Understanding Perimeter | 理解周长

📚 Understanding Perimeter | 理解周长

Perimeter is the total distance around the outside of a flat shape. In this lesson, we will learn how to measure perimeter by counting unit squares and by adding side lengths.

周长是一个平面图形外边缘的总长度。本课我们将学习如何通过数单位正方形和相加边长来测量周长。

1. What Is Perimeter? | 什么是周长？

Every flat shape has a boundary. Think of a triangle, a rectangle, or a hexagon: each one has an outline that separates the inside of the shape from the outside. The total length of this outline is called the perimeter.

每个平面图形都有一条边界。想一想三角形、长方形或六边形：每个图形都有一条轮廓线，把图形的内部和外部隔开。这条轮廓线的总长度就叫作周长。

Imagine walking around a rectangular playground. If you start at one corner and walk all the way around until you return to that same corner, the distance you walked is the perimeter of the playground. Perimeter is always a measure of length, so we use units such as centimetres (cm), metres (m), or inches (in).

想象你在一个长方形操场上散步。如果你从一个角落出发，绕一整圈又回到同一个角落，那么你所走的路程就是这个操场的周长。周长始终是长度的度量，所以我们使用厘米（cm）、米（m）或英寸（in）等单位。

2. Perimeter with Unit Squares | 用单位正方形求周长

A unit square is a square whose sides are exactly 1 unit long. When a shape is drawn on a grid made of unit squares, we can find its perimeter by counting the unit edges on the outside of the shape.

单位正方形是边长正好为 1 个单位的正方形。当一个图形画在由单位正方形组成的网格上时，我们可以通过数图形外侧的单位边来求出周长。

For example, imagine an L-shaped figure made from three unit squares. Look only at the edges that form the outer boundary. Count the unit segments along the bottom, the right side, the top, and the left side. The total number of unit edges is the perimeter.

例如，设想一个由三个单位正方形拼成的 L 形图形。只观察形成外边界的边。数一数底部、右侧、顶部和左侧的单位线段。单位边的总数就是周长。

Perimeter = Number of unit edges on the outer boundary

周长 = 外边界上单位边的数量

3. Counting Only Outer Edges | 只数外边缘

When counting unit edges on a grid, you must count only the edges that are on the outside of the shape. Edges that are shared between two squares inside the shape are not part of the perimeter.

在网格上数单位边时，必须只数图形外侧的边。图形内部两个正方形共用的边不属于周长。

Suppose three squares are arranged in a straight row. The outer boundary has three unit edges on the top, three unit edges on the bottom, one unit edge on the left, and one unit edge on the right. There are also two internal vertical edges where the squares touch, but we do not count them.

假设三个正方形排成一条直线。外边界有 3 条单位边在顶部，3 条单位边在底部，1 条单位边在左侧，1 条单位边在右侧。正方形相接处还有两条内部竖边，但我们不数它们。

Perimeter = 3 + 3 + 1 + 1 = 8 units

周长 = 3 + 3 + 1 + 1 = 8 个单位

4. Adding Side Lengths | 将边长相加

Sometimes a shape is drawn with the length of every side written beside it. In that case, the easiest way to find the perimeter is to add all the side lengths together.

有时图形会标注出每条边的长度。在这种情况下，求周长最简单的方法就是把所有边长加在一起。

For a polygon with four sides, the perimeter is the sum of all four side lengths. If the sides are a, b, c, and d, then the perimeter is a plus b plus c plus d.

对于一个四边形，周长是四条边长的总和。如果边长分别是 a、b、c 和 d，那么周长就是 a 加 b 加 c 加 d。

Perimeter = a + b + c + d

周长 = a + b + c + d

For example, a quadrilateral has sides of length 4 cm, 6 cm, 4 cm, and 6 cm. Its perimeter is 4 + 6 + 4 + 6 = 20 cm.

例如，一个四边形的边长分别为 4 cm、6 cm、4 cm 和 6 cm。它的周长是 4 + 6 + 4 + 6 = 20 cm。

5. Perimeter of Rectangles | 长方形的周长

A rectangle has two lengths and two widths. The opposite sides of a rectangle are equal, so the perimeter can be found by adding two lengths and two widths.

长方形有两条长和两条宽。长方形的对边相等，所以周长可以用两条长加两条宽来求出。

There is a useful shortcut. First add the length and the width, then multiply the sum by 2. Another way is to multiply the length by 2 and the width by 2, then add the two products.

这里有一个简便方法。先把长和宽相加，再把和乘以 2。另一种方法是先把长乘以 2，宽乘以 2，再把两个积相加。

Perimeter of a rectangle = 2 × (length + width)

长方形周长 = 2 × （长 + 宽）

If a rectangle has a length of 8 cm and a width of 3 cm, its perimeter is 2 × (8 + 3) = 2 × 11 = 22 cm.

如果一个长方形的长为 8 cm，宽为 3 cm，它的周长是 2 × (8 + 3) = 2 × 11 = 22 cm。

6. Perimeter of Squares | 正方形的周长

A square is a special rectangle in which all four sides are equal. Because every side of a square has the same length, we can multiply the length of one side by 4.

正方形是一种特殊的长方形，它的四条边都相等。因为正方形的每条边长度相同，我们把一条边的长度乘以 4 即可。

Perimeter of a square = 4 × side length

正方形周长 = 4 × 边长

For example, if one side of a square is 6 cm long, the perimeter is 4 × 6 = 24 cm. If the side length is 9 m, the perimeter is 4 × 9 = 36 m.

