Course: 9.14 涉及三角的复合形状区域

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涉及三角的复合形状区域
Jerry was hired to paint the front of a neighbor’s shed. Using the illustration below for dimensions , how many square feet of paint does he need?
::Jerry被雇用在邻居棚子前涂油漆。用下面的插图来说明他需要多少平方英尺的油漆?

In this concept, you will learn to find the area of .
::在这一概念中,你会学会找到...

Finding the Area of Composite Shapes
::寻找复合形状区域

The area of a rectangle is equal to its length times its width: $\begin{aligned} A = l w \end{aligned}$ . The area of a triangle is equal to its base times its height divided by 2: $\begin{aligned} A = \frac{b h}{2} \end{aligned}$ . In order to find the composite area of two or more shapes, simply find the area of each 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.
::矩形区域等于其宽度的长度乘以:A=lw。三角形区域等于其基数乘以其高度除以2:A=bh2。为了找到两个或两个以上形状的复合区域,只需找到每个形状的面积并将其相加在一起。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, if it’s not already done for you, divide the area into shapes you can work with.
::首先, 如果还没有为您完成, 请将区域分割成您可以工作的形状。

Next, find the area of the rectangle.
::下一个,找到矩形的区域。

$\begin{aligned} \begin{array}{rcl} A & = & l w \\ A & = & 9 (5) \\ A & = & 45 i n .^{2} \end{array} \end{aligned}$

Then, find the area of one of the triangles.
::然后找到三角形之一的区域

In this figure, you will need to find the base first.
::在此图中,您需要先找到基数。

To find the base of one triangle, subtract 9 from 22, then divide by 2.
::要找到一个三角形的底部, 从 22 中减去 9, 然后除以 2 。

$\begin{aligned} \begin{array}{rcl} 22 - 9 & = & 13 \\ 13 \div 2 & = & 6.5 \end{array} \end{aligned}$

The base of each triangle is 6.5 inches.
::每个三角的底部是6.5英寸

Next, calculate the area of one triangle.
::接下来,计算一个三角形的面积。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{b h}{2} \\ A & = & \frac{(6.5) (5)}{2} \\ A & = & 16.25 i n .^{2} \end{array} \end{aligned}$

The last step is to add the three areas together. Remember, there are two triangles.
::最后一步是将三个区域加在一起。记住, 有两个三角形。

$\begin{aligned} \begin{array}{rcl} Composite Area & = & Area of Triangle 1 + Area of Rectangle + Area of Triangle 2 \\ A & = & A_{T 1} + A_{R} + A_{T 2} \\ A & = & 16.25 + 45 + 16.25 \\ A & = & 77.5 \end{array} \end{aligned}$

::复合区域=三角区1+三角区2A=AT1+AR+AT2A=16.25+45+16.25A=77.5。

The answer is 77.5 square inches.
::答案是77.5平方英寸

Examples
::实例

Example 1
::例1

Earlier, you given a problem about Jerry, who was hired to paint the front of a neighbor’s shed.
::早些时候,你给Jerry带来了一个问题,Jerry被雇来画邻居棚子的前面。

He needs to figure out how many square feet of paint he’ll use according to the following diagram .
::他需要弄清楚根据下图他将使用多少平方英尺的油漆。

First, calculate the area of the rectangle.
::首先,计算矩形的区域。

$\begin{aligned} \begin{array}{rcl} A_{R} & = & l w \\ A_{R} & = & 6 \times 7.5 \\ A_{R} & = & 45 s q f t \end{array} \end{aligned}$

The area of the rectangle is 45 square feet.
::矩形区域为45平方英尺。

Next, calculate the area of the triangle.
::接下来,计算三角形的区域。

$\begin{aligned} \begin{array}{rcl} A_{T} & = & \frac{b h}{2} \\ A_{T} & = & \frac{(6) (4)}{2} \\ A_{T} & = & 12 f t^{2} \end{array} \end{aligned}$

Then, add the areas together.
::然后,将区域加在一起。

$\begin{aligned} \begin{array}{rcl} A & = & A_{R} + A_{T} \\ A & = & 45 s q f t + 12 s q f t \\ A & = & 57 s q f t \end{array} \end{aligned}$

The answer is 57 square feet. Jerry needs a can of paint that will cover 57 sq. feet.
::答案是57平方英尺杰瑞需要一罐油漆覆盖57平方英尺

Example 2
::例2

A figure is made up of three triangles. Each triangle has a base of 6 inches and a height of 4 inches. What is the combined area of all three triangles?
::数字由三个三角形组成。每个三角形的基数为 6 英寸,高度为 4 英寸。所有三个三角形的合并区域是什么 ?

