Section: 查找三角形的尺寸和面积 | 8.1 找出三角体的尺寸和面积

Section outline

Jessie saw this median as she rode to school. The height of each triangle in the median is 7 feet and the base is 5 feet. Jessie wondered about the area of each triangle. If there are seven triangles in a row, what is the total area of all seven triangles?
::Jessie在上学时看到这个中位数。中位数中每个三角的高度是7英尺,基点是5英尺。Jessie想知道每个三角的面积。如果每列有7个三角,那么所有7个三角的总面积是多少?

In this concept, you will learn to find the dimensions and area of triangles.
::在这个概念中,你会学会找到三角形的大小和区域。

Area of a Triangle
::三角三角区域区域

Area is the amount of two-dimensional space a figure covers. To find the area of a triangle, you multiply the dimensions, or sides, of the figure. In a triangle, those dimensions are its height, $\begin{align*}h\end{align*}$ , and its base, $\begin{align*}b\end{align*}$ . The area formula for triangles is:
::区域是数字覆盖的二维空间的大小。要找到三角形的区域,您要乘以图的尺寸或侧面。在三角形中,这些尺寸是其高度、h和基数,b。三角形的区域公式是:

$\begin{align*}A = \frac{1}{2}bh\end{align*}$
::A=12bh =12bh

The base is the area at the bottom of the triangle opposite the vertex or top point . When finding the area of triangles, remember that the height of a triangle is always perpendicular to the base. The height is not necessarily a side of the triangle; this happens only in right triangles , because the two sides joined by a right angle are perpendicular.
::基点是顶点或顶点对面三角形底部的区域。当找到三角形区域时, 请记住三角形的高度总是与基点垂直的。高度不一定是三角形的侧面; 这只在右三角形中发生, 因为右角与右角相连接的两边是垂直的。

You can see in the right triangle above that the left side is also the height of the triangle. It is perpendicular to the base. The has a dotted line to show you the measurement for the height of the triangle.
::您可以在右边三角形上方的右侧看到,左侧也是三角形的高度。它与底部垂直。它有一个点线以显示三角形高度的测量。

Let’s look at an example.
::让我们举个例子。

Find the area of the triangle below.
::查找下方三角形的区域。

First, substitute 11 for $\begin{align*}b\end{align*}$ (the base) and 16 for $\begin{align*}h\end{align*}$ (the height) into the formula for area.
::首先,将b(基数)改为11,将h(高)改为区域公式。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} b h \\ A & = & \frac{1}{2} (11) (16) \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} bh \\ A &=& \frac{1}{2}(11)(16) \end{array}\end{align*}$
Next, solve for $\begin{align*}A\end{align*}$ .
::A=12bhA=12(11)(16) 下一个, A的解答。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} (11) (16) \\ A & = & 88 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2}(11)(16) \\ A &=& 88 \end{array}\end{align*}$
::A=12(11)(16)(16)A=88

The answer is 88.
::答案是88岁

The area of the triangle is $\begin{align*}88 \ cm^2\end{align*}$ .
::三角形区域为88厘米2。

You can also use this same formula to figure out a missing dimension of the triangle. This is possible if you are given the area and one dimension to start with. Then you can use the formula, substitute the values, and solve for the missing dimension.
::您也可以使用相同的公式来解析三角形中缺少的维度。如果给您给此区域和一个维度以开始,这是可能的。然后您可以使用该公式,替换值,并解决缺失的维度。

Let’s take a look at an example.
::让我们举个例子。

A triangle has an area of $\begin{align*}44 \ m^2\end{align*}$ . The base of the triangle is $\begin{align*}8 \ m\end{align*}$ . What is its height?
::三角形的面积为44平方米,三角形的底部是8米。它的高度是多少?

First, substitute 8 for $\begin{align*}b \end{align*}$ (the base) and 44 for $\begin{align*}A\end{align*}$ (the area) into the formula for area.
::第一,将b(基数)改为8,将A(面积)改为A(面积)。

%20%5C%5C%0A44%20%26%3D%26%204h%0A%5Cend%7Barray%7D">
$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} b h \\ 44 & = & \frac{1}{2} (8) (h) \\ 44 & = & 4 h \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} bh \\ 44 &=& \frac{1}{2} (8)(h) \\ 44 &=& 4h \end{array}\end{align*}$

Next, divide both sides by 4 to solve for $\begin{align*}h\end{align*}$ .
::A=12bh44=12(844=4h 下一步,将两边除以4, 解决 h。

