Perimeter and Area | 周长与面积

📚 Perimeter and Area | 周长与面积

Perimeter and area are two of the most important measurements in geometry. They help us describe the size of a shape in two different ways: how long the boundary is, and how much space is inside. In this lesson, we will explore both concepts step by step, using rectangles, squares, and composite shapes built from them.

周长和面积是几何学中最重要的两种测量。它们从两个不同的角度帮助我们描述图形的大小：边界有多长，以及内部有多少空间。在本课中，我们将逐步探索这两个概念，研究对象包括长方形、正方形以及由它们组成的复合图形。

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1. What is Perimeter? | 什么是周长？

The perimeter of a shape is the total distance around its outer edge. Think of a fence built around a garden: the length of the fence is the perimeter. We measure perimeter in linear units such as millimetres (mm), centimetres (cm), metres (m) and kilometres (km).

图形的周长是其外边缘的总长度。可以想象花园周围修建的篱笆：篱笆的长度就是周长。我们用长度单位来测量周长，例如毫米（mm）、厘米（cm）、米（m）和千米（km）。

To find the perimeter of any polygon, we simply add the lengths of all its sides. This works for triangles, rectangles, squares and all other straight-sided shapes.

要求任意多边形的周长，我们只需把所有边的长度相加。这个方法适用于三角形、长方形、正方形以及其他所有直边图形。

Perimeter = Sum of all side lengths
周长 = 所有边长的总和

• Perimeter is measured in linear units (cm, m, etc.).
• 周长用长度单位测量（cm、m 等）。
• Every polygon has a perimeter, whether it is regular or irregular.
• 每个多边形都有周长，无论它是正多边形还是不规则图形。
• Perimeter is always a single dimension — it has no square part.
• 周长始终是一维的——它没有平方部分。

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2. Finding Perimeter of Rectangles and Squares | 求长方形和正方形的周长

A rectangle has two pairs of equal sides: the length (l) and the width (w). Because opposite sides are equal, we can use a shortcut formula instead of adding all four sides one by one.

长方形有两组相等的边：长（l）和宽（w）。因为对边相等，我们可以用快捷公式代替逐一相加四条边。

Rectangle perimeter: P = 2 × (l + w)
长方形周长：P = 2 ×（l + w）

A square has four equal sides. If the side length is s, then the perimeter is simply four times s.

正方形的四条边都相等。如果边长为 s，那么周长就是 s 的四倍。

Square perimeter: P = 4 × s
正方形周长：P = 4 × s

For example, a rectangle with length 8 cm and width 5 cm has perimeter P = 2 × (8 + 5) = 2 × 13 = 26 cm. A square with side 6 m has perimeter P = 4 × 6 = 24 m. Notice that the unit stays as cm or m — we do not square it for perimeter.

例如，一个长 8 cm、宽 5 cm 的长方形，周长 P = 2 ×（8 + 5）= 2 × 13 = 26 cm。一个边长为 6 m 的正方形，周长 P = 4 × 6 = 24 m。注意单位仍然是 cm 或 m——求周长时我们不需要加平方。

Shape 图形	Given 已知条件	Perimeter 周长
Rectangle 长方形	l = 8 cm, w = 5 cm	2 × (8 + 5) = 26 cm
Square 正方形	s = 6 m	4 × 6 = 24 m

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3. Perimeter of Composite Shapes | 复合图形的周长

A composite shape is made by joining two or more simple shapes together. To find its perimeter, we do not add the sides that are inside the shape — we only count the sides on the outside boundary.

复合图形由两个或多个简单图形拼接而成。求其周长时，我们不计算图形内部的边——只计算外边界的边。

Consider an L-shaped figure made from two rectangles. The best strategy is to draw the shape, mark every outer side with its length, and then add them all. If some side lengths are not given, we can use the known lengths to find them by addition or subtraction.

考虑一个由两个长方形组成的 L 形图形。最佳策略是画出图形，在每条外边上标出长度，然后把它们全部加起来。如果某些边长没有给出，我们可以利用已知长度通过加法或减法求出。

For example, suppose an L-shape has known outer sides of 10 cm, 8 cm, 6 cm and 2 cm. The missing side is found by subtracting 8 − 6 = 2 cm, and the remaining side by 10 − 2 = 8 cm. The perimeter is 10 + 8 + 6 + 2 + 2 + 8 = 36 cm.

例如，假设一个 L 形图形的已知外边长为 10 cm、8 cm、6 cm 和 2 cm。其中一条未知边通过 8 − 6 = 2 cm 求出，另一条通过 10 − 2 = 8 cm 求出。周长就是 10 + 8 + 6 + 2 + 2 + 8 = 36 cm。

• Only count the outer boundary — never count inner division lines.
• 只计算外边界——绝不计算内部分割线。
• Use given lengths to find missing side lengths.
• 利用已知长度求出未知边长。
• Add carefully in order around the shape.
• 沿着图形一周按顺序仔细相加。

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4. What is Area? | 什么是面积？

Area is the amount of space inside a two-dimensional shape. While perimeter is measured in linear units, area is measured in square units, such as square centimetres (cm²), square metres (m²), and square kilometres (km²).

面积是二维图形内部所包含的空间大小。周长以长度单位度量，而面积以平方单位度量，例如平方厘米（cm²）、平方米（m²）和平方千米（km²）。

Think of covering a table with square tiles: the number of square tiles needed to cover the whole table equals the area of the table top. Each tile represents one square unit.

想象用方形瓷砖铺满一张桌子：铺满整个桌面所需的瓷砖数量就等于桌面的面积。每块瓷砖代表一个平方单位。

Area is measured in square units.
面积以平方单位度量。

The square unit tells us how many unit squares fit inside the shape. For example, an area of 20 cm² means 20 squares of size 1 cm × 1 cm would fit inside.

平方单位告诉我们图形内部能放下多少个单位正方形。例如，20 cm² 的面积意味着能放下 20 个边长为 1 cm × 1 cm 的小正方形。

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5. Finding Area of Rectangles and Squares | 求长方形和正方形的面积

The area of a rectangle is found by multiplying its length by its width. This works because rows and columns of unit squares fill the entire rectangle.

长方形的面积等于长乘以宽。这是因为单位正方形按行和列排列，可以填满整个长方形。

Rectangle area: A = l × w
长方形面积：A = l × w

For a square, the length and width are equal, so the area is the side length multiplied by itself.

对于正方形，长和宽相等，所以面积等于边长自乘。

Square area: A = s × s = s²
正方形面积：A = s ×

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