First, use the formula to find the area of one triangle.
::首先,使用公式查找一个三角形的区域。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{b h}{2} \\ A & = & \frac{(6) (4)}{2} \end{array} \end{aligned}$

Next, calculate.
::下一个,计算。

$\begin{aligned} A = 12 s q i n \end{aligned}$
::A=12 平方英寸

Then, remember that there are three triangles so multiply your answer times 3.
::那么,记住,有三个三角形, 使回答乘以3。

$\begin{aligned} \begin{array}{rcl} A_{T} & = & 12 s q i n \times 3 \\ A_{T} & = & 77.5 s q i n \end{array} \end{aligned}$

The answer is 77.5 square inches.
::答案是77.5平方英寸

Example 3
::例3

A figure is made up of two triangles and one rectangle. Each triangle has a base of 5 inches and a height of 3 inches. The rectangle has a length of 4 inches and a width of 3 inches. What is the total area of the figure?
::数字由两个三角形和一个矩形组成。每个三角形的基数为5英寸,高度为3英寸。矩形的长度为4英寸,宽度为3英寸。图的总面积是多少?

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

$\begin{aligned} \begin{array}{rcl} A_{R} & = & l w \\ A_{R} & = & 4 (3) \\ A_{R} & = & 12 i n .^{2} \end{array} \end{aligned}$

Next, find the area of one of the triangles.
::下一个,找到一个三角形的区域。

$\begin{aligned} \begin{array}{rcl} A_{T} & = & \frac{b h}{2} \\ A_{T} & = & \frac{(5) (3)}{2} \\ A_{T} & = & 7.5 i n .^{2} \end{array} \end{aligned}$

Then, add the three areas together. Remember, there are two triangles.
::然后,将三个区域加在一起。记住,有两个三角形。

$\begin{aligned} \begin{array}{rcl} A & = & A_{T 1} + A_{R} + A_{T 2} \\ A & = & 7.5 + 12 + 7.5 \\ A & = & 27 \end{array} \end{aligned}$

The answer is 27 square inches.
::答案是27平方英寸

Example 4
::例4

A figure is made up of one triangle and one square. The square and the triangle have the same base length of 8 feet. The height of the triangle is 7 feet. What is the total area of the figure?
::数字由一个三角形和一个方形组成。方形和三角形的基长为8英尺。三角形的高度为7英尺。图的总面积是多少?

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

$\begin{aligned} \begin{array}{rcl} A_{S} & = & s^{2} \\ A_{S} & = & 8^{2} \\ A_{S} & = & 64 f t .^{2} \end{array} \end{aligned}$

Next, find the area of the triangle.
::下一个,找到三角形的区域。

$\begin{aligned} \begin{array}{rcl} A_{T} & = & \frac{b h}{2} \\ A_{T} & = & \frac{(8) (7)}{2} \\ A_{T} & = & 28 f t .^{2} \end{array} \end{aligned}$

Then, add the areas together.
::然后,将区域加在一起。

$\begin{aligned} \begin{array}{rcl} A & = & A_{S} + A_{T} \\ A & = & 64 s q f t + 28 s q f t \\ A & = & 92 s q f t \end{array} \end{aligned}$

The answer is 92 square feet.
::答案是92平方英尺

Example 5
::例5

Find the area of the composite figure below.
::将综合图的面积查找如下。

First, calculate the area of the rectangle.
::首先,计算矩形的区域。

$\begin{aligned} \begin{array}{rcl} A_{R} & = & l w \\ A_{R} & = & 17 \times 8.5 \\ A_{R} & = & 144.5 s q i n \end{array} \end{aligned}$

The area of the rectangle is 144.5 square inches.
::矩形区域为144.5平方英寸。

Next, calculate the area of the triangle. In order to do this, you first need to recognize that the height is equal to 17 inches minus 8.5 inches, or 8.5 inches.
::接下来,计算三角形的区域。要做到这一点,您首先需要认识到高度等于17英寸负8.5英寸,或8.5英寸。

Then, use the formula to find the area of the triangle.
::然后,使用公式找到三角形的区域。

$\begin{aligned} \begin{array}{rcl} A_{T} & = & \frac{b h}{2} \\ A_{T} & = & \frac{(17) (8.5)}{2} \\ A_{T} & = & 72.25 i n^{2} \end{array} \end{aligned}$

Then, add the areas together.
::然后,将区域加在一起。

$\begin{aligned} \begin{array}{rcl} A & = & A_{R} + A_{T} \\ A & = & 144.5 s q i n + 72.25 s q i n \\ A & = & 216.75 s q i n \end{array} \end{aligned}$

The answer is 216.75 square inches.
::答案是216.75平方英寸

Review
::回顾

Find the area of each combined figure.
::查找每个合并数字的面积。

A figure is made up of a triangle and a square. The square and the triangle have the same base of 7 inches. The triangle has a height of 5 inches, what is the total area of the figure?
::数字由三角形和方形组成。方形和三角形的基数相同,为7英寸。三角形的高度为5英寸,图的总面积是多少?

