三角和四边形区域或周边区域

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Perimeter
::周边

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Perimeter is the distance around a shape. In other words, the total boundary length of a closed two-dimensional figure is called its perimeter . To find the perimeter of any two dimensional shape, find the sum of the lengths of all the sides.
::周边是形状周围的距离。 换句话说, 封闭的二维图的总边界长度被称为它的周界。 要找到任何两维形状的周界, 请找到所有两维形状的长度的和 。

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Area
::地区面积

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Area is the amount of surface enclosed by a closed two-dimensional figure. It is measured by the number of unit square s it takes to cover a two-dimensional shape. For example, if you count the small squares, you will find there are 15 of them. Therefore, the area is $\begin{array}{r}3\cdot 5\end{array}$ or $\begin{array}{r}15{\text{unit}}^2.\end{array}$
::区域是封闭的二维图封装的表面数量。它用它覆盖二维形状所需的单位方形数来测量。例如,如果数小方形,你会发现其中有15个。因此,区域是 3°5 或 15 单位2。

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Area of Rectangle
::矩形区域

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A rectangle is a very basic shape for area calculation. The area of a rectangle is base times height .
::矩形是计算区域的一个非常基本的形状。矩形区域是基时高度。

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$\begin{array}{r}{\text{Area}}_{\text{rectangle}}=bh\end{array}$
::面积角=bh

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Area of Parallelogram
::平行面积

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Any can be turned into a rectangle without changing its area.
::任何物体都可变成一个矩形,而不改变其面积。

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Therefore, the area of a parallelogram is also base times height .
::因此,平行图的面积也是基时高度。

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Note: To calculate the , the base and height (altitude) of a parallelogram must be perpendicular . However, the lateral sides (slant height) of a parallelogram are not perpendicular to the base. Therefore, height may need to be calculated before the area can be found.
::注:要计算 ,平行图的基数和高度(高度)必须是垂直的。但是,平行图的平面(倾角高度)与基数并不垂直。因此,在找到区域之前,可能需要计算高度。

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$\begin{array}{r}{\text{Area}}_{\text{parallelogram}}=bh\end{array}$
::平方平方图=bh

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Is it possible to find the area of a parallelogram by just multiplying the lengths of two adjacent sides ?
::能否通过仅仅乘以相邻两侧的长度来找到平行图的区域?

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It is to find the area of a parallelogram that does not have 90-degree corners by multiplying the lengths of two adjacent sides.
::以相邻两侧的长度乘以90度角的平行图区域。

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Area of Triangle
::三角三角区域

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You can think of any triangle as half a parallelogram . If you rotate a triangle $\begin{array}{r}{180}^{\circ }\end{array}$ about the midpoint of one of its sides, the original triangle and the new triangle will be a parallelogram.
::您可以将任何三角区想象成半个平行图。如果在三角区中间点左右旋转一个三角区180,则原来的三角区和新的三角区将是一个平行图。

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Therefore, the area of a triangle is base times height divided by two.
::因此,三角形区域是基数高度除以2。

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Remember that any of the three sides can be the base. Also remember that the height must be perpendicular to the base and extend to the highest point of the triangle.
::记住, 任何三边都可以是基点。 还要记住, 高度必须与基点垂直, 并延伸到三角形的最高点 。

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$\begin{array}{r}{\text{Area}}_{\text{triangle}}=\frac{bh}{2}=\frac{1}{2}bh\end{array}$
::矩形=bh2=12bh

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Area of Trapezoid
::草原面积

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A trapezoid can be thought of as half a parallelogram by rotating it $\begin{array}{r}{180}^{\circ }\end{array}$ about the midpoint of one of its non- parallel sides.
::通过旋转它180°C 大约其非平行边之一的中点, 捕捉类可以被看成是半个平行图。

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The base of this parallelogram is $\begin{array}{r}{b}_1+b_2\end{array}$ and the height is $\begin{array}{r}h\end{array}$ . The area of the parallelogram is $\begin{array}{r}(b_1+b_2)h\end{array}$ . Therefore, the area of the trapezoid is $\begin{array}{r}\frac{(b_1+b_2)h}{2}\end{array}$ .
::此平行图的底部是 b1+b2, 高度是 h。 平行图的面积是 (b1+b2) h。 因此, 捕捉类的面积是 (b1+b2) h2。

