Section: 三角棱晶表面面积 | 10.5 三角棱晶表面面积 | The Way To Learn

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Section outline

Max bought a light prism for his mom for Mother's Day. He wants to figure out how much wrapping paper is needed to wrap it. To do this, he needs to figure out the of the gift. The triangular end has a base of 3 cm and height of 4 cm. The length of each side is 6 cm and the height of the prism (length of the rectangular side) is 8 cm. What is the surface area of the gift that Max bought for his mom?
::麦克斯在母亲节为妈妈买了光棱镜。他想弄清楚包装纸需要多少包装纸。要做到这一点, 他需要弄清楚礼物的底部。三角形的底部是3厘米和4厘米的高度。每边的长度是6厘米, 棱柱的高度( 矩形侧的长度)是8厘米。 Max给他妈妈买的礼物的表面面积是多少?

In this concept, you will learn how to calculate the surface area of a triangular prism .
::在此概念中, 您将学习如何计算三角棱镜的表面区域。

Finding the Surface Area of a Triangular Prism
::查找三角棱晶表面区域

Area is the space that is contained in a two-dimensional figure . Surface area is the total area of all of the sides and faces of a three-dimensional figure. To find the surface area, the area of each face is calculated and then add these areas together.
::面积是二维图中包含的空间。表面面积是所有侧面的总面积和三维图的面孔。要找到表面积,每个面的面积都要计算,然后将这些区域加在一起。

One way to do this is to use a , since a net is a two-dimensional representation of a three-dimensional solid , or an unfolded picture of a solid.
::其中一个方法就是使用一个,因为一个网是三维固体的二维表示法,或是一个固体的展露图。

What is the surface area of the figure below?
::下图的表面积是多少?

The net for this triangular prism is as follows:
::这一三角棱镜的网图如下:

Now, let's fill in the measurements for the sides of each face in order to calculate their area. Triangular prisms have their own formula for finding surface area because they have two triangular faces opposite each other .
::现在,让我们填满每面两面的测量结果, 以计算它们的面积。三角棱镜有它们自己寻找表面积的公式, 因为它们有两个三角面对立的公式。

The formula $\begin{aligned} A = \frac{1}{2} b h \end{aligned}$ is used to find the area of the top and bases triangular faces, where A = area, b = base, and h = height. The formula $\begin{aligned} A = l w \end{aligned}$ is used to find the area of the three rectangular side faces, where A = area, l = length, and w = width.
::公式 A= 12bh 用于查找上方的区域和三角面基底的面积,其中 A = 区域, b = 基数, h = 高度。公式 A = lw 用于查找三个矩形面的区域, 其中 A = 区域, l = 长度, w = 宽度。

Plugging in the measurements that are given in the net, calculate the area of each face. Remember to use the correct area formula for the triangles and .
::插入网中给出的测量数据, 计算每个面孔的面积。记住要对三角形和三角形使用正确的区域公式。

$\begin{aligned} Bottom face & Top face & side & side & side \\ A = \frac{1}{2} b h & A = \frac{1}{2} b h & A = S_{1} \cdot H & A = S_{2} \cdot H & A = S_{3} \cdot H \\ \frac{1}{2} (16) (6) + & \frac{1}{2} (16) (6) + & 17 \times 10 + & 17 \times 10 + & 17 \times 16 \\ 48 + & 48 + & 170 + & 170 + & 272 \\ = & 708 c m^{2} \end{aligned}$

::底面面面面侧面A=12bhA=12bhA=12bhA=S1}HA=S2}HA=S3H12(16)(6)+12(16)(6)+17×10+17×10+17×10+17×1648+48+170+170+170+272=708厘米2

When you add these values together, you get a surface area of 708 square centimeters for this triangular prism.
::当将这些值加在一起时, 这个三角棱柱的表面面积为708平方厘米。