$\begin{aligned} \begin{array}{rcl} 44 & = & 4 h \\ \frac{44}{4} & = & \frac{4 h}{4} \\ h & = & 11 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 44 &=& 4h \\ \frac{44}{4} &=& \frac{4h}{4} \\ h &=& 11 \end{array}\end{align*}$
The answer is 11.
::44=4h444=4h4h4h=11 答案是11

The height of the triangle is $\begin{align*}11 \ cm\end{align*}$ .
::三角形的高度是11厘米。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the triangles.
::早些时候,有人给了你一个关于三角形的问题。

Each of the seven triangles has a base length of 5 feet and a height of 7 feet.
::7个三角形中,每个三角形的基长为5英尺,高度为7英尺。

First, substitute 5 for $\begin{align*}b\end{align*}$ (the base) and 7 for $\begin{align*}h\end{align*}$ (the height) into the formula for area.
::第一,将b(基数)改为5,将h(高)改为区域公式。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} b h \\ A & = & \frac{1}{2} (5) (7) \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} bh \\ A &=& \frac{1}{2} (5)(7) \end{array}\end{align*}$

Next, solve for $\begin{align*}A\end{align*}$ .
::下一个是A

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} (5) (7) \\ A & = & 17.5 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2}(5)(7) \\ A &=& 17.5 \end{array}\end{align*}$
::A=12(5)(7)-A=17.5

Then, since there are 7 triangles, multiply this area by 7.
::然后,既然有7个三角形, 这个区域乘以7。

$\begin{aligned} \begin{array}{rcl} Total Area & = & (17.5) (7) \\ Total Area & = & 122.5 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{Total Area} &=& (17.5)(7) \\ \text{Total Area} &=& 122.5 \end{array} \end{align*}$
::总面积=122.5

The answer is 122.5.
::答案是122.5

The total area of the media is $\begin{align*}122.5 \ ft^2\end{align*}$ .
::媒体的总面积为122.5平方英尺。

Example 2
::例2

First, substitute 4.5 for $\begin{align*}b\end{align*}$ (the base) and 12 for $\begin{align*}h\end{align*}$ (the height) into the formula for area.
::第一,将b(基数)改为4.5,将h(高度)改为12(高度)。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} b h \\ A & = & \frac{1}{2} (4.5) (12) \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} bh \\ A &=& \frac{1}{2} (4.5)(12) \end{array} \end{align*}$

Next, solve for $\begin{align*}A\end{align*}$ .
::下一个是A

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} (4.5) (12) \\ A & = & 27 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} (4.5)(12) \\ A &=& 27 \end{array}\end{align*}$
::A=12(4.5)(12)A=27

The answer is 27.
::答案是27个

The area of the triangle is $\begin{align*}27 \ cm^2\end{align*}$ .
::三角形区域为27厘米2。

Example 3
::例3

Find the area of each triangle given the $\begin{align*}\text{base} = 6 \ \text{inches}\end{align*}$ and the $\begin{align*}\text{height} = 4 \ \text{inches}\end{align*}$ .
::根据基数=6英寸和高度=4英寸,查找每个三角形的区域。

First, substitute 6 for $\begin{align*}b\end{align*}$ (the base) and 4 for $\begin{align*}h\end{align*}$ (the height) into the formula for area.

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} b h \\ A & = & \frac{1}{2} (6) (4) \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} bh \\ A &=& \frac{1}{2} (6)(4) \end{array} \end{align*}$ Next, solve for $\begin{align*}A\end{align*}$ .
::首先,在区域公式中,将b(基数)和h(高度)替换为6。A=12bhA=12(6)(4)下一步,A的解答。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} (6) (4) \\ A & = & 12 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} (6)(4) \\ A &=& 12 \end{array}\end{align*}$ The answer is 12.
::A=12(6)(4)A=12 答案是12

The area of the triangle is $\begin{align*}12 \ in^2\end{align*}$ .
::三角形区域为12英寸2。

Example 4
::例4

Find the area of each triangle given the $\begin{align*}\text{base} = 3.5 \ \text{feet}\end{align*}$ and the $\begin{align*}\text{height} = 4 \ \text{feet}\end{align*}$ .
::根据基数=3.5英尺和高度=4英尺,查找每个三角形的区域。

First, substitute 3.5 for $\begin{align*}b\end{align*}$ (the base) and 4 for $\begin{align*}h\end{align*}$ (the height) into the formula for area.
::首先,将b(基数)改为3.5,将h(高度)改为区域公式。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} b h \\ A & = & \frac{1}{2} (3.5) (4) \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} bh \\ A &=& \frac{1}{2} (3.5)(4) \end{array} \end{align*}$ Next, solve for $\begin{align*}A\end{align*}$ .

