Course: 6.6 三角棱晶表面区域-interactive

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum

三角棱晶表面面积

In This Lesson
::在本课程中

You will be working with the of triangular prisms. In , you learned that the surface area of a polyhedron is the sum of the areas of all of its faces, including any bases. This lesson will help you practice finding the surface area of triangular prisms through real-world examples. Triangular prisms have a triangle base, you will start by reviewing how to find the area of a triangle.
::您将会与三角棱晶一起工作。在其中, 您学会了多面形的表面区域是其所有面部区域的总和, 包括任何基数。此课将有助于您通过真实世界实例来练习如何找到三角棱晶的表面区域。三角棱晶有一个三角基。您将首先审查如何找到三角形区域。

Triangles
::三角三角形

Recall that the formula for the area of a triangle is $\begin{align*}A=\tfrac{1}{2}bh\end{align*}$ , where $\begin{align*}b \end{align*}$ is the length of the base, and $\begin{align*}h \end{align*}$ is the height of the triangle. Remember, that the height of a triangle is also called the , and goes from the vertex at the top of the triangle to the base and is perpendicular .
::提醒注意三角形区域的公式是 A=12bh, 其中 b 是基数长度, h 是三角形的高度。记住, 三角形的高度也称为 , 从三角形顶部的顶部的顶部转到基数, 并且是垂直的。

Three triangles illustrating different placements of height <img class= — Heights of Triangles

In all of the triangles above, h is the same, but in very different locations. For an obtuse triangle , the height could be outside of the triangle. For a right triangle , the height is one of the legs because they are already perpendicular to each other. Let's say that $\begin{align*}h = 7 \end{align*}$ and $\begin{align*}b = 6. \end{align*}$ The area of the triangle would be, $\begin{align*}A=\tfrac{1}{2}(7)(6)=7(3)=21\end{align*}$ .
::在以上所有三角形中, h 是相同的, 但在非常不同的位置。对于隐形三角形, 高度可以是三角形之外。对于右三角形, 高度是腿部之一, 因为腿部已经相互垂直。假设 h=7 和 b=6 。三角形的面积是 A=12(7) (6)=7(3)=21 。

To find the surface area of a triangular based prism , just find and add the areas of all of the faces. However, here, the two bases are triangles. So, the surface area will be the area of three different and the area of two congruent triangles.
::要找到三角棱柱的表面区域, 只需找到并添加所有面孔的区域。但是, 这里, 两个基座是三角形。因此, 地表区域将是三个不同的区域, 两个相似的三角形的区域。

A-Frame Home
::A-Frame家园

An A-frame house is the shape of an isosceles triangular prism and is very popular in areas where it snows a lot. The roof goes to the ground, making it difficult for snow to stick. They are also very easy to build and affordable. Sven and Helga want to build a second home in Turino, Italy. They want to build an A-frame home and the building site is an 800 ft ² rectangle , where the width must be between 28 ft and 18 ft . Building regulations say that the maximum height cannot be greater than 27 feet. What is the surface area of the roofing needed for their home?
::A-框架房屋是三边立体棱柱形的形状,在下雪很多的地方非常流行。屋顶落到地上, 使雪很难粘住。它们也非常容易建造, 价格也非常低廉。 Sven和Helga想要在意大利都灵建造第二个房子。他们想要建造一个 A-框架房屋, 建筑工地是一个800英尺长的矩形, 宽度必须在28英尺到18英尺之间。建筑条例规定, 最高高度不能大于27英尺。房顶的面积是多少?

Use the interactive to maximize the size of the home with the building site and the roof height. You may assume that the roofing goes all the way to the ground. Determine the length and width of the home and the size of the roof. The front door is on one of the bases and the length of the home is greater than the width.
::使用互动关系使房屋与建筑工地和屋顶高度的大小最大化。您可以假设屋顶一直延伸到地面。确定房屋的长度和宽度以及屋顶的大小。前门位于一个底座上, 房屋的长度大于宽度。

In this example, you found the surface area of a triangular prism using the lengths of all the sides of the bases (the , in this case), the length of the house, and the overall height). Note, the base of a triangular prism could also be equilateral, right, or scalene.
::在此示例中,您发现三角棱镜的表面区域使用了基底两侧的长度(这里指的是底部的长度 ) , 房子的长度, 以及整个高度。注意, 三角棱镜的底部也可以是等边的、右的或缩放的。

Would You Like a Slice a Pizza?
::你想吃切片披萨吗?

Vinny's By the Slice sells pizza by the slice. He has special to-go boxes that hold one slice of pizza for his customers that don't want a whole pizza. The boxes themselves are equilateral triangular prisms and he has three different sizes; small, medium, and large. The measurements are in the table below.
::Vinny's by the Slits's by the Slits's sells pizza by the piece. 他有特别的比萨饼箱,为那些不想要整份比萨的顾客保管一块披萨。盒子本身是等边三角棱镜,他有三个不同的大小:小、中、大。测量在下面的表格中。

Pizza Size	Box Edge Length	Box Height	Triangle Height	Total Surface Area
Small	x in	2 in	3.5 in	38 in ²
Medium	x + 2 in	2 in	5.2 in	67.2 in ²
Large	2 x in	2 in	6.9 in	103.2 in ²

The interactive is 3 equilateral triangular prisms where the height is fixed at 2 inches. There is a slider so that you can adjust the value of x for all three prisms. For each prism, the equation will be written that represents the surface area. What is the value of x?
::交互式是3 个等边三角棱镜, 高度固定在 2 英寸。有一个滑块可以调整所有 3 个棱镜的 x 值。对于每个棱镜, 方程式将被写入代表表面积的方程式。 x 值是多少 ?

Open pizza box showing folded design, illustrating triangular prism shape for surface area discussion.

Discussion Question
::讨论问题

In real life, a pizza box of this style would actually have rectangular faces/sides that are attached to the bottom of the box and then 2 additional rectangular faces/sides that are attached to the top. How would this change the total surface area of the box?
::在现实生活中,这种风格的比萨饼盒实际上会有附在盒子底部的长方形面/侧面,然后有另外两个附在盒子顶部的长方形面/侧面。这将如何改变盒子的表面总面积?

for some examples of surface areas of prisms!
::以一些棱镜表面区域为例!

Summary
::摘要

The surface are of a prism is the sum of the areas of all of its faces.
::表面是棱镜表面是其面孔的面积的总和

Triangular prisms have two (congruent) triangular bases and three rectangular sides.
::三角棱镜有两个(相容的)三角基点和三个矩形边。

The type of triangle the base of a triangular prism is can tell you a lot about the sides of the prism:
::三角棱镜基底的三角形类型可以告诉你许多关于棱镜两边的情况:

If the bas e is an isosceles triangles, two of the rectangular faces will be congruent.
::如果基座是等分形三角形,则两个矩形面将相容。

If the base is an equilateral triangles, all three of the rectangular faces will be congruent.
::如果基座是一个等边三角形,则所有三个长方形的面孔都是相同的。

Section outline

General

三角棱晶表面面积

In This Lesson ::在本课程中

Triangles ::三角三角形

A-Frame Home ::A-Frame家园

Would You Like a Slice a Pizza? ::你想吃切片披萨吗?

Summary ::摘要

In This Lesson
::在本课程中

Triangles
::三角三角形

A-Frame Home
::A-Frame家园

Would You Like a Slice a Pizza?
::你想吃切片披萨吗?

Summary
::摘要