Perimeter and Area of Plane Figures | 平面图形的周长与面积

📚 Perimeter and Area of Plane Figures | 平面图形的周长与面积

Perimeter and area are two fundamental concepts in geometry that help us measure and understand the world around us. From fencing a garden to tiling a floor, these calculations appear in everyday life and form a critical part of the IGCSE mathematics syllabus.

周长和面积是几何学中两个基本概念，帮助我们测量和理解周围的世界。从围花园篱笆到铺设地板，这些计算出现在日常生活中，是 IGCSE 数学课程的重要组成部分。

1. What is Perimeter? | 什么是周长？

Perimeter is the total distance around the outside of a two-dimensional shape. It is measured in linear units such as centimetres (cm), metres (m), or kilometres (km).

周长是二维图形外边缘的总距离。它用线性单位来测量，如厘米（cm）、米（m）或千米（km）。

To find the perimeter of any polygon, simply add together the lengths of all its sides. For example, a triangle with sides of 3 cm, 4 cm, and 5 cm has a perimeter of 3 + 4 + 5 = 12 cm.

要找到任何多边形的周长，只需将所有边的长度相加。例如，一个边长为 3 cm、4 cm 和 5 cm 的三角形，其周长为 3 + 4 + 5 = 12 cm。

For shapes with equal sides, such as squares or regular polygons, you can multiply the side length by the number of sides. A regular hexagon with side length 6 cm has a perimeter of 6 × 6 = 36 cm.

对于具有相等边的形状，例如正方形或正多边形，可以将边长乘以边数。一个边长为 6 cm 的正六边形的周长为 6 × 6 = 36 cm。

2. What is Area? | 什么是面积？

Area measures the amount of space inside a two-dimensional shape. It is measured in square units such as square centimetres (cm²), square metres (m²), or square kilometres (km²).

面积测量二维图形内部的空间量。它用平方单位来测量，如平方厘米（cm²）、平方米（m²）或平方千米（km²）。

Imagine covering a rectangle with unit squares—each square represents one square unit. The total number of squares that fit inside the shape gives its area. For example, a rectangle 5 cm long and 3 cm wide contains 5 × 3 = 15 unit squares, so its area is 15 cm².

想象用单位正方形覆盖一个长方形——每个正方形代表一个平方单位。能够放入图形内的正方形总数即为面积。例如，一个长 5 cm、宽 3 cm 的长方形包含 5 × 3 = 15 个单位正方形，因此其面积为 15 cm²。

Different shapes require different formulas for area, and understanding when to use each formula is a key skill for IGCSE exams.

不同的形状需要不同的面积公式，理解何时使用每个公式是 IGCSE 考试中的关键技能。

3. Squares and Rectangles | 正方形与长方形

The rectangle is the simplest shape for area calculation. If a rectangle has length l and width w, its perimeter and area are given by the following formulas.

长方形是最简单的面积计算形状。若一个长方形的长为 l、宽为 w，其周长和面积由以下公式给出。

Perimeter of rectangle = 2 × (l + w)

Area of rectangle = l × w

For a square, all four sides are equal. If the side length is s, then the perimeter is 4s and the area is s².

对于正方形，四条边都相等。若边长为 s，则周长为 4s，面积为 s²。

Example: A rectangular field is 20 m long and 15 m wide. Its perimeter is 2 × (20 + 15) = 70 m, and its area is 20 × 15 = 300 m².

示例：一个长方形场地长 20 m、宽 15 m。其周长为 2 × (20 + 15) = 70 m，面积为 20 × 15 = 300 m²。

4. Triangles | 三角形

For any triangle with base b and perpendicular height h, the area is half the product of base and height. The perpendicular height is measured at a right angle to the base.

对于任何底边为 b、垂直高度为 h 的三角形，面积是底与高乘积的一半。垂直高度是指与底边成直角测量的高度。

Area of triangle = ½ × b × h

The perimeter of a triangle is simply the sum of its three sides: P = a + b + c.

三角形的周长是其三边之和：P = a + b + c。

Example: A triangle has a base of 8 cm and a height of 5 cm. Its area is ½ × 8 × 5 = 20 cm².

示例：一个三角形的底边为 8 cm、高度为 5 cm。其面积为 ½ × 8 × 5 = 20 cm²。

When the triangle is right-angled, the two shorter sides can serve as base and height. For other triangles, you may need to draw the height carefully or use alternative methods such as Heron’s formula.

当三角形为直角三角形时，两条较短的边可以作为底和高。对于其他三角形，可能需要仔细画出高，或使用海伦公式等替代方法。

5. Parallelograms | 平行四边形

A parallelogram has two pairs of parallel sides. To find its area, multiply the base by the perpendicular height—not by the slant side length.

平行四边形有两对平行边。求其面积时，用底乘以垂直高度——不能乘以斜边长度。

Area of parallelogram = b × h

For example, a parallelogram with a base of 12 cm and a perpendicular height of 4 cm has an area of 12 × 4 = 48 cm².

