📚 IGCSE Maths Topic 5: Mensuration - Area, Perimeter, Volume and Surface Area | IGCSE 数学 第5单元：测量法——面积、周长、体积与表面积
Mensuration is one of the most formula-driven topics in IGCSE Mathematics. It covers perimeter, area, surface area and volume of common 2D and 3D shapes, and it regularly appears in both non-calculator and calculator papers.
测量法是 IGCSE 数学中公式运用最多的主题之一。它涵盖常见二维和三维图形的周长、面积、表面积与体积，经常在非计算器和计算器试卷中出现。
This article focuses on the key formulas, unit conversions and exam strategies you need for Topic 5 Mensuration. Each section pairs an English explanation with a Chinese explanation so you can study both versions side by side.
本文聚焦第5单元测量法的核心公式、单位换算和考试策略。每个小节都先给出英文讲解，再给出中文讲解，方便你对照学习。
The topic rewards accuracy rather than cleverness: if you know the formula, convert units correctly and round at the right moment, the marks come quickly. Most lost marks come from using the slanted side as a height, mixing cm with m, or leaving a value rounded too early.
这个主题奖励的是准确而不是聪明：只要公式记对、单位换算正确、在正确的时机取近似，分数来得很快。失分大多来自把斜边当成高、把 cm 和 m 混用，或者过早四舍五入。
1. What Mensuration Covers and How It Is Examined | 测量法的考查范围与命题方式
Mensuration questions ask you to find a length, an area, a volume or a surface area, or to work backwards from one of these to find a missing dimension. They appear as short two-mark questions and as multi-step structured questions worth six marks or more.
测量法题目要求你求长度、面积、体积或表面积，或者反过来由这些量求未知的边长。它们既会以 2 分的短题出现，也会以 6 分以上的多步结构题出现。
nd the numbers are chosen to cancel nicely. In a calculator paper you use the π button and you are normally expected to round your final answer to 3 significant figures unless the question says otherwise.
在非计算器试卷中，题目通常会告诉你取 π = 3.142 或 π = 22 ÷ 7，数字是精心设计过的、便于约分。在计算器试卷中你可以用 π 键，通常要求把最终答案保留 3 位有效数字，除非题目另有说明。
Some boards print a short formula sheet on the paper and some do not, so check your own specification. Either way, you must know the area formulas for a triangle, parallelogram and trapezium, the volume formulas for a prism, cylinder, pyramid, cone and sphere, and the surface area formulas for a cylinder and a sphere.
有些考试局会在试卷上给出简短的公式表，有些则不给，所以要核对你自己的考纲。无论如何，你必须掌握三角形、平行四边形、梯形的面积公式，棱柱、圆柱、棱锥、圆锥、球的体积公式，以及圆柱和球的表面积公式。
- Always write the formula you are using before substituting numbers. Method marks are awarded even if the arithmetic goes wrong.
- Always give units with your answer, and use the units the question used: cm² for area, cm³ for volume.
- Do not round intermediate values. Keep full calculator accuracy and round only the final answer.
- 代入数字前先写出所用公式。即使算错，公式分依然可以拿到。
- 答案必须带单位，并且使用题目给定的单位：面积用 cm²，体积用 cm³。
- 中间步骤不要取近似。保留计算器的完整精度，只对最终答案取近似。
2. Perimeter and Area of 2D Shapes | 二维图形的周长与面积
Perimeter is the total distance around the outside of a shape. Area measures the space inside a closed shape and is always expressed in square units such as cm², m² or mm².
周长是围绕图形外部的总距离。面积测量的是封闭图形内部的空间，单位始终是平方单位，例如 cm²、m² 或 mm²。
½(a + b)h. In every triangle and parallelogram, height must be perpendicular to the base.
在 IGCSE 核心中，你必须掌握以下面积公式：长方形 A = lw、三角形 A = ½bh、平行四边形 A = bh、梯形 A = ½(a + b)h。在三角形和平行四边形中，高必须与底边垂直。
Do not confuse the slanted side of a triangle or parallelogram with its height. The perpendicular height is the shortest distance from the base to the opposite vertex or side.
不要把三角形或平行四边形的斜边误认为高。垂直高是从底边到对顶点或对边的最短距离。
Rectangle: P = 2(l + w), A = lw | Triangle: A = ½bh | Parallelogram: A = bh | Trapezium: A = ½(a + b)h
For a trapezium, a and b are the two parallel sides and h is the perpendicular distance between them; it does not matter which parallel side you call a and which you call b, because you add them. For a compound shape, add the perimeters of the outer edges only - internal dividing lines are not part of the perimeter.
