📚 Shape and Space 6: Area, Volume and Surface Area | 形状与空间6：面积、体积与表面积
Welcome to the sixth instalment of our Shape and Space revision series. In this article, we will master the calculations of perimeter, area, surface area and volume for common 2D and 3D shapes, which is a core component of the Edexcel IGCSE Mathematics syllabus. We’ll also explore unit conversions and problem-solving strategies to help you secure top marks in the exam.
欢迎来到我们“形状与空间”复习系列的第六篇文章。在这篇文章中，我们将掌握Edexcel IGCSE数学考纲中常见的二维和三维图形的周长、面积、表面积及体积的计算。我们还会探讨单位换算和解题策略，帮助你在考试中锁定高分。
1. Units and Conversions | 单位与换算
Before calculating any area or volume, you must ensure that all measurements share the same unit. Area is measured in square units such as millimetres squared (mm²), centimetres squared (cm²), metres squared (m²) and kilometres squared (km²). Volume is measured in cubic units like mm³, cm³, m³ and km³.
在计算任何面积或体积之前，必须确保所有测量值使用相同的单位。面积使用平方单位，如平方毫米（mm²）、平方厘米（cm²）、平方米（m²）和平方千米（km²）。体积使用立方单位，如mm³、cm³、m³和km³。
To convert from one square unit to another, you multiply or divide by the square of the linear conversion factor. For example, since 1 m = 100 cm, 1 m² = 100 cm × 100 cm = 10,000 cm². Similarly, since 1 m = 1000 mm, 1 m² = 1,000,000 mm² and 1 m³ = 1,000,000,000 mm³.
要在不同平方单位之间换算，需要乘以或除以线性换算因数的平方。例如，因为1 m = 100 cm，所以1 m² = 100 cm × 100 cm = 10,000 cm²。同理，由于1 m = 1000 mm，1 m² = 1,000,000 mm²，1 m³ = 1,000,000,000 mm³。
| Length 1 m = 100 cm = 1000 mm |
| Area 1 m² = 10⁴ cm² = 10⁶ mm² |
| Volume 1 m³ = 10⁶ cm³ = 10⁹ mm³ |
When converting between units of volume, remember that volume is three-dimensional, so the linear factor is cubed. A common exam trap is treating volume conversion like area conversion. Always double-check whether you are working in 2D or 3D.
换算体积单位时，记住体积是三维的，因此线性因数要立方。常见的考试陷阱是把体积换算当成面积换算。务必检查你处理的是二维还是三维问题。
2. Perimeter of 2D Shapes | 二维图形的周长
The perimeter of a shape is the total distance around its boundary. For polygons, simply add the lengths of all sides. For example, the perimeter of a rectangle with length l and width w is 2(l + w).
图形的周长是其边界的总长度。对多边形来说，只需把所有边的长度相加。例如，长为l、宽为w的矩形，周长为2(l + w)。
For a regular polygon with n sides, each of length s, the perimeter is n × s. For a circle, the circumference is πd or 2πr, where d is the diameter and r is the radius.
对于有n条边、每条边长为s的正多边形，周长为n × s。对于圆，周长是πd或2πr，其中d是直径，r是半径。
When dealing with compound shapes, the perimeter is the sum of the outer edges only. Do not include internal lines that are not part of the boundary. A common error is to add all visible lines, so draw a clear path around the shape before calculating.
处理组合图形时，周长只计算外部边界的总和。不要包括不属于边界的内部线段。一个常见错误是把所有可见线段都加进去，所以计算前先沿着图形外沿画一条清晰的路径。
3. Area of 2D Shapes (Part 1) | 二维图形的面积（第一部分）
Area measures the amount of space inside a 2D shape. The following formulas are essential for IGCSE and should be memorised:
面积是二维图形内部所占的空间大小。以下公式对IGCSE非常关键，必须牢记：
- Rectangle: A = l × w
- Square: A = s²
- Triangle: A = ½ × b × h (where b is base and h is perpendicular height)
- Parallelogram: A = b × h (where h is perpendicular height)
- Trapezium: A = ½ × (a + b) × h (where a and b are the parallel sides)
In a triangle, the height must be perpendicular to the base. In the trapezium formula, the two parallel sides are denoted as a and b. As long as you can identify the parallel sides, the formula remains consistent.
在三角形中，高必须垂直于底边。在梯形公式中，两条平行边分别记为a和b。只要你能正确识别平行边，公式就总是成立。
For a parallelogram, the height is the vertical distance between the two bases, not the slant height. Many students incorrectly use the length of a slanted side as the height, which leads to an overestimation of the area.
对于平行四边形，高是两条底边之间的垂直距离，不是斜边长度。许多学生错误地把斜边长当作高，这会导致面积偏大。
4. Area of 2D Shapes (Part 2: Circles and Sectors) | 二维图形的面积（第二部分：圆与扇形）
The area of a circle is A = πr². For a sector with central angle θ (in degrees), the area is θ/360 × πr².
圆的面积为A = πr²。对于圆心角为θ（度）的扇形，其面积为θ/360 × πr²。
Similarly, the arc length of a sector is θ/360 × 2πr. The perimeter of a sector includes the two radii plus the arc length: 2r + arc length.
类似地，扇形的弧长为θ/360 × 2πr。扇形的周长等于两条半径加上弧长：2r + 弧长。
When θ = 360°, the sector becomes a full circle, and the formulas reduce to the standard circle formulas. Always check whether the angle is in degrees or radians; in the IGCSE syllabus, degrees are the default unless specified otherwise.