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These and other area formulas that are good to know are shown below.
::下文列出了这些公式和其他值得了解的公式。

Shape	Area Formula	Picture
Rectangle	$\begin{array}{r}A=bh\end{array}$
Parallelogram	$\begin{array}{r}A=bh\end{array}$
Triangle	$\begin{array}{r}A=\frac{bh}{2}\end{array}$
Trapezoid	$\begin{array}{r}A=\frac{(b_1+b_2)h}{2}\end{array}$
Rhombus	$\begin{array}{r}A=bh\end{array}$ or $\begin{array}{r}A=\frac{d_1{d}_2}{2}\end{array}$
Kite	$\begin{array}{r}A=\frac{d_1{d}_2}{2}\end{array}$
Square	$\begin{array}{r}A=s^2\end{array}$

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Calculating Area and Perimeter of Quadrilaterals
::四边形的计算面积和周边

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Adjust the dimensions of the quadrilateral given below by dragging the vertex and observe how the perimeter and area change.
::调整以下四边形的尺寸,拖曳顶点并观察周界和面积的变化。

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Rhombus
::滚轮

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Finding the Area and Perimeter
::寻找区域和周边区域

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1. Find the area of the trapezoid.
::1. 找出捕鼠类的区域。

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Given : " data-mouseout="dehighlightheight" data-smartphrase="true" role="term" tabindex="0"> height "> $\begin{array}{r}(h)\end{array}$ $\begin{array}{r}=8\end{array}$ , base $\begin{array}{r}(b_1)\end{array}$ $\begin{array}{r}=6\end{array}$ , base $\begin{array}{r}(b_2)\end{array}$ $\begin{array}{r}=10\end{array}$
::给定: 高度 = 8, 基数(b1) = 6, 基数(b2) = 10

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$\begin{array}{rl}& \text{Area of a trapezoid}\\ & =\frac{1}{2}h(b_1+b_2)\\ & =\frac{1}{2}(8)(6+10)\\ & =\frac{1}{2}(8)(16)\\ & =64{\text{units}}^2\end{array}$
::区域=12h(b1+b2)=12(8)(6+10)=12(8)(8)(16)=64单位2

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2. Find the area and perimeter of a square with side length 10 inches.
::2. 找出一个有10英寸侧长的广场的面积和周界。

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Perimeter is the distance around the shape. A square has four congruent sides, so each side is 10 inches.
::周围是形状周围的距离。 一个广场有四个相似的边, 所以每边有10英寸。

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$\begin{array}{rl}\text{Perimeter of a square}& =4\cdot \text{sides}\\ \text{Perimeter of a square}& =4(10)=40\text{in}\end{array}$
::平方 = 4°X = 40 英寸

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Area is the number of square units it takes to cover the shape.
::区域是覆盖形状所需的平方单位数。

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$\begin{array}{rl}\text{Area of a square}& ={\text{sides}}^2\\ \text{Area of a square}& ={10}^2=100{\text{in}}^2.\end{array}$
::平方面积=2 平方面积=102=100英寸2。

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Examples
::实例实例实例实例

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Example 1
::例1

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For both a rhombus and a , area can be found if you know the lengths of the diagonals . What is special about the diagonals of these shapes?
::如果知道对角线的长度,则可以找到对角线和对角线的区域。这些形状的对角线有什么特别之处?