对于梯形，a 和 b 是两条平行边，h 是它们之间的垂直距离；哪条平行边叫 a、哪条叫 b 无所谓，因为你要把它们相加。对于组合图形，只把外边缘的周长相加——内部分割线不属于周长。
3. Circles: Circumference, Area, Arcs and Sectors | 圆：周长、面积、弧与扇形
a is A = πr². If a question gives the diameter, divide by 2 before using the area formula.
圆的周长就是它的周界。使用 C = 2πr 或 C = πd 均可，因为直径是半径的两倍，两者结果相同。面积公式是 A = πr²。如果题目给出的是直径，在使用面积公式前要先除以 2。
er, substitute π = 3.142 in a non-calculator paper or use the π button on your calculator.
许多 IGCSE 题目要求给出含 π 的精确答案。此时答案中要保留 π，例如 36π cm²。如果要求小数答案，在非计算器试卷中代入 π = 3.142，或用计算器上的 π 键。
A semicircle has area ½πr² and a curved edge of length πr. A quarter circle has area ¼πr² and a curved edge of length ½πr, and its perimeter also includes the two radii.
半圆的面积是 ½πr²，曲边长 πr。四分之一圆的面积是 ¼πr²，曲边长 ½πr，其周长还要加上两条半径。
Arc length = (θ ÷ 360) × 2πr | Sector area = (θ ÷ 360) × πr²
or is a quarter of the circle, so the fraction θ ÷ 360 must be ¼. If a question gives the arc length and asks for the angle, rearrange the arc formula rather than guessing.
这里的 θ 是圆心角的度数。快速自检：如果 θ = 90，扇形是圆的四分之一，所以 θ ÷ 360 必须等于 ¼。如果题目给出弧长要求圆心角，请对弧长公式进行变形，不要凭感觉猜。
4. Unit Conversions for Length, Area and Volume | 长度、面积与体积的单位换算
Unit conversion is where many otherwise correct answers lose a mark. The rule is simple: linear conversions are multiplied by 10, 100 or 1000, area conversions are squared, and volume conversions are cubed.
单位换算是许多本来正确的答案丢掉一分的地方。规则很简单：长度换算乘以 10、100 或 1000，面积换算要平方，体积换算要立方。
| Quantity | Conversion |
| Length | 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m |
| Area | 1 cm² = 100 mm², 1 m² = 10 000 cm² |
| Volume | 1 cm³ = 1000 mm³, 1 m³ = 1 000 000 cm³ |
| Capacity | 1 litre = 1000 ml = 1000 cm³, 1 m³ = 1000 litres |
Before you substitute into any formula, convert every length into the same unit. A common trap is a question that gives a length in metres and a width in centimetres; convert first, then multiply.
在代入任何公式之前，先把所有长度换成同一单位。常见的陷阱是题目给出的长用米、宽用厘米；先换算，再相乘。
The capacity link is worth memorising because it is used in real-life questions about tanks, bottles and containers. A cuboid measuring 20 cm by 10 cm by 5 cm has volume 1000 cm³, which is exactly 1 litre.
体积与容积的换算关系值得背下来，因为水箱、瓶子和容器类的应用题常会用到。一个 20 cm × 10 cm × 5 cm 的长方体体积是 1000 cm³，正好是 1 升。
5. Compound Shapes and Shaded Regions | 组合图形与阴影区域
Most IGCSE area questions are not a single triangle or rectangle but a compound shape. There are two reliable strategies: split the shape into rectangles, triangles and parts of circles, or treat it as a large shape with a smaller shape removed.
大多数 IGCSE 面积题不是单一三角形或长方形，而是组合图形。有两种可靠策略：把图形拆成长方形、三角形和圆的一部分，或者把它看作一个大图形挖去一个小图形。
For a shaded region, first decide which is easier. If the shaded part is an awkward L-shape, subtract: total area minus unshaded area. If the shaded part is a simple shape sitting inside something bigger, add the pieces you can see.
对于阴影区域，先判断哪种方法更简单。如果阴影部分是别扭的 L 形，就用减法：总面积减去非阴影面积。如果阴影部分是放在大图形里的简单图形，就把能看清的部分相加。
- Draw extra lines on the diagram to show how you split the shape.
- Label each section with its own area before adding them up.
- For circle parts, use fractions of πr² and keep π until the last line if an exact answer is required.
- 在图上画辅助线，表明你是如何分割图形的。
- 先给每一块标出各自的面积，再相加。
- 涉及圆的部分时，用 πr² 的分数形式；若要求精确答案，就把 π 保留到最后一步。
A useful habit is to estimate the answer first. If a shape fits inside a 10 cm by 10 cm square, no shaded area can exceed 100 cm²; an answer of 250 cm² tells you immediately that something has gone wrong.
