📚 Volume and Surface Area of Prisms and Cylinders | 棱柱与圆柱的体积和表面积
Volume and surface area are core mensuration skills in IGCSE Mathematics. A clear understanding of cross-sections, units and formula rearrangement will help you answer both straightforward and multi-step questions confidently.
体积和表面积是 IGCSE 数学中测量学的核心技能。清楚理解横截面、单位以及公式变形，将帮助你自信地解答直接和多步骤题目。
This revision guide covers the definitions of a prism and a cylinder, the standard volume and surface area formulas, composite and hollow solids, reverse calculations, unit conversions and the exam mistakes that cost marks most often. Every formula is developed from first principles so that you never have to rely on memory alone.
本篇复习指南涵盖棱柱与圆柱的定义、标准体积与表面积公式、组合体与空心立体、逆向计算、单位换算，以及最容易失分的考试错误。每个公式都从基本原理推导，因此你不必只靠死记硬背。
1. Key Definitions and Units | 核心定义与单位
A prism is a three-dimensional solid with the same cross-section from one end to the other. The end faces are congruent, and the length is the perpendicular distance between them.
棱柱是一种从一端到另一端横截面都相同的三维立体。端面全等，长度是两端面之间的垂直距离。
A cylinder is a prism with a circular cross-section. Its radius r is the distance from the centre to the circumference, and its height h is the perpendicular distance between the circular faces.
圆柱是一种横截面为圆形的棱柱。它的半径 r 是从圆心到圆周的距离，高 h 是两个圆形面之间的垂直距离。
Volume measures the space inside a solid and uses cubic units. Surface area measures the total area of all outer faces and uses square units.
体积测量的是立体内部的空间，使用立方单位。表面积测量的是所有外表面的总面积，使用平方单位。
Prisms are named after their cross-section: a triangular prism, a rectangular prism (cuboid), a pentagonal prism, a hexagonal prism and so on. The cross-section must stay exactly the same all the way along, so a pyramid and a cone are not prisms, and neither is a sphere.
棱柱按横截面命名：三棱柱、矩形棱柱（长方体）、五棱柱、六棱柱等。横截面必须沿全长保持完全一致，因此棱锥和圆锥都不是棱柱，球也不是。
Two useful vocabulary points: a face is a flat surface, an edge is where two faces meet, and a vertex is a corner. For a cylinder the curved surface is not a face in the strictest sense, but exam questions still talk about the curved surface area.
两个常用术语：面是平面，棱是两个面相交的线，顶点是角。对圆柱而言，曲面严格来说不算平面意义上的面，但考试中仍然会讨论曲面面积。
| Unit conversion | 单位换算 |
|---|---|
| 1 m = 100 cm | 1 米 = 100 厘米 |
| 1 m² = 10 000 cm² | 1 平方米 = 10 000 平方厘米 |
| 1 m³ = 1 000 000 cm³ | 1 立方米 = 1 000 000 立方厘米 |
| 1 litre = 1000 cm³ | 1 升 = 1000 立方厘米 |
| 1 m³ = 1000 litres | 1 立方米 = 1000 升 |
2. Volume of a Prism | 棱柱的体积
The volume of any prism is found by multiplying the area of its cross-section by its length. Because the cross-section never changes, the solid is effectively a stack of identical slices.
任何棱柱的体积都等于横截面积乘以长度。由于横截面始终不变，这个立体实际上就是由无数个相同薄片堆叠而成。
V = A × l
Here V is the volume, A is the area of the cross-section and l is the perpendicular length between the two end faces. The cross-section is usually a triangle, a rectangle, a trapezium or a regular polygon, so you may need to recall those area formulas first.
其中 V 是体积，A 是横截面积，l 是两个端面之间的垂直长度。横截面通常是三角形、矩形、梯形或正多边形，因此你可能需要先回忆这些图形的面积公式。
long. The cross-sectional area is ½ × 6 × 4 = 12 cm², so the volume is 12 × 10 = 120 cm³.