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The diagonals of both rhombuses and are perpendicular. This means that when the shapes are broken down into triangles, one diagonal can be the base of the triangle and a portion of the other diagonal will be the height.
::双角和直角的对角。 这意味着, 当形状分解成三角形时, 一个对角可以是三角的底部, 而另一个对角的一部分将是高度 。

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Example 2
::例2

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The diagonals of a rhombus bisect each other. The diagonals are also perpendicular. Derive the area formula for a rhombus $\begin{array}{r}A=\frac{d_1{d}_2}{2}\end{array}$ .
::圆柱形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形对角形形对角形对角形形对角形形形对角形形形形形形对角形形形形对角形形形形对角形形形形形形对角形形形对角形形对角形形对角形对角形形形对角形对角形形形对角形形形形对角形形形形对角形对角形A=D1d22。

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The diagonals divide the rhombus into four . For each of the triangles, the base and height are $\begin{array}{r}\frac{d_1}{2}\end{array}$ and $\begin{array}{r}\frac{d_2}{2}\end{array}$ .
::三角形的底部和高度是 d12 和 d22 。

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$\begin{array}{rl}& \text{Area of rhombus}ABCD\\ & =\mathrm{△}AOB+\mathrm{△}BOC+\mathrm{△}COD+\mathrm{△}DOA\\ & =\left(\frac{1}{2}\cdot AO\cdot BO\right)+\left(\frac{1}{2}\cdot BO\cdot CO\right)+\left(\frac{1}{2}\cdot CO\cdot DO\right)+\left(\frac{1}{2}\cdot DO\cdot AO\right)\\ & =\left(\frac{1}{2}\cdot \frac{d_1}{2}\cdot \frac{d_2}{2}\right)+\left(\frac{1}{2}\cdot \frac{d_1}{2}\cdot \frac{d_2}{2}\right)+\left(\frac{1}{2}\cdot \frac{d_1}{2}\cdot \frac{d_2}{2}\right)+\left(\frac{1}{2}\cdot \frac{d_1}{2}\cdot \frac{d_2}{2}\right)\end{array}$
::ABCD-AOBO+(12)-(12)-(12)-(12)-(12)-(12)-(12)-(12)-(12)-(12)-(12)-(12)-(12)-(12)-(d)-(12)+(12)-(d)-12-(d)-(12)+(12)-(d)-12-(d)-(12)+(12)-(12)-(d)-12)-(d)-(12)+(12)-(12)-(d)-12)-(d)-(12)+(12)-(12)-(d)-(12)+(12)-(12)-(d)-(d)-(12)+(12)-(12)-(12)-(d)-(d)-(d)

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The rhombus is made up of four triangles with the same area $\begin{array}{r}4\left(\frac{d_1{d}_2}{8}\right)=\frac{d_1{d}_2}{2}\end{array}$ .
::圆柱由4个三角形组成,同一区域为4(d1d28)=d1d22。

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Example 3
::例3

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Explain why the formula $\begin{array}{r}A=bh\end{array}$ works to find the area of a rhombus.
::解释为什么公式A=bh 工作寻找暴风车的区域。

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$\begin{array}{r}A=bh\end{array}$ is the formula for the area for any parallelogram. Since a rhombus is a parallelogram, this area formula works for a rhombus.
::A=bh 是任何平行图的区域的公式。 由于 rhombus 是平行图, 此区域公式为 rhombus 工作 。

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CK-12 PLIX Interactive
::CK-12 PLIX 交互式互动

Summary

播放段落 Perimeter is the distance around a shape, found by summing the lengths of all sides. ::周围是形状周围的距离, 通过折射所有侧面的长度而发现。 播放段落 Area is the amount of surface enclosed by a closed two-dimensional figure. ::区域是指封闭的二维图中封闭的表面数量。 播放段落 The area of a rectangle or parallelogram is calculated: $\begin{array}{r}A=bh\end{array}$   ::矩形或平行图的面积计算为: A=bh 播放段落 The area of a triangle is calculated: $\begin{array}{r}A=\frac{bh}{2}\end{array}$   ::三角形区域的计算结果为: A=bh2 播放段落 The area of a trapezoid is calculated: $\begin{array}{r}A=\frac{(b_1+b_2)h}{2}\end{array}$   ::捕捉类动物的区域计算为:A=(b1+b2)h2

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Review
::审查审查审查审查

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1. Find the area and perimeter of a rectangle with a length of 12 inches and a height of 15 inches.
::1. 找出长12英寸、高15英寸的矩形区域和周界。

