Course: 1.14矩形的面积和周长

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矩形的面积和周长
Kathryn is getting new carpeting for the three bedrooms in her house. She decided to get the same type of carpeting for each room. The carpeting she picked costs $3 per square foot. In order to place an order for the carpeting, she needs to know how much she will need. Luckily, each of her rooms are . The first bedroom is 15 feet by 12 feet. The second bedroom is 12 feet by 11 feet. The third bedroom is 9 feet by 10 feet. How many square feet of carpeting will Kathryn need to order? How much will it cost?
::Kathryn正在为她家的三间卧室获得新的地毯。她决定为每间房间获得同样的地毯。她为每平方英尺挑选地毯的费用是3美元。为了订购地毯, 她需要知道她需要多少。幸运的是, 她的每个房间都是。第一个卧室是15英尺乘12英尺。第二个卧室是12英尺乘11英尺。第三个卧室是9英尺乘10英尺。 Kathryn需要多少平方英尺的地毯? 要多少钱?

In this concept, you will learn how to use formulas to find the perimeter and area of squares and rectangles.
::在这个概念中,你将学会如何使用公式来找到方形和矩形的周界和面积。

Finding Perimeter and Area of Rectangles
::查找周边和矩形区域

Perimeter is the distance around a figure.
::周边是数字周围的距离。

A rectangle is a four-sided shape that has two lengths ( $\begin{aligned} l \end{aligned}$ ) that are equal and two widths ( $\begin{aligned} w \end{aligned}$ ) that are equal. The perimeter of a rectangle can be found using the formula below.
::矩形是四面形,其两长(l)相等,两宽(w)相等。矩形的周边可使用以下公式找到。

$\begin{aligned} Rectangle Perimeter : P = 2 l + 2 w \end{aligned}$

A square is a four-sided shape that has four equal sides ( $\begin{aligned} s \end{aligned}$ ). Therefore, it is possible to find the perimeter of a square by multiplying one of the side lengths by 4. The perimeter of a square can be found using the formula below.
::方形是四面形,有四面相等的方形(s)。因此,可以通过将侧长之一乘以4来找到方形的周界。可用以下公式找到方形的周界。

$\begin{aligned} Square Perimeter : P = 4 s \end{aligned}$

Area is the number of square units contained inside a figure. The areas of rectangles and squares can be found using the formulas below.
::区域是图中包含的平方单位数。矩形和方形区域可使用以下公式查找。

$\begin{aligned} \begin{array}{rcl} Rectangle Area : A & = & l w \\ Square Area : A & = & s^{2} \end{array} \end{aligned}$

Here is an example.
::举一个例子。

Find the perimeter and area of the rectangle below.
::找出下面矩形的周界和地区

First, note the length and width of the rectangle. One of the sides is 12 inches and another is 7 inches. This will be your length and width. It does not actually matter which one you assign as length and which one you assign as width.
::首先,请注意矩形的长度和宽度。边之一是12英寸,另一边是7英寸。这将是你的长度和宽度。实际上, 您指定哪一条为长度和哪一条为宽并不重要。

$\begin{aligned} l = 12 i n; w = 7 i n \end{aligned}$
::l = 12 i n; w = 7 i n

Next, find the perimeter by substituting the values for $\begin{aligned} l \end{aligned}$ and $\begin{aligned} w \end{aligned}$ into the perimeter formula and evaluating.
::接下来,通过将I值和W值替换为I值和W值来找到周边值,然后在周边公式和评估中找到周边值。

$\begin{aligned} \begin{array}{rcl} P & = & 2 l + 2 w \\ P & = & 2 (12) + 2 (7) \\ P & = & 24 + 14 \\ P & = & 38 i n \end{array} \end{aligned}$

Notice that your units for perimeter are inches because the original dimensions of the rectangle were in inches. The units for perimeter will always be the same as the units of the dimensions.
::请注意,你们的周边单位是英寸,因为矩形的最初尺寸是英寸。周边单位总是与大小单位相同。

