligned} 72 = 24 + 2 W \end{aligned}$

::72=24+2W

Then, subtract 24 from both sides to get your variable alone on the right side.
::然后,从两边减去24, 让你的变量单独在右边。

$\begin{aligned} \begin{array}{rcl} 72 - 24 & = & 24 - 24 + 2 W \\ 48 & = & 2 W \end{array} \end{aligned}$

::72-24=24-24+2W48=2W

Then, multiply both sides by the reciprocal of 2 to isolate your variable.
::然后将两边乘以二对一的乘数隔开你的变数

$\begin{aligned} \begin{array}{rcl} \frac{1}{2} \times \frac{48}{1} & = & \frac{1}{2} \times \frac{2 W}{1} \\ \frac{48}{2} & = & W \\ W & = & 24 \end{array} \end{aligned}$

::12×481=12×2W1482=WW=24

The answer is 24.
::答案是24个。

The width of the rectangle is 24 feet.
::矩形宽度是24英尺

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Raj and his garden fence.
::之前 Raj和他的花园篱笆让你出了点问题

He knows that the area of the land is 240 square feet, the length of one side is 15 feet, and needs to know the width of the land.
::他知道这片土地面积为240平方英尺,一边长15英尺,需要了解土地的宽度。

First, substitute the values for the area and the side length into the area formula.
::首先,将区域公式中的区域值和侧长值替换为区域公式。

$\begin{aligned} \begin{array}{rcl} A & = & L \times W \\ 240 f t^{2} & = & 15 f t \times W \end{array} \end{aligned}$

::A=LxW240 平方2=15 平方xW

Next, multiply both sides by the reciprocal of 15 in order to isolate the variable $\begin{aligned} W \end{aligned}$ .
::其次,将双方乘以对等15,以孤立变数W。

$\begin{aligned} \begin{array}{rcl} \frac{1}{15} \times \frac{240}{1} & = & \frac{1}{15} \times \frac{15}{1} \times W \\ W & = & 16 \end{array} \end{aligned}$

::115x2401=115x151x16

The answer is 16.
::答案是16岁

The width of Raj’s garden is 16 ft. Therefore the dimensions of the garden are 16 ft by 15 ft.
::Raj的花园宽度是16英尺,因此花园的尺寸是16英尺乘15英尺。

Example 2
::例2

A square has a perimeter of 196 inches. Determine the length of one side of the square.
::方形的周界为196英寸,确定方形一侧的长度。

First, substitute the value for the perimeter $\begin{aligned} (P) \end{aligned}$ into the perimeter formula.
::首先,将周边值(P)改为周边公式。

$\begin{aligned} 196 = 4 s \end{aligned}$

::196=4s

Next, multiply both sides by the reciprocal of 4 to isolate your variable.
::接下来,将两边乘以4乘以4 来孤立你的变数。

$\begin{aligned} \begin{array}{rcl} \frac{1}{4} \times \frac{196}{1} & = & \frac{1}{4} \times \frac{4 s}{1} \\ \frac{196}{4} & = & s \\ s & = & 49 \end{array} \end{aligned}$

::14x1961=14x4s11964=ss=49

The answer is 49.
::答案是49个

Example 3
::例3

Find the perimeter of the following square if the side length is 4.5 inches.
::如果侧长为4.5英寸,则查找以下方形的周界。

First, substitute the value for the side length into the perimeter formula.
::首先,用周边方程式的侧长值替代值。

$\begin{aligned} \begin{array}{rcl} P & = & 4 s \\ P & = & 4 (4.5) \end{array} \end{aligned}$

::P=4sP=4(4.5)

Next, multiply by 4 to solve for the perimeter.
::接下来,乘以4 解决周界。

$\begin{aligned} \begin{array}{rcl} P & = & 4 (4.5) \\ P & = & 18 \end{array} \end{aligned}$

::P=4(4.5)P=18

The answer is 18.
::答案是18岁

Example 4
::例4

Can you find the area of the square in Example 1?
::您能否在例1中找到方形区域?