例题：一个三棱柱的横截面是底 6 cm、垂直高 4 cm 的三角形，柱长 10 cm。横截面积是 ½ × 6 × 4 = 12 cm²，所以体积是 12 × 10 = 120 cm³。
This is reassuring: the familiar cuboid formula is a special case of the general prism formula, not a separate rule.
长方体其实就是矩形棱柱，因此同一个公式给出 V = l × w × h。这很令人安心：熟悉的长方体公式只是一般棱柱公式的特例，而不是另一条独立规则。
3. Surface Area of a Prism | 棱柱的表面积
The surface area of a prism has two parts: the two congruent end faces, and the lateral surface, which unrolls into a rectangle whose width is the perimeter of the cross-section and whose length is l.
棱柱的表面积由两部分组成：两个全等的端面，以及侧面。侧面展开后是一个矩形，其宽为横截面的周长，长为 l。
SA = P × l + 2A
e cross-section. The term P × l is the lateral surface area and 2A accounts for both ends.
公式中 P 是横截面的周长，l 是长度，A 是横截面积。P × l 这一项是侧面积，2A 对应两个端面。
Using the triangular prism from the last section: the other two sides of the triangle are found by Pythagoras, since it is a right-angled triangle with legs 6 cm and 4 cm. The hypotenuse is √6² + 4² = √52 eral area is 17.21 × 10 = 172.1 cm², and adding the two ends gives 172.1 + 24 = 196.1 cm².
继续用上一节的三棱柱：由于它是直角边为 6 cm 和 4 cm 的直角三角形，另外两条边可用勾股定理求出。斜边为 √6² + 4² = √52 ≈ 7.21 cm，所以周长约为 6 + 4 + 7.21 = 17.21 cm。侧面积为 17.21 × 10 = 172.1 cm²，再加上两个端面得 172.1 + 24 = 196.1 cm²。
Always check whether the question wants the total surface area or only the lateral or curved surface area. A lidless box, for example, has only one end face, so you would add A rather than 2A.
务必检查题目问的是总表面积，还是只有侧面（曲面）面积。例如一个无盖的盒子只有一个端面，此时应加 A 而不是 2A。
4. Volume of a Cylinder | 圆柱的体积
Because a cylinder is a prism whose cross-section is a circle, the general prism formula becomes the familiar cylinder formula.
因为圆柱是横截面为圆的棱柱，一般棱柱公式就变成了熟悉的圆柱公式。
V = πr²h
The area of the circular cross-section is πr², and it is multiplied by the height h. Volume is always in cubic units: cm³, m³, mm³.
圆形横截面的面积是 πr²，再乘以高 h。体积单位永远是立方单位：cm³、m³、mm³。
en V = π × 4² × 15 = 240π ≈ 754 cm³. Leave the answer in terms of π when the question says "give your answer in terms of π", and use π ≈ 3.142 otherwise.
例题：一个罐子半径 4 cm、高 15 cm。则 V = π × 4² × 15 = 240π ≈ 754 cm³。当题目要求"用 π 表示答案"时，就把答案写成含 π 的形式；否则用 π ≈ 3.142 计算。
is 10 cm, then r = 5 cm, and forgetting to halve it makes the volume four times too large.
一个常见陷阱是题目给出的是直径而不是半径。若直径是 10 cm，则 r = 5 cm；忘记除以 2 会使体积变成正确的四倍。
5. Surface Area of a Cylinder | 圆柱的表面积
Cut a hollow cylinder down one side and unroll it: the curved surface becomes a rectangle of width 2πr (the circumference) and height h. Add the two circular ends of area πr² each.
把空心圆柱沿一条母线剪开并展开：曲面变成一个宽为 2πr（圆周长）、高为 h 的矩形。再加上两个面积为 πr² 的圆形端面。
SA = 2πrh + 2πr² = 2πr(r + h)
The term 2πrh is the curved surface area, and 2πr² is the area of the two circular ends combined. The factorised form 2πr(r + h) is quicker to evaluate and easier to remember.
其中 2πrh 是曲面面积，2πr² 是两个圆形端面面积之和。因式分解形式 2πr(r + h) 计算更快、也更容易记住。
.5 cm². Notice that the numerical value of the lateral area is larger than the ends here because the can is tall compared with its radius.