A figure is made up of a triangle and a rectangle. The triangle has a height of 8 inches and a base of 9 inches. The rectangle has dimensions of $\begin{aligned} 7 inches \times 9 inches \end{aligned}$ . What is the area of the figure?
::数字由三角形和矩形组成。三角形的高度为8英寸,基数为9英寸。矩形的尺寸为7英寸×9英寸。图的面积是什么?

A figure is made up of four triangles. Each triangle has a base of 7 inches and a height of 9 inches. What is the total area of the figure?
::数字由四个三角形组成。每个三角形的基数为7英寸,高度为9英寸。图的总面积是多少?

A figure is made up of three triangles. Each triangle has a base of 5 feet and a height of 4.5 feet. What is the total area of the figure?
::数字由三个三角形组成。每个三角形的基数为5英尺,高度为4.5英尺。图的总面积是多少?

A figure is made up of two triangles and a square. The triangles and the square have the same base length of 5 feet. The triangles have a height of 4 feet. What is the total area of the figure?
::数字由两个三角形和一个正方形组成。三角形和正方形基长为5英尺。三角形和正方形的基长为4英尺。图的总面积是多少?

A figure is made up of two triangles and a square. The triangles and the square have the same base length of 8 feet. The triangles have a height of 7 feet. What is the total area of the figure?
::数字由两个三角形和一个方形组成。三角形和方形的基长为8英尺。三角形和方形的基长为7英尺。图的总面积是多少?

A figure is made up of one square and one triangle. The square has a side length of 9 feet. The triangle has a base of 7 feet and a height of 6 feet. What is the total area of the figure?
::数字由一个方形和一个三角形组成。方形的侧长为9英尺。三角形的底部为7英尺, 高度为6英尺。图的总面积是多少?

A figure is made up of two triangles. The triangles have the same base of 15 inches. One triangle has a height of 9 inches and one has a height of 11 inches. What is the total area of the figure?
::数字由两个三角形组成。三角形的基数相同,为15英寸。一个三角形的高度为9英寸,一个三角形的高度为11英寸。图的总面积是多少?

A figure is made up of a triangle and a rectangle. The triangle has a height of 8.5 inches and a base of 10 inches. The rectangle has dimensions of $\begin{aligned} 9 inches \times 10 inches \end{aligned}$ . What is the area of the figure?
::数字由三角形和矩形组成。三角形的高度为8.5英寸,基数为10英寸。矩形的尺寸为9英寸×10英寸。图的面积是什么?

A figure is made up of a triangle and a rectangle. The triangle has a height of 11 inches and a base of 9.5 inches. The rectangle has dimensions of $\begin{aligned} 12 inches \times 14 inches \end{aligned}$ . What is the area of the figure?
::数字由三角形和矩形组成。三角形的高度为11英寸,基数为9.5英寸。矩形的尺寸为12英寸×14英寸。图的面积是什么?

Solve each problem.
::解决每一个问题。

Julius drew a triangle that had a base of 15 inches and a height of 11 inches. What is the area of the triangle Julius drew?
::朱利叶斯画了一个三角形,基数15英寸,高度11英寸。三角形朱利叶斯画的面积是多少?

A triangle has an area of 108 square centimeters. If its height is 9 cm, what is its base?
::一个三角形的面积为108平方厘米。如果它的高度是9厘米,那么它的底部是什么?

What is the height of a triangle whose base is 36 inches and area is 234 square inches?
::三角形的高度是36英寸面积是234平方英寸

Tina is painting a triangular sign. The height of the sign is 32 feet. The base is 27 feet. How many square feet will Tracy paint?
::Tina画的是三角形的标志标志的高度是32英尺基地是27英尺 Tracy要画多少平方英尺?

Tina drew a picture of a triangle with a base of 6 inches and a height of 5.5 inches. What is the area of the triangle?
::蒂娜画了一个三角形的图片,基数为6英寸,高度为5.5英寸。三角形的面积是多少?

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

涉及三角的复合形状区域

Finding the Area of Composite Shapes ::寻找复合形状区域

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 Shapes
::寻找复合形状区域

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