First, substitute the value for the side length into the area formula.
::首先,将区域公式的侧长值替换为值。

$\begin{aligned} \begin{array}{rcl} A & = & s \times s \\ A & = & 4.5 \times 4.5 \end{array} \end{aligned}$

::A=sxsA=4.5x4.5

Next, multiply to solve for the area.
::下一个,乘数解决区域问题。

$\begin{aligned} \begin{array}{rcl} A & = & 4.5 \times 4.5 \\ A & = & 20.25 \end{array} \end{aligned}$

::A=4.5x4.5A=20.25

The answer is 20.25.
::答案是20.25。

The area of the square is $\begin{aligned} 20.25 i n^{2} \end{aligned}$ .
::广场面积为20.25英寸2。

Example 5
::例5

A square has an area of 144 sq. meters. What is the side length?
::方形面积为144平方米,侧长多少?

First, substitute the value for the area into the area formula.
::首先,将区域值替换为区域公式。

$\begin{aligned} \begin{array}{rcl} A & = & s \times s \\ 144 m^{2} & = & s \times s \end{array} \end{aligned}$

::A=sxs 144 m2=sxs =sxs A=sxs 144 m2=sxs

Next, take the square root of the area to isolate $\begin{aligned} s \end{aligned}$ . Remember that the opposite of square is square root.
::下一步, 选择区域的平方根来隔离 s. 记住正方根对面是平方根。

$\begin{aligned} \begin{array}{rcl} 144 & = & s \times s \\ 144 & = & s^{2} \\ s & = & \sqrt{144} \\ s & = & 12 \end{array} \end{aligned}$

::144=sxs144=s2s=144s=12

The answer is 12.
::答案是12岁

Review
::回顾

Find the area and perimeter of each square or rectangle using formulas and the given dimensions. Each problem will have two answers.
::使用公式和给定尺寸查找每个正方形或矩形的面积和周边。每个问题将有两个答案。

1. A square with a side length of 5 inches.
::1. 侧长5英寸的广场。

2. A rectangle with a length of 5 inches and a width of 3 inches.
::2. 长5英寸宽3英寸长的矩形。

3. A rectangle with a length of 8 cm and a width of 6 cm.
::3. 长8厘米和宽6厘米的矩形。

4. A square with a side length of 11 feet.
::4. 侧长11英尺的广场。

5. A rectangle with a length of 9 inches and a width of 4.5 inches.
::5. 长9英寸长和宽4.5英寸长的矩形。

6. A square with a side length of 7 feet.
::6. 边长7英尺的广场。

7. A rectangle with a length of 12 meters and a width of 11 meters.
::7. 长12米、宽11米的矩形。

8. A square with a side length of 13 meters.
::8. 一个侧长13米的广场。

9. A rectangle with a length of 15 feet and a width of 8 feet.
::9. 长15英尺宽8英尺长的矩形。

10. A square with a side length of 12.5 feet.
::10. 侧长12.5英尺的广场。

Find the missing side length given the area of each square.
::根据每个广场的面积查找缺失的侧长。

11. $\begin{aligned} A = 64 i n^{2} \end{aligned}$
::11. A=64英寸2

12. $\begin{aligned} A = 36 i n^{2} \end{aligned}$
::12. A=36英寸2

13. $\begin{aligned} A = 81 m^{2} \end{aligned}$
::13. A=81平方米

14. $\begin{aligned} A = 100 i n^{2} \end{aligned}$
::14. A=100英寸2

15. $\begin{aligned} A = 144 f t^{2} \end{aligned}$
::15. A=144英尺2

16. $\begin{aligned} A = 121 c m^{2} \end{aligned}$
::16. A=121厘米2

17. $\begin{aligned} A = 4 m m^{2} \end{aligned}$
::17. A=4毫米2

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。

章节大纲

Perimeter and Area of Squares and Rectangles ::周边和广场和矩形区域

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)

Perimeter and Area of Squares and Rectangles
::周边和广场和矩形区域

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)