对于 r = 4 cm、h = 15 cm 的罐子：SA = 2π × 4 × 15 + 2π × 4² = 120π + 32π = 152π ≈ 477.5 cm²。注意这里侧面积的数值比端面大，因为罐子相对于半径来说比较高。
A solid cylinder has two ends, so it uses 2πr². A pipe or a tube that is open at both ends has no ends, so its surface area is just 2πrh. Read the wording carefully: "closed at one end" means adding only πr².
实心圆柱有两个端面，所以用 2πr²。两端开口的管子没有端面，表面积就只有 2πrh。要仔细读题："一端封闭"意味着只需加 πr²。
6. Composite and Hollow Solids | 组合体与空心立体
Many exam questions combine prisms and cylinders, or drill a hole through a solid. The strategy is always the same: split the solid into simple pieces, then add or subtract.
许多考题会把棱柱与圆柱组合，或者在立体中打一个洞。策略始终一样：把立体拆成简单部分，然后相加或相减。
For composite solids, add the volumes of the separate parts to get the total volume. Add the surface areas only for faces that are actually on the outside, because hidden internal faces do not count.
对于组合体，把各部分体积相加得到总体积。表面积只计算真正位于外部的面，因为内部隐藏的面不计入。
ter radius R and inner radius r, both with height h, has volume πR²h − πr²h = πh(R² − r²).
对于空心立体，用外部立体减去内部立体。一个外半径 R、内半径 r、高均为 h 的空心管，体积为 πR²h − πr²h = πh(R² − r²)。
A metal washer is a nice two-dimensional version of the same idea: its area is πR² − πr², and the volume is that area times the thickness. Recognising this saves a great deal of time.
金属垫圈是同一思路的二维版本：它的面积是 πR² − πr²，体积就是该面积乘以厚度。识别出这一点可以节省大量时间。
7. Working Backwards: Rearranging the Formulas | 逆向计算：公式变形
Questions often give the volume or the surface area and ask for a missing dimension. You must rearrange the formula before substituting numbers.
题目常常给出体积或表面积，要求求出缺失的尺寸。你必须先对公式变形，再代入数值。
| Unknown | Formula rearranged | 待求量 |
|---|---|---|
| Height h | h = V ÷ (πr²) | 高 h |
| Radius r | r = √(V ÷ (πh)) | 半径 r |
| Prism length l | l = V ÷ A | 棱柱长度 l |
54 ≈ 6.37 cm. Always keep a few extra digits during the working and round only at the end.
例题：某圆柱体积为 500 cm³，半径为 5 cm。则 h = 500 ÷ (π × 5²) = 500 ÷ 78.54 ≈ 6.37 cm。计算过程中始终多保留几位小数，只在最后一步取近似值。
When the unknown is the radius, remember to take the square root at the end. Forgetting the square root is one of the most frequent errors in this topic.
当未知量是半径时，记住最后要开平方。忘记开平方是本专题中最常见的错误之一。
8. Unit Conversions and Consistency | 单位换算与一致性
Never mix units inside a single calculation. Convert everything to the same unit first, then compute, then give the answer in the unit the question asks for.
绝不要在同一个计算中混用单位。先把所有量换算成同一单位，再计算，最后按题目要求的单位给出答案。
Lengths convert by powers of 10, areas by powers of 100 and volumes by powers of 1000. This is because a volume has three length dimensions, so each one must be converted separately.
长度按 10 的幂换算，面积按 100 的幂换算，体积按 1000 的幂换算。这是因为体积有三个长度维度，每一个都要单独换算。
1 m³ = 1000 dm³ = 1 000 000 cm³ = 1000 litres
m³, which you can immediately use in V = πr²h after converting any lengths to centimetres.
容积问题极为常见。若容器能装 2 升，即 2000 cm³，把所有长度换算为厘米后就可以直接代入 V = πr²h。
Watch out for questions that mix metres and centimetres, such as a cylinder of radius 20 cm and height 1.5 m. Convert the height to 150 cm first, or convert the radius to 0.2 m, but never substitute 20 and 1.5 together.