当θ = 360°时，扇形就变成了完整圆，公式退化为标准圆的公式。始终检查角度是度数还是弧度；在IGCSE考纲中，默认使用度数，除非特别说明。
5. Surface Area of 3D Shapes | 三维图形的表面积
The surface area of a 3D shape is the total area of all its faces. For a cuboid, the surface area is 2(lw + lh + wh), where l, w and h are the length, width and height.
三维图形的表面积是所有面的总面积。对于长方体，表面积为2(lw + lh + wh)，其中l、w和h分别是长、宽和高。
For a cylinder with radius r and height h, the surface area is the sum of the two circular ends and the curved surface: 2πr² + 2πrh.
对于半径为r、高为h的圆柱体，表面积等于两个圆形底面积加侧面积：2πr² + 2πrh。
A triangular prism requires you to sum the areas of two triangular bases and three rectangular faces. A sphere has the elegant formula 4πr². For a cone, the total surface area is πr² + πrl, where l is the slant height.
三棱柱需要计算两个三角形底面和三个矩形侧面的面积之和。球的表面积公式为4πr²。对于圆锥，总表面积为πr² + πrl，其中l是斜高。
When asked to find the “curved surface area” of a cylinder, you only calculate 2πrh. Always read the question carefully; it may ask for the total surface area or just the curved part.
当题目要求圆柱的“侧面积”时，你只需计算2πrh。务必仔细读题，可能问的是总表面积或仅侧面积。
6. Volume of 3D Shapes | 三维图形的体积
Volume measures the space inside a 3D object. For a cuboid, V = lwh. For any prism (including a cylinder), the volume is V = Area of cross-section × length.
体积是三维物体内部所占空间的大小。长方体的体积为V = lwh。对于任何棱柱（包括圆柱），体积为V = 底面积 × 高。
Thus, a cylinder has volume πr²h. A triangular prism with base area A and length l has volume A × l. A sphere has volume 4/3 πr³.
因此，圆柱的体积为πr²h。底面积为A、长为l的三棱柱体积为A × l。球的体积为4/3 πr³。
The volume of a cone is 1/3 πr²h, and the volume of a pyramid is 1/3 × base area × height. These “one-third” formulas apply when the shape tapers to a point from a base.
圆锥的体积为1/3 πr²h，棱锥的体积为1/3 × 底面积 × 高。这些“三分之一”公式适用于从底面逐渐变尖到一点的形状。
Do not confuse the slant height l with the perpendicular height h in cones and pyramids. The volume formula always requires the perpendicular height, not the slant height.
不要混淆圆锥和棱锥中的斜高l与垂直高h。体积公式中使用的永远是垂直高，而不是斜高。
7. Working with Composite Shapes | 组合图形的解题技巧
Composite shapes are made by combining basic shapes. When finding the area or volume, break the shape into known parts, calculate each part separately, then add or subtract as required.
组合图形是由基本图形组合而成的。求面积或体积时，将图形拆分成已知的部分，分别计算，然后根据需要进行加法或减法运算。
For example, the area of a semicircle is ½ πr². If a rectangle has a semi-circular cap, the total area is the sum of the rectangle and the semicircle. Conversely, if a circular hole is cut out of a rectangle, subtract the circle’s area from the rectangle’s area.
例如，半圆的面积为½ πr²。如果一个矩形上方有一个半圆盖，总面积等于矩形面积加半圆面积。相反，如果矩形上挖去一个圆洞，则从矩形面积中减去圆的面积。
In 3D, for a cylinder with a cone on top (like a ice cream cone), the total volume is πr²h_cylinder + 1/3 πr²h_cone. When dealing with compound volumes, draw a clear diagram and label all known dimensions before starting.
在三维情形下，对于顶部有圆锥的圆柱体（类似冰激凌蛋筒），总体积为πr²h_圆柱 + 1/3 πr²h_圆锥。处理组合体积时，先画出清晰的示意图并标注所有已知尺寸，再开始计算。
8. Common Pitfalls and Exam Tips | 常见误区与考试技巧
One of the most frequent errors is using inconsistent units. Always convert all dimensions to the same unit before performing multiplication. If a question gives a length in metres and another in centimetres, convert one before calculating.
最常见的错误之一是单位不统一。在执行乘法之前，始终把所有尺寸转换为相同单位。如果题目给的长度有米和厘米混合，先换算再计算。
Another pitfall is confusing the diameter and the radius in circle formulas. A standard circular field may have a diameter of 20 m, but the formula requires the radius, so you must use 10 m. Similarly, the height in a triangle must be perpendicular to the base.
另一个误区是在圆公式中混淆直径和半径。一个标准圆形池塘直径可能是20 m，但公式需要半径，所以必须用10 m。类似地，三角形的高必须垂直于底边。
Finally, always read the question for the required degree of accuracy. The Edexcel IGCSE often asks answers to 1 decimal place or 3 significant figures. If a question says “give your answer in terms of π”, keep π as a symbol, not a decimal. Write your answer step by step, and check your units in the final answer.
最后，务必关注题目对精度的要求。Edexcel IGCSE常要求答案保留1位小数或3位有效数字。如果题目说“答案用π表示”，则保留π符号，不展开成小数。按步骤作答，并在最终答案中检查单位。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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