$\begin{aligned} S A = b h + (S_{1} + S_{2} + S_{3}) H \end{aligned}$
You can also use one formula to calculate the surface area of a triangular prism which can save time over the process of using a net to derive the areas:
::SA=bh+(S1+S2+S3)HH 您也可以使用一个公式来计算三角棱柱的表面区域,这样在使用网来获取区域的过程中可以节省时间 :

where b = base; h = height of the triangle ; $\begin{aligned} S_{1}, S_{2}, S_{3} = L \end{aligned}$ the length of each side of the triangle base, and H = the height of the prism (which is the length of the rectangles).
::此处 b = 基数; h = 三角形的高度; S1, S2, S3 = L 三角形基数的每侧长度; H = 棱晶的高度( 即矩形的长度)。

First, find the area of the two triangular faces. Each face will have an area of $\begin{aligned} \frac{1}{2} b h \end{aligned}$ . Remember, you can use a formula to calculate the area of a pair of faces. Therefore, you can double this formula to find the area of both triangular faces at once which results in the formula $\begin{aligned} 2 (\frac{1}{2} b h) \end{aligned}$ . The 2 multiplied by the $\begin{aligned} \frac{1}{2} \end{aligned}$ equals 1, or cancels each other out, and you're left with $\begin{aligned} b h \end{aligned}$ .
::首先, 找到两个三角形面的面积。每张脸的面积将为 12bh 。记住, 您可以使用一个公式来计算一对面的面积。因此, 您可以加倍使用这个公式来找到两个三角形面的面积, 从而得出公式2( 12bh ) 。 2 乘以 12 等于 1 , 或者相互取消 , 剩下的是 bh 。

Next, you need to calculate the area of each of the rectangular side faces. The length of each rectangle is the same as the height of the prism, so call this $\begin{aligned} H \end{aligned}$ . The width of each rectangle is actually the same as the sides of the triangular base. Instead of multiplying the length and width for each rectangle, you can combine this information. Since there are 3 rectangular widths (all equal to the sides of the triangles), multiply the perimeter of the triangular base by the height of the rectangles, $\begin{aligned} H \end{aligned}$ , which will give the area of all three rectangles.
::接下来, 您需要计算每个矩形侧面的面积。每个矩形的长度与棱柱的高度相同, 请调用此 H。每个矩形的宽度实际上与三角形基底的两侧相同。您不必为每个矩形乘以长度和宽度, 您可以将这些信息组合起来。因为三角形的宽度有三个矩形宽度( 均等于三角形的两侧) , 三角形底部的周长会乘以矩形的高度, H, 这将提供所有三个矩形的面积。

If you put these pieces together—the area of the bases and the area of the side faces—you get this formula:
::如果你把这些碎片——基地的面积和侧面的面积——放在一起,你就会得到这个公式:

$\begin{aligned} S A = b h + (s 1 + s 2 + s 3) H \end{aligned}$

::SA=bh+(s1+s2+s3)H

where bh = the area of the triangle top and base, and (s1 + s2 + s3)H = the area of the rectangular side faces.
::bh = 三角形顶部和底部的区域,和 (s1 + s2 + s3)H = 矩形侧面的区域。

Remember that the height of the triangular base is not necessarily the same as the height of the prism (H).
::记住三角基点的高度不一定与棱晶(H)的高度相同。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the gift Max bought for his mom.
::之前有人给你一个问题关于Max给他妈妈买的礼物

The triangular end has a base of 3 cm and height of 4 cm. The length of each side is 6 cm and the height of the prism (length of the rectangle) is 8 cm. What is the surface area of this triangular prism?
::三角形端的基数为3厘米,高度为4厘米。每边的长度为6厘米,棱柱的高度(矩形的长度)为8厘米。三角形棱柱的表面面积是多少?