例如，如果正方形的一条边长是 6 cm，周长是 4 × 6 = 24 cm。如果边长是 9 m，周长是 4 × 9 = 36 m。

7. Perimeter of Irregular Shapes | 不规则图形的周长

Not every shape is a rectangle or a square. A polygon can have sides of many different lengths. Even for an irregular shape, the rule is the same: add the lengths of all the outer sides.

并非每个图形都是长方形或正方形。多边形可以有很多不同长度的边。即使是不规则图形，规则也是一样的：把所有外边的长度相加。

For example, a five-sided figure has side lengths 3 m, 4 m, 7 m, 5 m, and 2 m. Its perimeter is 3 + 4 + 7 + 5 + 2 = 21 m.

例如，一个五边形的边长分别为 3 m、4 m、7 m、5 m 和 2 m。它的周长是 3 + 4 + 7 + 5 + 2 = 21 m。

Always check that you have included every side. A useful habit is to start at one vertex and go around the shape in one direction, listing every side as you go.

要时刻检查是否把每条边都算进去了。一个好习惯是：从一个顶点出发，沿一个方向绕图形走一圈，边走边列出每条边。

8. Finding Missing Side Lengths | 求未知边长

Sometimes you know the perimeter of a shape and the lengths of most of its sides, but one side is missing. To find the missing side, subtract the sum of the known sides from the total perimeter.

有时我们知道一个图形的周长以及大多数边的长度，但有一条边未知。要求未知边长，用总周长减去已知边长的和即可。

Missing side = Perimeter – Sum of known sides

未知边长 = 周长 – 已知边长的和

For example, a polygon has a perimeter of 30 cm. The known sides are 8 cm, 7 cm, and 5 cm. The sum of the known sides is 8 + 7 + 5 = 20 cm. The missing side is 30 – 20 = 10 cm.

例如，一个多边形的周长是 30 cm。已知边长为 8 cm、7 cm 和 5 cm。已知边长的和是 8 + 7 + 5 = 20 cm。未知边长是 30 – 20 = 10 cm。

9. Real-World Applications | 生活中的周长

Perimeter appears in many everyday situations. If you want to build a fence around a garden, you need to know the perimeter of the garden. If you want to put a border around a picture frame, you need the perimeter of the frame.

周长出现在许多日常情境中。如果你想在菜园周围建一圈栅栏，你需要知道菜园的周长。如果你想给相框四边镶上花边，你需要知道相框的周长。

People who lay tiles, design running tracks, or wrap gifts all use perimeter. When a teacher puts tape around a bulletin board, the length of tape needed is equal to the perimeter of the board.

铺地砖的人、设计跑道的人和包装礼物的人都会用到周长。当老师用胶带围绕布告栏粘贴时，所需胶带的长度就等于布告栏的周长。

10. Common Mistakes | 常见错误

Counting internal edges inside a grid shape: only outside edges are counted. 在网格图形中数了内部边：只有外边缘才算。
Forgetting to add one or more sides: check that every side is included. 忘记加某一条或多条边：检查是否每条边都已包含。
Confusing perimeter with area: perimeter is a length, while area is the amount of space inside the shape. 混淆周长和面积：周长是一种长度，而面积是图形内部的空间大小。
Using the wrong formula for a rectangle: remember that opposite sides are equal. 使用了错误的长方形公式：记住对边相等。

11. Practice Problems | 练习题

Try these problems on your own, then check your answers. Use the correct units in your final answer.

请先自己尝试解决这些问题，再对照答案。最终答案要使用正确的单位。

Problem 题目	Answer 答案
1. Find the perimeter of a square with side length 5 cm. 求边长为 5 cm 的正方形的周长。	20 cm
2. Find the perimeter of a rectangle with length 9 m and width 4 m. 求长为 9 m、宽为 4 m 的长方形的周长。	26 m
3. A triangle has sides 6 cm, 7 cm, and 8 cm. Find its perimeter. 一个三角形的三条边分别为 6 cm、7 cm 和 8 cm，求它的周长。	21 cm
4. A polygon has a perimeter of 40 cm. Known sides are 12 cm, 10 cm, and 8 cm. Find the missing side. 一个多边形周长是 40 cm，已知边为 12 cm、10 cm 和 8 cm，求未知边长。	10 cm
5. A rectangle has a perimeter of 30 cm and a length of 9 cm. Find its width. 一个长方形周长是 30 cm，长是 9 cm，求宽。	6 cm

12. Summary | 小结

Perimeter is the total distance around the outside of a shape. To find it, add the lengths of all outer sides. On a grid, count only the unit edges on the outer boundary. For rectangles, use 2 × (length + width). For squares, use 4 × side length.

周长是一个图形外边缘的总长度。求周长时，把所有外边的长度相加。在网格上，只数外边界上的单位边。对于长方形，使用 2 × (长 + 宽)。对于正方形，使用 4 × 边长。

If a side length is missing, subtract the sum of the known sides from the total perimeter. Practice with both regular and irregular shapes so that the idea of perimeter becomes natural and clear.

如果有一条边未知，用总周长减去已知边长的和即可。多练习规则图形和不规则图形，这样周长的概念就会变得自然而清晰。

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Understanding Perimeter | 理解周长