Now, find the area by substituting the values for $\begin{aligned} l \end{aligned}$ and $\begin{aligned} w \end{aligned}$ into the area formula and evaluating.
::现在,通过将l和w的值替换为l和w的值来找到区域,然后在区域公式和评估中找到区域。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & (12) (7) \\ A & = & 84 i n^{2} (or 84 square inches) \end{array} \end{aligned}$

Notice that your units for area are square inches. The units for area will always be the square of the units of the dimensions. This is because area is a calculation of the number of square units it takes to cover a figure.
::请注意您所在区域的单位为平方英寸。区域单位将始终是各维单位的平方。这是因为区域是用来计算数字的平方单位数。

The answer is the perimeter of the rectangle is 38 in and the area of the rectangle is $\begin{aligned} 84 i n^{2} \end{aligned}$ .
::答案是矩形周边是38,矩形区域是84 i n 2。

Here is another example.
::下面是另一个例子。

Find the perimeter and area of the square below.
::找出下方广场的周界和地区。

First, note that only one dimension is given. This is because all four sides of a square are equal in length.
::第一,注意只给出一个维度,这是因为平方的所有四面长度相等。

$\begin{aligned} s = 14 f t \end{aligned}$
::s = 14 f t = 14 f t

Next, find the perimeter by substituting the value for $\begin{aligned} s \end{aligned}$ into the perimeter formula and evaluating.
::接下来,通过在周边公式和评估中以 s 值替代 s 值来找到周边。

$\begin{aligned} \begin{array}{rcl} P & = & 4 s \\ P & = & 4 (14) \\ P & = & 56 f t \end{array} \end{aligned}$

Now, find the area by substituting the value for $\begin{aligned} s \end{aligned}$ into the area formula and evaluating.
::现在,通过将 s 的值替换成区域公式并进行评估来找到区域。

$\begin{aligned} \begin{array}{rcl} A & = & s^{2} \\ A & = & 14^{2} \\ A & = & 196 f t^{2} (or 196 square feet) \end{array} \end{aligned}$

The answer is the perimeter of the square is 56 in and the area of the square is $\begin{aligned} 196 i n^{2} \end{aligned}$ .
::答案是广场的周边是56, 广场的面积是196 i n 2 。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Kathryn and her new carpets.
::早些时候,你被问及凯瑟琳和她的新地毯的问题。

Kathryn is ordering new carpeting for the three bedrooms in her house. The carpeting she picked costs $3 per square foot. The bedrooms are $\begin{aligned} 15 f t \times 12 f t, 12 f t \times 11 f t \end{aligned}$ , and $\begin{aligned} 9 f t \times 10 f t \end{aligned}$ . Kathryn needs to know how many square feet of carpeting to order and how much it will cost.
::Kathryn为她家的三间卧室订了新的地毯,她每平方英尺选的地毯费用为3美元。卧室面积为15英尺的t × 12英尺的t,12英尺的t × 11英尺的t,9英尺的t × 10英尺的t。Kathryn需要知道需要多少平方英尺的地毯,以及需要多少钱。

First, find the area of each bedroom. Because the bedrooms are rectangles, you can use the formula $\begin{aligned} A = l w \end{aligned}$ to find the area of each.
::首先,找到每个卧室的面积。由于卧室是矩形,你可以使用公式A = l = y 来找到每个卧室的面积。

First Bedroom:
::第一个卧室:

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & (15) (12) \\ A & = & 180 f t^{2} \end{array} \end{aligned}$

Second Bedroom:
::第二卧室:

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & (12) (11) \\ A & = & 132 f t^{2} \end{array} \end{aligned}$

Third Bedroom:
::第三卧室:

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & (9) (10) \\ A & = & 90 f t^{2} \end{array} \end{aligned}$

Next, add up the three areas in order to find the total amount of carpet that Kathryn will need to order.
::接下来,将这三个区域加起来,以便找到凯瑟琳需要订购的地毯总量。