First, substitute the given values into the formula.
::首先,将给定值替换为公式。

$\begin{aligned} S A & = b h + (s 1 + s 2 + s 3) H \\ S A & = 3 (4) + (3 + 6 + 6) (8) \end{aligned}$

::SA=bh+(s1+s2+s2+s3)HSA=3(4)+(3+6+6+6(8))

Next, multiply the base times height for the area of the triangles (bh), and add their three sides (s1 + s2 + s3).
::下一步,乘以三角形(bh)区域的基准时间高度,并增加其三边(s1 + s2 + s3)。

$\begin{aligned} S A & = 12 + 15 (8) \end{aligned}$

::SA=12+15(8)

Then, multiply the sum of the triangle sides by the height of the prism (H) and add the values together for the answer, remembering to include the appropriate unit of measurement.
::然后,将三角边的和数乘以棱晶(H)的高度,并将答案的值加在一起,记住包括适当的计量单位。

$\begin{aligned} S A & = 12 + 120 \end{aligned}$

$\begin{aligned} S A & = 132 c m^{2} \end{aligned}$

::SA=12+120SA=132厘米2

The answer is the surface area of the gift Max bought for his mom is 132 square centimeters.
::答案是麦克斯给他妈妈买的礼物的表面面积是132平方厘米

Example 2
::例2

Calculate the surface area of the following triangular prism:
::计算以下三角棱镜的表面区域 :

First, substitute the given values into the formula.
::首先,将给定值替换为公式。

$\begin{aligned} S A & = b h + (s 1 + s 2 + s 3) H \\ S A & = 4 (6) + (4 + 7 + 7) (12) \end{aligned}$

::SA=bh+(s1+s2+s3+s3) HSA=4(6)+(4+7+7+7)(12)

Next, multiply the base times height for the area of the triangles (bh) and add the three sides of the triangle (s1 + s2 + s3).
::下一步,乘以三角形(bh)区域的基准时间高度,并添加三角形的三边(s1 + s2 + s3)。

$\begin{aligned} S A & = 24 + 18 (12) \end{aligned}$

::SA=24+18(12)

Then, multiply the sum of the triangle sides by the height of the prism (H) and add the values together for the answer, making sure to include the appropriate unit of measurement.
::然后,将三角边的和数乘以棱晶(H)的高度,并将答案的值相加,以确保包含适当的计量单位。

$\begin{aligned} S A & = 24 + 216 \end{aligned}$

::SA=24+216

$\begin{aligned} S A & = 240 c m^{2} \end{aligned}$

::SA=240厘米2

The answer is the surface area of the above triangular prism is 240 square centimeters.
::答案是上述三角棱镜的表面面积为240平方厘米。

Example 3
::例3

A triangular prism has a triangular end with a base of 5 inches and a height of 4 inches. The length of each side is 8 inches and the width of each side is 6 inches. What is the surface area of the prism?
::三角棱晶有一个三角端,底部5英寸,高度4英寸。每侧长度为8英寸,每侧宽度为6英寸。棱晶的表面面积是多少?

First, substitute the given values into the formula.
::首先,将给定值替换为公式。

$\begin{aligned} S A & = b h + (s 1 + s 2 + s 3) H \\ S A & = 5 (4) + (5 + 6 + 6) (8) \end{aligned}$

::SA=bh+(s1+s2+s2+s3)HSA=5(4)+(5+6+6+6(8))

Next, multiply the base times height for the area of the triangles (bh), and add their three sides (s1 + s2 + s3).
::下一步,乘以三角形(bh)区域的基准时间高度,并添加其三边(s1 + s2 + s3)。

$\begin{aligned} S A & = 20 + 17 (8) \end{aligned}$

::SA=20+17(8)

Then, multiply the sum of the triangle sides by the height of the prism (H) and add the values together for the answer, remembering to include the appropriate unit of measurement.
::然后,将三角边的和数乘以棱晶(H)的高度,并将答案的值加在一起,记住包括适当的计量单位。

$\begin{aligned} S A & = 20 + 136 \end{aligned}$

::SA=20+136

$\begin{aligned} S A & = 156 i n^{2} \end{aligned}$

::SA=156英寸2

The answer is the surface area of the above triangular prism is 156 square inches.
::答案是上述三角棱镜的表面面积为156平方英寸。

Example 4
::例4

A triangular prism has a triangular end with a base of 8 feet and a height of 6 feet. The length of each side is 10 feet and the width of each side is 6 feet. What is the surface area of the prism?
::三角棱晶有一个三角端, 底部为8英尺, 高度为6英尺。每边的长度为10英尺, 每边宽度为6英尺。棱晶的表面面积是多少?