例如，一个底为 12 cm、垂直高度为 4 cm 的平行四边形，其面积为 12 × 4 = 48 cm²。

The perimeter of a parallelogram is found by adding all four sides. If the side lengths are a and b, then P = 2 × (a + b).

平行四边形的周长是四条边之和。若两边长分别为 a 和 b，则 P = 2 × (a + b)。

Be careful: the perpendicular height is always shorter than the slant side unless the parallelogram is a rectangle. Using the slant side in the area formula is a common mistake.

注意：除非平行四边形是长方形，否则垂直高度总是短于斜边。在面积公式中使用斜边长度是常见错误。

6. Trapeziums | 梯形

A trapezium (also called a trapezoid in some countries) has at least one pair of parallel sides. The parallel sides are called bases, and the distance between them is the height.

梯形（在某些国家称为 trapezoid）至少有一对平行边。平行边称为底边，它们之间的距离为高度。

Area of trapezium = ½ × (a + b) × h

Here, a and b are the lengths of the two parallel sides, and h is the perpendicular distance between them.

这里，a 和 b 是两条平行边的长度，h 是它们之间的垂直距离。

Example: A trapezium has parallel sides of 10 cm and 6 cm, with a height of 4 cm. Its area is ½ × (10 + 6) × 4 = 32 cm².

示例：一个梯形的平行边分别为 10 cm 和 6 cm，高度为 4 cm。其面积为 ½ × (10 + 6) × 4 = 32 cm²。

To find the perimeter of a trapezium, add all four sides together. If the non-parallel sides are c and d, then P = a + b + c + d.

求梯形周长时，将四条边相加即可。若非平行边为 c 和 d，则 P = a + b + c + d。

7. Circles | 圆

A circle is a special shape defined by its radius r (the distance from the centre to any point on the edge) and its diameter d (twice the radius). The perimeter of a circle is called its circumference.

圆是一种特殊形状，由半径 r（从圆心到圆周上任意一点的距离）和直径 d（半径的两倍）定义。圆的周长称为圆周（circumference）。

Circumference = 2 × π × r 或 C = π × d

Area of circle = π × r²

π (pi) is approximately 3.14159. In IGCSE questions, you may be asked to leave your answer in terms of π or to use a numerical approximation such as 3.14 or 22/7.

π（圆周率）约等于 3.14159。在 IGCSE 题目中，你可能被要求将答案保留为 π 的形式，或使用数值近似值如 3.14 或 22/7。

Example: A circle has a radius of 7 cm. Its circumference is 2 × π × 7 = 14π ≈ 43.98 cm, and its area is π × 7² = 49π ≈ 153.94 cm².

示例：一个圆的半径为 7 cm。其圆周为 2 × π × 7 = 14π ≈ 43.98 cm，面积为 π × 7² = 49π ≈ 153.94 cm²。

Remember to square the radius before multiplying by π. Forgetting to square, or using diameter in place of radius, are two frequent errors students make.

记住先平方半径再乘以 π。忘记平方，或用直径代替半径，是学生常犯的两个错误。

8. Compound Shapes | 组合图形

Many real-world objects do not have a single simple shape—they are combinations of rectangles, triangles, circles, and other figures. These are called compound shapes or composite figures.

许多现实物体并非单一简单形状——它们由长方形、三角形、圆和其他图形组合而成。这些被称为组合图形或复合图形。

To find the area of a compound shape, split it into simpler shapes whose areas you can calculate, then add or subtract the areas as needed.

求组合图形面积时，将其分割成可计算面积的简单图形，然后根据需要相加或相减面积。

For example, a letter “L” shape can be divided into two rectangles. The area of the whole shape is the sum of the areas of the two rectangles.

例如，字母 “L” 形状可以分成两个长方形。整个图形的面积是两个长方形面积之和。

When dealing with a shape that has a hole cut out, such as a circular hole in a rectangle, subtract the area of the hole from the area of the larger shape.

处理有挖空部分的图形时，例如长方形中的圆孔，要从大图形面积中减去孔的面积。

For perimeter, trace around the outer edge of the compound shape and add each segment length. Do not include internal boundaries where two shapes meet.

求周长时，沿组合图形的外边缘走一圈，将每段长度相加。不要包括两个图形接触的内部分界线。

9. Units of Measurement | 测量单位

Consistent use of units is essential in all geometry problems. Area always uses square units, while perimeter uses linear units.

在所有几何问题中，单位使用的一致性至关重要。面积始终使用平方单位，而周长使用线性单位。

Common conversions between units of length and area include:

长度和面积单位之间的常见换算包括：

Length	Equivalent	Area	Equivalent
1 m	100 cm	1 m²	10 000 cm²
1 cm	10 mm	1 cm²	100 mm²
1 km	1000 m	1 km²	1 000 000 m²

When converting area units, remember that the conversion factor must be squared. For example, since 1 m = 100 cm, 1 m² = 100 × 100 = 10 000 cm².