一个有用的习惯是先估算答案。如果图形能放进一个 10 cm × 10 cm 的正方形里，任何阴影面积都不可能超过 100 cm²；如果算出 250 cm²，你立刻就知道出错了。
6. Volume of Prisms and Cylinders | 棱柱与圆柱的体积
A prism is any solid with a constant cross-section along its length. The volume of any prism is the cross-sectional area multiplied by the length, which is why the same formula covers cuboids, triangular prisms and cylinders.
棱柱是指沿着长度方向横截面保持不变的立体。任何棱柱的体积都等于横截面积乘以长度，因此同一个公式适用于长方体、三棱柱和圆柱。
Prism: V = Across-section × length | Cuboid: V = lwh | Cylinder: V = πr²h
For a cylinder, the cross-section is a circle, so the volume is πr² multiplied by the height. Remember that the height must be measured perpendicular to the circular face; if the cylinder is lying on its side, the height is the length along the table.
对于圆柱，横截面是圆，因此体积等于 πr² 乘以高。记住高必须垂直于圆面测量；如果圆柱侧躺，那么"高"就是沿桌面方向的长度。
For a triangular prism, find the area of the triangle first, then multiply by the length of the prism. Do not multiply the three side lengths together - that gives a different, meaningless number.
对于三棱柱，先求三角形的面积，再乘以棱柱的长度。不要把三条边长直接相乘——那得到的是另一个没有意义的数。
If a question asks for capacity, convert your volume into litres at the end. Volume in cm³ divided by 1000 gives litres, so 2500 cm³ is 2.5 litres.
如果题目问容积，最后要把体积换算成升。以 cm³ 为单位的体积除以 1000 得到升，所以 2500 cm³ 就是 2.5 升。
7. Pyramids, Cones and Spheres: Volume and Surface Area | 棱锥、圆锥与球：体积与表面积
Pyramids and cones taper to a point, so their volume is one third of the prism or cylinder that would just contain them.
棱锥和圆锥收拢到一个顶点，因此它们的体积等于刚好能容纳它们的棱柱或圆柱体积的三分之一。
Pyramid: V = ⅓ × base area × h | Cone: V = ⅓πr²h | Sphere: V = (4 ÷ 3)πr³
n that gives the slant height l, find the perpendicular height first using Pythagoras: h = √l² − r².
垂直高 h 是从顶点竖直向下量到基座中心的距离，不是沿着斜边的长度。在给出斜高 l 的圆锥题中，先用毕达哥拉斯定理求垂直高：h = √l² − r²。
face of πrl plus a circular base of πr², so its total surface area is πrl + πr². A pyramid's surface area is the base area plus the total area of all its triangular faces.
表面积方面，球最简单：SA = 4πr²。圆锥的曲面面积为 πrl，加上底的面积 πr²，总表面积是 πrl + πr²。棱锥的表面积等于底面积加上所有三角形侧面的面积之和。
- Sphere: SA = 4πr², V = (4 ÷ 3)πr³.
- Cone: curved SA = πrl, total SA = πrl + πr², V = ⅓πr²h.
- Hemisphere: curved SA = 2πr², total SA = 3πr², V = (2 ÷ 3)πr³.
- 球：SA = 4πr²，V = (4 ÷ 3)πr³。
- 圆锥：侧面积 = πrl，总表面积 = πrl + πr²，V = ⅓πr²h。
- 半球：曲面积 = 2πr²，总表面积 = 3πr²，V = (2 ÷ 3)πr³。
8. Surface Area of Prisms and Cylinders | 棱柱与圆柱的表面积
Surface area is the total area of every face of a solid. The safest method is to draw the net, label each face with its area, and add them all up.
表面积是立体所有面的面积总和。最稳妥的方法是画出展开图，给每个面标上面积，然后把它们全部相加。
For a cuboid, the surface area is 2(lw + lh + wh), because there are three pairs of identical faces. For a closed cylinder, the two circular ends contribute 2πr² and the curved surface, when unrolled, is a rectangle of width 2πr and height h.
对于长方体，表面积是 2(lw + lh + wh)，因为有三对完全相同的面。对于封闭圆柱，两个圆形端面贡献 2πr²，而展开后的曲面是一个宽为 2πr、高为 h 的长方形。
Cuboid: SA = 2(lw + lh + wh) | Closed cylinder: SA = 2πr² + 2πrh
Read the question carefully to see whether the cylinder is open at one or both ends. A pipe or a tin without a lid removes one or both circles, so you would use 2πrh alone, or πr² + 2πrh.
仔细读题，判断圆柱是一端开口还是两端开口。一根管子或一个没有盖子的罐头会去掉一个或两个圆面，此时你只需用 2πrh，或者 πr² + 2πrh。
9. Similar Shapes and Scale Factors | 相似图形与比例因子
Two shapes are similar if one is an enlargement of the other, with every corresponding length in the same ratio. That ratio is the linear scale factor k.