First, substitute the given values into the formula.
::首先,将给定值替换为公式。

$\begin{aligned} S A & = b h + (s 1 + s 2 + s 3) H \\ S A & = 8 (6) + (8 + 6 + 6) (10) \end{aligned}$

::SA=bh+(s1+s2+s3+s3) HSA=8(6)+(8+6+6)(10)

Next, multiple the base times height for the area of the triangles (bh), and add the three sides of the triangle (s1 + s2 + s3).
$\begin{aligned} S A & = 48 + 20 (10) \end{aligned}$

::其次,将三角形区域(bh)的基时高度乘以多个倍数,并添加三角形的三边(s1 + s2 + s3),SA=48+20(10)

Then, multiply the sum of the triangle sides by the height of the prism (H) and add the values together for the answer, making sure to include the appropriate unit of measurement.
::然后,将三角边的和数乘以棱晶(H)的高度,并将答案的值相加,以确保包含适当的计量单位。

$\begin{aligned} S A & = 48 + 200 \end{aligned}$

$\begin{aligned} S A & = 248 f t^{2} \end{aligned}$

::SA=48+200 SA=248平方英尺

The answer is the surface area of the above triangular prism is 248 square feet.
::答案是上述三角棱镜的表面面积是248平方英尺。

Example 5
::例5

A triangular prism has a triangular end with a base of 12 inches and a height of 9 inches. The length of each side is 14 inches and the width of each side is 9 inches. What is the surface area of the prism?
::三角棱晶有一个三角端,底部12英寸,高度9英寸。每侧长度为14英寸,每侧宽度为9英寸。棱晶的表面面积是多少?

First, substitute the given values into the formula.
::首先,将给定值替换为公式。

$\begin{aligned} S A & = b h + (s 1 + s 2 + s 3) H \\ S A & = 12 (9) + (9 + 9 + 9) (14) \end{aligned}$

::SA=bh+(s1+s2+s3+s3) HSA=12(9)+(9+9+9+9)+(14)

Next, multiply the base times height for the area of the triangles (bh), and add the three sides of the triangle (s1 + s2 + s3).
$\begin{aligned} S A & = 108 + 27 (14) \end{aligned}$

::下一步,乘以三角形(bh)区域的基时高度,加上三角形的三边(S1+S2+S3),SA=108+27(14)

Then, multiply the sum of the triangle sides by the height of the prism (H) and add the values together for the answer, making sure to include the appropriate unit of measurement.
::然后,将三角边的和数乘以棱晶(H)的高度,并将答案的值相加,以确保包含适当的计量单位。

$\begin{aligned} S A & = 108 + 378 \end{aligned}$

$\begin{aligned} S A & = 486 i n^{2} \end{aligned}$

::SA=108+378 SA=486 英寸2

The answer is the surface area of the above triangular prism is 486 square inches.
::答案是上述三角棱镜的表面面积是486平方英寸。

Review
::回顾

Use the following figure to answer each question.
::使用下图回答每个问题。

What unit are the measurements of this figure?
::这个数字的测量单位是什么?

What is the height (or length) of the whole prism?
::整个棱镜的高度(长度)是多少?

What is the total length of the triangular base?
::三角基点的总长度是多少?

What is the base and height length of the triangular face?
::三角形面的基数和高度长度是多少?

What is the formula for finding the surface area of a triangular prism?
::找到三角棱镜表面积的公式是什么?