换算面积单位时，记住换算因子必须平方。例如，由于 1 m = 100 cm，所以 1 m² = 100 × 100 = 10 000 cm²。

10. Problem-Solving Strategies | 解题策略

Success in perimeter and area problems depends on a clear, step-by-step approach. First, identify what the question is asking for—is it perimeter or area? Then choose the correct formula based on the shape given.

解决周长和面积问题的关键在于清晰、分步骤的方法。首先，确定题目要求什么——是周长还是面积？然后根据给定的形状选择正确的公式。

Here are some steps to follow:

以下是一些可以遵循的步骤：

Read the question carefully and underline all given measurements. 仔细阅读题目，并在所有给定测量值下划线。
Draw a diagram if one is not provided, labelling all known sides. 如果题目未提供图形，请画出示意图，并标注所有已知边。
Select the appropriate formula for the shape. 为该形状选择合适的公式。
Substitute the values into the formula and calculate. 将数值代入公式并计算。
Check that your units are correct and write the final answer with the correct unit. 检查单位是否正确，并用正确单位写出最终答案。

For compound shapes, consider whether you should split the shape or subtract parts. Sketching dotted lines to show how you are dividing the shape helps both you and the examiner.

对于组合图形，考虑应该分割图形还是减去部分。用虚线画出如何分割图形，有助于你和阅卷者理解。

Always ask: does my answer make sense? A perimeter cannot be smaller than the longest side, and an area cannot be negative. These simple sanity checks catch many careless errors.

始终自问：我的答案合理吗？周长不能小于最长边，面积不能为负。这些简单的合理性检查能发现许多粗心错误。

11. Worked Examples | 例题精讲

Let us apply these concepts through two worked examples.

让我们通过两道例题来应用这些概念。

Example 1: A rectangular garden is 12 m long and 8 m wide. A path 2 m wide runs around the outside of the garden. Find the area of the path.

例 1：一个长方形花园长 12 m、宽 8 m。花园外围有一条宽 2 m 的小路。求小路的面积。

The outer rectangle has length 12 + 2 + 2 = 16 m and width 8 + 2 + 2 = 12 m. Its area is 16 × 12 = 192 m². The garden itself has area 12 × 8 = 96 m². Path area = 192 − 96 = 96 m².

外长方形的长为 12 + 2 + 2 = 16 m，宽为 8 + 2 + 2 = 12 m。其面积为 16 × 12 = 192 m²。花园本身的面积为 12 × 8 = 96 m²。小路面积为 192 − 96 = 96 m²。

Example 2: A semicircle of radius 10 cm is attached to the side of a rectangle measuring 20 cm by 10 cm. Find the total area of the combined shape.

例 2：一个半径为 10 cm 的半圆附在长 20 cm、宽 10 cm 的长方形一侧。求组合图形的总面积。

The rectangle has area 20 × 10 = 200 cm². The semicircle has area ½ × π × 10² = 50π ≈ 157.08 cm². Total area = 200 + 50π ≈ 357.08 cm².

长方形面积为 20 × 10 = 200 cm²。半圆面积为 ½ × π × 10² = 50π ≈ 157.08 cm²。总面积为 200 + 50π ≈ 357.08 cm²。

Notice that in Example 2, the diameter of the semicircle (20 cm) matches the length of the rectangle, so the shape fits together perfectly.

注意在例 2 中，半圆的直径（20 cm）与长方形的长相等，因此两个形状完美贴合。

12. Common Mistakes and Tips | 常见错误与提示

Students often make predictable mistakes when working on perimeter and area problems. Recognising these traps can help you avoid them in exams.

学生在处理周长和面积问题时经常犯可预见的错误。识别这些陷阱可以帮助你在考试中避免它们。

Confusing perimeter and area: perimeter is a distance (1D), area is a measure of surface (2D). Use different units and different formulas. 混淆周长和面积：周长是距离（一维），面积是表面量度（二维）。使用不同的单位和不同的公式。
Using the wrong height in triangles and parallelograms: always use the perpendicular height, not the slant side. 在三角形和平行四边形中使用错误的高度：始终使用垂直高度，而不是斜边。
Forgetting to square the radius in circle area: area of circle uses πr², not πr. 忘记在圆面积中平方半径：圆面积使用 πr²，不是 πr。
Adding internal lines when calculating perimeter of compound shapes: only the outer boundary counts. 计算组合图形周长时加上内部分割线：只有外边界才计入。
Mixing units: convert all measurements to the same unit before calculating. 混合单位：计算前将所有测量值转换为相同单位。

Practice is the best way to master these skills. Work through a variety of problems, from simple single shapes to complex compound figures, and check your answers step by step.

练习是掌握这些技能的最佳方式。从简单的单一图形到复杂的组合图形，做各种不同类型的题目，并逐步检查你的答案。

Remember to show all your working in exams. Even if your final answer is wrong, you may still earn marks for choosing the correct formula or substituting values correctly.

记住在考试中展示所有步骤。即使最终答案有误，选择正确公式或正确代入数值仍可能得分。

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Perimeter and Area of Plane Figures | 平面图形的周长与面积