如果两个图形中一个是另一个的放大，且所有对应长度之比相同，它们就是相似图形。这个比值就是长度比例因子 k。
Length: k | Area: k² | Volume: k³
If two similar solids have a length ratio of 3, then their areas are in the ratio 9 and their volumes are in the ratio 27. This is the fastest route through many 3D mensuration questions, because it avoids recalculating everything from scratch.
如果两个相似立体的长度比是 3，那么它们的面积比是 9，体积比是 27。这是许多三维测量题最快的解法，因为它省去了从头重算的麻烦。
To move from area back to length, take the square root of the area ratio. To move from volume back to length, take the cube root of the volume ratio. If two similar containers hold 8 litres and 27 litres, the length ratio is the cube root of 27 ÷ 8, which is 3 ÷ 2.
要从面积回到长度，取面积比的平方根；要从体积回到长度，取体积比的立方根。如果两个相似的容器分别能装 8 升和 27 升，长度比就是 27 ÷ 8 的立方根，即 3 ÷ 2。
10. Density, Mass and Volume Problems | 密度、质量与体积的应用题
Some IGCSE mensuration questions link volume to mass through density. Density is mass per unit volume, and the triangle formula is easy to remember.
有些 IGCSE 测量法题目通过密度把体积与质量联系起来。密度是单位体积的质量，这个三角关系式很好记。
Density = Mass ÷ Volume | Mass = Density × Volume | Volume = Mass ÷ Density
Typical units are g/cm³ for solids and liquids, and kg/m³ for gases and large quantities. If the density is in g/cm³, keep the volume in cm³; if it is in kg/m³, convert the volume to m³ first.
常见的单位是固体和液体用 g/cm³，气体和大量物质用 kg/m³。如果密度以 g/cm³ 给出，体积就要保持 cm³；如果密度以 kg/m³ 给出，要先把体积换算成 m³。
- Step 1: calculate the volume of the solid using the correct mensuration formula.
- Step 2: choose the correct rearrangement of the density relationship.
- Step 3: check that the units of density and volume match before substituting.
- 第一步：用正确的测量公式求出立体的体积。
- 第二步：选择密度关系的正确变形形式。
- 第三步：代入前检查密度与体积的单位是否匹配。
11. Working Backwards to Find a Missing Dimension | 逆运算求未知边长
Many higher-tier questions give you the area or volume and ask for a length, radius or height. The method is always the same: write the formula, substitute the known values, then rearrange.
许多高阶题目给出面积或体积，要求你求长度、半径或高。方法始终相同：写出公式，代入已知量，然后变形求解。
² and you need the radius, start from A = πr², so r² = A ÷ π = 50 ÷ 3.142 = 15.91, and r = √15.91 = 3.99 cm. Keep the full value on your calculator display and round only at the end.
如果一个圆的面积是 50 cm²，要求半径，就从 A = πr² 出发，得 r² = A ÷ π = 50 ÷ 3.142 = 15.91，所以 r = √15.91 = 3.99 cm。把完整数值留在计算器上，只在最后一步取近似。
es. Always sanity-check: a height of 6.37 cm for a 500 cm³ cylinder with a 5 cm radius is reasonable, because the base area is about 78.5 cm².
如果一个圆柱的体积是 500 cm³、半径是 5 cm，那么 500 = π × 5² × h，所以 h = 500 ÷ (25π) = 6.37 cm（3 位有效数字）。一定要合理性检查：底面积约 78.5 cm²，半径为 5 cm 的圆柱要装下 500 cm³，高约 6.37 cm 是合理的。
When the unknown appears squared or cubed, undo it with a square root or cube root at the end, and remember that a length must be positive.
当未知量是平方或立方形式时，最后要用平方根或立方根还原，并牢记长度必须取正值。
12. Exam Technique and Common Errors | 考试技巧与常见错误
Nearly all lost marks in mensuration come from a small number of recurring mistakes, and every one of them is avoidable with a routine.
测量法中几乎所有的失分都来自少量反复出现的错误，而只要养成习惯，每一个都可以避免。
- Using the slanted side as the height of a triangle, parallelogram or cone.
- Mixing units, such as multiplying a length in metres by a width in centimetres.
- Forgetting to divide the diameter by 2 before using A = πr².
- Rounding mid-calculation, which shifts the final answer outside the accepted range.
- Giving an area in cm³ or a volume in cm², or omitting units altogether.
- Leaving π in the answer when a decimal was requested, or the reverse.
- 把斜边当成三角形、平行四边形或圆锥的高。
- 单位混用，例如用米作单位的长度乘以用厘米作单位的宽度。
- 使用 A = πr² 前忘记把直径除以 2。
- 计算中途取近似，导致最终答案超出可接受范围。
- 面积写成 cm³、体积写成 cm²，或者干脆不写单位。
- 题目要求小数答案却保留 π，或者反过来。
A good routine for every mensuration question is: write the
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