What is the surface area of this figure?
::这个数字的表面面积是多少?

Find the surface area of each triangular prism. Round your answer to the nearest hundredths place where needed.
::查找每个三角棱镜的表面区域。将您的答案绕到最接近的百个位置。

A triangular prism has a triangular base of 6 inches and a triangular height of 4 inches. The length of each side is 6 inches, 5 inches and 5 inches. The height of the prism is 3 inches. What is the surface area of the prism?
::三角棱晶的三角底部为6英寸,三角高度为4英寸。每边的长度为6英寸、5英寸和5英寸。棱晶的高度为3英寸。棱晶的表面面积是多少?

A triangular prism has a triangular base of 5 feet and a triangular height of 4 feet. The length of each side is 5 feet, 5 feet and $\begin{aligned} \sqrt{20} \end{aligned}$ feet. The height of the prism is 5 feet. What is the surface area of the prism?
::三角棱镜的三角基底为5英尺,三角高为4英尺。每边的长度为5英尺、5英尺和20英尺。棱镜的高度为5英尺。棱镜的表面面积是多少?

A triangular prism has a triangular base of 4 inches and a triangular height of 3 inches. The length of each side is 3 inches, 4 inches and 5 inches. The height of the prism is 3 inches. What is the surface area of the prism?
::三角棱晶的三角基点为 4 英寸,三角高度为 3 英寸。每边的长度为 3 英寸, 4 英寸和 5 英寸。棱晶的高度为 3 英寸。棱晶的表面面积是多少?

A triangular prism has a triangular base of 6 inches and a triangular height of 5 inches. The length of each side is 6 inches, $\begin{aligned} \sqrt{34} \end{aligned}$ inches and $\begin{aligned} \sqrt{34} \end{aligned}$ inches. The height of the prism is 5 inches. What is the surface area of the prism?
::三角棱晶的三角底部为6英寸,三角高度为5英寸。每边的长度为6英寸、34英寸和34英寸。棱晶的高度为5英寸。棱晶的表面面积是多少?

A triangular prism has a triangular base of 15 inches and a triangular height of 10 inches. The length of each side is 15 inches, $\begin{aligned} \sqrt{200} \end{aligned}$ inches and $\begin{aligned} \sqrt{125} \end{aligned}$ inches. The height of the prism is 10 inches. What is the surface area of the prism?
::三角棱晶的三角基点为15英寸,三角高度为10英寸。每边的长度为15英寸、200英寸和125英寸。棱晶的高度为10英寸。棱晶的表面面积是多少?

A triangular prism has a triangular base of 9 inches and a triangular height of $\begin{aligned} 4.5 \sqrt{3} \end{aligned}$ inches. The length of all three sides is 9 inches. The height of the prism is 7 inches. What is the surface area of the prism?
::三角棱晶的三角基点为9英寸,三角高4.53英寸。所有三边的长度为9英寸。棱晶的高度为7英寸。棱晶的表面面积是多少?

A triangular prism has a triangular base of 8 inches and a triangular height of 4 inches. The length of each side is 8 inches, 5 inches and $\begin{aligned} \sqrt{137} \end{aligned}$ inches. The height of the prism is 6 inches. What is the surface area of the prism?
::三角棱晶的三角基点为8英寸,三角高4英寸。每边的长度为8英寸、5英寸和137英寸。棱晶的高度为6英寸。棱晶的表面面积是多少?

A triangular prism has a triangular base of 5.5 feet and a triangular height of 4 feet. The length of each side is 5.5 feet, 4 feet and $\begin{aligned} \sqrt{46.25} \end{aligned}$ feet. The height of the prism is 5 feet. What is the surface area of the prism?
::三角棱镜的三角底部为5.5英尺,三角高度为4英尺。每边的长度为5.5英尺、4英尺和46.25英尺。棱镜的高度为5英尺。棱镜的表面面积是多少?

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。

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::资源

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