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} (3.5) (4) \\ A & = & 7 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} (3.5)(4) \\ A &=& 7 \end{array}\end{align*}$ The answer is 7.
::A=12(3.5)(4)A=7 答案是7

The area of the triangle is $\begin{align*}7 \ ft^2\end{align*}$ .
::三角形区域为 7 平方英尺。

Example 5
::例5

Find the area of each triangle given the $\begin{align*}\text{base} = 8 \ mm\end{align*}$ and the $\begin{align*}\text{height} = 9 \ mm\end{align*}$ .
::根据基数=8毫米和高度=9毫米,查找每个三角形的区域。

First, substitute 8 for $\begin{align*}b\end{align*}$ (the base) and 9 for $\begin{align*}h\end{align*}$ (the height) into the formula for area.
::第一,将b(基数)改为8,将h(高度)改为区域公式。

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} b h \\ A & = & \frac{1}{2} (8) (9) \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} bh \\ A &=& \frac{1}{2} (8)(9) \end{array} \end{align*}$

Next, solve for $\begin{align*}A\end{align*}$ .
::下一个是A

$\begin{aligned} \begin{array}{rcl} A & = & \frac{1}{2} (8) (9) \\ A & = & 36 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2} (8)(9) \\ A &=& 36 \end{array}\end{align*}$ The answer is 36.
::A=12(8)(9)A=36 答案是36

The area of the triangle is $\begin{align*}36 \ mm^2\end{align*}$ .
::三角形区域为36毫米2。

Review
::回顾

Find the area of each triangle described below. Round your answer to the nearest hundredths place if needed.
::查找下面描述的每个三角形的区域。如果需要, 将您的答复四舍五入到最近的一百个位置。

1. $\begin{align*}b = 10 \ \text{inches}, h = 5 \ \text{inches}\end{align*}$
::1. b=10英寸,h=5英寸

2. $\begin{align*}b = 7 \ \text{inches}, h = 5.5 \ \text{inches}\end{align*}$
::2.=7英寸,h=5.5英寸

3. $\begin{align*}b = 8 \ \text{feet}, \text{height} = 6 \ \text{feet}\end{align*}$
::3. b=8英尺,高度=6英尺

4. $\begin{align*}b = 9 \ \text{feet}, \text{height} = 7.5 \ \text{feet}\end{align*}$
::4. b=9英尺,高度=7.5英尺

5. $\begin{align*}b = 12 \ \text{meters}, h = 9 \ \text{meters}\end{align*}$
::5. b=12米,h=9米

6. $\begin{align*}b = 15 \ \text{feet}, h = 12 \ \text{feet}\end{align*}$
::6. b=15英尺,h=12英尺

7. $\begin{align*}b = 12.5 \ \text{feet}, h = 3.5 \ \text{feet}\end{align*}$
::7. b=12.5英尺,h=3.5英尺

8. $\begin{align*} b = 15.25 \ \text{feet}, h = 8.5 \ \text{feet}\end{align*}$
::8. b=15.25英尺,h=8.5英尺

9. $\begin{align*}b = 25.75 \ \text{feet}, h = 13.5 \ \text{feet}\end{align*}$
::9. b=25.75英尺,h=13.5英尺

Find the missing dimension for each triangle given the area and one other dimension.
::查找每个三角形的缺失维度, 给区域和其他一个维度。

10. $\begin{align*}A = 4.5 \ sq.\ in , b = 4.5 \ in , h =?\end{align*}$
::10.A=4.5平方米,b=4.5英寸,h=?

11. $\begin{align*}A = 21 \ sq.\ ft, b = 7 \ ft, h =?\end{align*}$
::11. A=21平方英尺,b=7平方英尺,h=?

12. $\begin{align*}A = 60 \ sq. \ in, h = 10 \ in, b =?\end{align*}$
::12. A=60 sq. in, h=10 in, b=?

13. $\begin{align*}A = 97.5 \ sq. \ ft, h = 13 \ ft, b =?\end{align*}$
::13. A=97.5平方英尺,h=13平方英尺,b=?

14. $\begin{align*}A = 187 \ sq. \ ft, b = 22 \ ft, h =?\end{align*}$
::14.A=187平方英尺,b=22平方英尺,h=?

15. $\begin{align*}A = 405 \ sq.\ ft, b = 30 \ ft, h =?\end{align*}$
::15. A=405平方英尺,b=30平方英尺,h=?

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

Area of a Triangle ::三角三角区域区域

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

Area of a Triangle
::三角三角区域区域

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