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2. Find the area and perimeter of a right triangle with legs 3 cm and 4 cm and hypotenuse 5 cm.
::2. 找出右三角形的面积和周界,右三角形的腿3厘米、4厘米和下角5厘米。

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3. Find the area of a trapezoid with bases 4 cm and 12 cm and height 9 cm.
::3. 找出基地为4厘米和12厘米、高度为9厘米的捕捉类动物区域。

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4. The perimeter of a rectangle is 150 cm. The length is 4 times more than the width. What is the length of the rectangle?
::4. 矩形的周边为150厘米。长度是宽度的4倍。矩形的长度是多少?

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5. The area of a triangle is $\begin{array}{r}30{\text{cm}}^2\end{array}$ . The base is twice as long as the height. What is the height of the triangle?
::5. 三角形的面积为30厘米。基数是高度的两倍。三角形的高度是多少?

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6. The perimeter of a right triangle is 24 in. The area is $\begin{array}{r}24{\text{in}}^2\end{array}$ . The hypotenuse is 4 inches longer than the base. What are the lengths of the sides of the triangle?
::6. 右三角的周边是24英寸,面积是24英寸,下限比基数长4英寸,三角形两侧的长度是多少?

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7. Why is it necessary to use square units (such as $\begin{array}{r}{\text{in}}^2\end{array}$ ) when referring to area?
::7. 为什么在提及面积时必须使用平方单位(如英寸)?

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8. Why don't you use square units when referring to perimeter?
::8. 为什么你在提到周边时不使用平方单位?

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9. When finding the area of a trapezoid, does it matter which sides you label as $\begin{array}{r}{b}_1\end{array}$ and $\begin{array}{r}{b}_2\end{array}$ ?
::9. 在找到一个区域时,你将哪一边标为b1和b2是否重要?

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10. Find the area of a square with a diagonal of 12 in.
::10. 找出方形区域,对角线为12英寸。

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11. Explain in your own words why you divide by two in the formula for the area of a triangle.
::11. 用你自己的话来解释为什么三角形区域公式除以两个。

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12. Explain in your own words why you divide by two in the formula for the area of a trapezoid.
::12. 用你自己的话来解释为什么在方程式中将一个区域除以两个。

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13. Find the area of a kite with diagonals 10 cm and 18 cm.
::13. 找到对角10厘米和18厘米的风筝区域。

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14. Derive the formula $\begin{array}{r}A=s^2\end{array}$ for the area of a square.
::14. 生成方形面积的公式A=s2。

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15. The diagonals of a kite are perpendicular and one diagonal bisects the other diagonal. Derive the formula $\begin{array}{r}A=\frac{d_1{d}_2}{2}\end{array}$ for the area of a kite.
::15. 风筝的对角线是垂直的,一个对角的对角线是另一对角线的对角线,产生风筝区域A=d1d22的公式。

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16. Sketch and assign dimensions to a rectangle and parallelogram (one that isn't a rectangle) such that they have the same area.
::16. 向矩形和平行图(不是矩形的图)拉伸和分配尺寸,使其具有相同的区域。

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17. Sketch and assign dimensions to a rhombus and a rectangle (neither of which are squares) and give them dimensions such that they have the same area.
::17. 向圆柱形和矩形(两者均不是正方形)拉伸和分配维度,并给其尺寸,使其具有相同的区域。

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18. Sketch and assign dimensions to a trapezoid and a triangle such that they have the same area.
::18. 将尺寸伸展和分配到一个三角形和一个三角形上,使其具有相同的区域。

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19. Sketch and assign dimensions to a rectangle. Find its area and perimeter. Double the dimensions to create a new rectangle, and find the area and perimeter. What do you observe? Why is this the case? Experiment with other figures and other scale factors to see if these results hold true.
::19. 向矩形伸展和指定尺寸,找出其面积和周界,将尺寸翻倍以创造新的矩形,并找到区域和周界,你观察到什么情况?为什么会这样?与其他数字和其他规模因素实验,看这些结果是否真实。

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Review (Answers)
::审查(答复)

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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 。