$\begin{aligned} \begin{array}{rcl} Total Area & = & 180 + 132 + 90 \\ Total Area & = & 402 f t^{2} \end{array} \end{aligned}$

The first answer is that Kathryn will need to order 402 square feet of carpeting.
::第一个答案是凯瑟琳需要订购402平方英尺的地毯

Now, find the total cost. The carpet costs $3 per square foot. Multiply the total amount of carpeting by $3 to find the total cost.
::现在,找到总成本。地毯每平方英尺花费3美元。将地毯的总额乘以3美元以找出总成本。

$\begin{aligned} \begin{array}{rcl} Cost & = & 402 \cdot $ 3 \\ Cost & = & $ 1206 \end{array} \end{aligned}$

The answer is it will cost $1206 for Kathryn to get new carpeting for the bedrooms in her house.
::答案是Kathryn要花1206美元才能为她家的卧室买新的地毯

Example 2
::例2

Marcy has purchased a new rug for her office. The rug measures $\begin{aligned} 6 f t \times 9 f t \end{aligned}$ . Given these measurements, what is the perimeter of the rug? What is the area?
::Marcy为她的办公室买了一块新的地毯。地毯的面积是6英尺t×9英尺。根据这些测量结果,地毯的周界是多少?面积是多少?

First, notice that you were given the dimensions $\begin{aligned} 6 f t \times 9 f t \end{aligned}$ for the rug. From this you can deduce that the rug is a rectangle. The length is 6 feet and the width is 9 feet.
::首先,请注意您得到了地毯的尺寸 6 f t × 9 f t。您可以从中推断地毯是一个矩形。长度为 6 英尺, 宽度为 9 英尺。

Next, find the perimeter of the rug using the perimeter of a rectangle formula.
::接下来,用矩形方形的周界找到地毯的周界

$\begin{aligned} \begin{array}{rcl} P & = & 2 l + 2 w \\ P & = & 2 (6) + 2 (9) \\ P & = & 12 + 18 \\ P & = & 30 f t \end{array} \end{aligned}$

Then, find the area of the rug using the area of a rectangle formula.
::然后用矩形公式的面积找到地毯的面积

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & (6) (9) \\ A & = & 54 f t^{2} \end{array} \end{aligned}$

The answer is that the perimeter of the rug is 30 feet and the area of the rug is 54 square feet.
::答案是地毯的周长是30英尺地毯的面积是54平方英尺

Example 3
::例3

Find the perimeter and area of a rectangle with a width of 10 inches and a length of 12 inches.
::寻找长10英寸和长12英寸长的矩形的周边和区域。

First, note that for this problem you have:
::首先,请注意,对于这个问题,你们有:

$\begin{aligned} l = 12 i n; w = 10 i n \end{aligned}$
::l = 12 i n; w = 10 i n

Now, find the perimeter by substituting the values for $\begin{aligned} l \end{aligned}$ and $\begin{aligned} w \end{aligned}$ into the perimeter formula and evaluating.
::现在,通过将I值和W值替换到周边公式和评估来找到周边。

$\begin{aligned} \begin{array}{rcl} P & = & 2 l + 2 w \\ P & = & 2 (12) + 2 (10) \\ P & = & 24 + 20 \\ P & = & 44 i n \end{array} \end{aligned}$

Next, find the area by substituting the values for $\begin{aligned} l \end{aligned}$ and $\begin{aligned} w \end{aligned}$ into the area formula and evaluating.
::接下来,通过将l和w的值替换为l和w的值来查找区域,然后在区域公式和评估中找到区域。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & (12) (10) \\ A & = & 120 i n^{2} \end{array} \end{aligned}$

The answer is the perimeter of the rectangle is 44 in and the area of the rectangle is $\begin{aligned} 120 i n^{2} \end{aligned}$ .
::答案是矩形周边是44,矩形区域是120 i n 2。

Example 4
::例4

Find the perimeter of a square with a side length of 4.5 inches.
::找到方形的周边,侧长4.5英寸。

First, note that only one dimension was given. This is because all four sides of a square are equal in length.
::第一,注意只给出一个维度,这是因为平方的所有四面长度相等。

$\begin{aligned} s = 4.5 i n \end{aligned}$
::s = 4.5 i n = 4.5 i n

Next, find the perimeter by substituting the value for $\begin{aligned} s \end{aligned}$ into the perimeter formula and evaluating.
::接下来,通过在周边公式和评估中以 s 值替代 s 值来找到周边。

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

The answer is the perimeter of the square is 18 in.
::答案是广场周边是18英寸

Example 5
::例5

Find the perimeter of a rectangle with a length of 15 feet and a width of 12 feet.
::找到长15英尺宽12英尺长的矩形周围。

First, note that for this problem you have:
::首先,请注意,对于这个问题,你们有:

$\begin{aligned} l = 15 f t; w = 12 f t \end{aligned}$
::l = 15 吨;w = 12 吨

Next, find the perimeter by substituting the values for $\begin{aligned} l \end{aligned}$ and $\begin{aligned} w \end{aligned}$ into the perimeter formula and evaluating.
::接下来,通过将I值和W值替换为I值和W值来找到周边值,然后在周边公式和评估中找到周边值。

$\begin{aligned} \begin{array}{rcl} P & = & 2 l + 2 w \\ P & = & 2 (15) + 2 (12) \\ P & = & 30 + 24 \\ P & = & 54 f t \end{array} \end{aligned}$

The answer is the perimeter of the rectangle is 54 ft.
::答案是矩形的周边是54英尺

Review
::回顾

Find the area and perimeter of each rectangle using the given dimensions.
::使用给定尺寸查找每个矩形的面积和周界。

Length = 10 in, width = 5 in
::长度 = 10 英寸,宽 = 5 英寸

Length = 12 ft, width = 8 feet
::长度 = 12英尺,宽 = 8英尺

Length = 11 ft, width = 5 feet
::长度 = 11英尺,宽 = 5英尺

Length = 17 miles, width = 18 miles
::长度 = 17 英里,宽 = 18 英里

Length = 22 ft, width = 20 feet
::长度 = 22英尺,宽 = 20英尺

Length = 8 cm, width = 6 cm
::长度 = 8 厘米,宽 = 6 厘米

Length = 20 cm, width = 17 cm
::长度 = 20 厘米,宽 = 17 厘米

Length = 3 feet, width = 2 feet
::长度 = 3英尺,宽 = 2英尺

Length = 15 yards, width = 11 yards
::长 = 15 码,宽 = 11 码

Length = 10 yards, width = 6 yards
::长度=10码,宽=6码

Find the area and perimeter of each square using the given dimensions.
::使用给定的尺寸查找每个广场的面积和周界。

$\begin{aligned} s = 6 f t \end{aligned}$
::s s = 6 f t = 6 f t

$\begin{aligned} s = 8 f t \end{aligned}$
::s s = 8 f t = 8 f t

$\begin{aligned} s = 9 i n \end{aligned}$
::s = 9 i n = 9 i n

$\begin{aligned} s = 4 i n \end{aligned}$
::s = 4 i n = 4 i n

$\begin{aligned} s = 12 i n \end{aligned}$
::s = 12 i n = 12 i n

$\begin{aligned} s = 7 f t \end{aligned}$
::s = 7 f t = 7 f t

$\begin{aligned} s = 5 c m \end{aligned}$
::= 5 cm = 5 cm

$\begin{aligned} s = 3 m \end{aligned}$
::=3米 = 3米

$\begin{aligned} s = 10 m \end{aligned}$
::s = 10米

$\begin{aligned} s = 11 y a r d s \end{aligned}$
::= 11 y a r d s

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

General

矩形的面积和周长

Finding Perimeter and Area of Rectangles ::查找周边和矩形区域

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)

Resources ::资源

Finding Perimeter and Area of Rectangles
::查找周边和矩形区域

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)

Resources
::资源