要警惕混用米和厘米的题目，例如半径 20 cm、高 1.5 m 的圆柱。先把高换算成 150 cm，或把半径换算成 0.2 m，但绝不能把 20 和 1.5 直接一起代入。
9. Problem-Solving Strategy and Common Errors | 解题策略与常见错误
A reliable routine for any mensuration question: identify the solid, write down the formula, list the known values with units, convert to consistent units, substitute, then round and state the unit.
任何测量学题目都适用的可靠流程：识别立体类型，写出公式，列出已知量及单位，统一单位，代入计算，最后取近似值并写出单位。
Sketching the solid and labelling the radius, height or length is worth the thirty seconds it takes, because most errors come from misreading the diagram or the text.
画出立体并标注半径、高或长度，值得花上三十秒，因为大多数错误都源于误读图形或文字。
- Using the diameter as the radius, giving a volume four times too large. | 把直径当作半径使用，导致体积变成四倍。
- Forgetting the two circular ends, so the surface area is too small. | 忘记两个圆形端面，导致表面积偏小。
- Mixing cm and m in the same calculation. | 在同一计算中混用厘米和米。
- Rounding π to 3 in the middle of a long calculation. | 在长计算中间把 π 取成 3。
- Giving volume in cm² or area in cm³, which loses the unit mark. | 把体积写成 cm² 或把面积写成 cm³，从而丢掉单位分。
- Forgetting to take the square root when finding a radius. | 求半径时忘记开平方。
10. Worked Exam-Style Examples | 考试风格例题精讲
Example 1. A closed cylindrical tin has radius 7 cm and height 20 cm. Find its volume and its total surface area, giving each answer to 3 significant figures.
例 1：一个封闭的圆柱形铁罐，半径 7 cm、高 20 cm。求它的体积与总表面积，结果均保留 3 位有效数字。
78π ≈ 1190 cm². Keeping the answers in terms of π first makes the arithmetic much cleaner.
体积：V = π × 7² × 20 = 980π ≈ 3080 cm³。表面积：SA = 2π × 7 × 20 + 2π × 7² = 280π + 98π = 378π ≈ 1190 cm²。先保留含 π 的形式，计算会干净得多。
Example 2. A prism 25 cm long has a trapezium cross-section with parallel sides 6 cm and 10 cm and perpendicular distance 4 cm between them. Find the volume.
例 2：一个长 25 cm 的棱柱，横截面为梯形，平行边长分别为 6 cm 和 10 cm，两平行边之间的垂直距离为 4 cm。求体积。
ote that the trapezium formula ½(a + b)h must be used before multiplying by the prism length.
横截面积：A = ½ × (6 + 10) × 4 = 32 cm²。体积：V = 32 × 25 = 800 cm³。注意必须先使用梯形公式 ½(a + b)h，再乘以棱柱长度。
Example 3. A water tank in the shape of a cylinder has a base radius of 0.5 m and a height of 1.2 m. How many litres does it hold when full?
例 3：一个圆柱形水箱，底面半径 0.5 m、高 1.2 m。装满时能装多少升水？
t 942 litres. Working in metres first and converting at the end is safer than converting 0.5 m to 50 cm and 1.2 m to 120 cm, which also works but produces larger numbers.
V = π × 0.5² × 1.2 = 0.3π ≈ 0.942 m³。由于 1 m³ = 1000 升，水箱约可装 942 升水。先用米计算、最后再换算，比把 0.5 m 换成 50 cm、1.2 m 换成 120 cm 更安全；后者虽然也可行，但数字更大、更易出错。
11. Quick Revision Summary | 快速复习总结
Keep this table in mind on the night before the exam. If you can reproduce all four formulas and the two unit rules, most mensuration questions become routine.
考前一晚请记住这张表。只要能默写出全部四个公式和两条单位规则，大多数测量学题目都会变成常规题。
| Quantity | Formula | 量 |
|---|---|---|
| Volume of a prism | V = A × l | 棱柱体积 |
| Surface area of a prism | SA = P × l + 2A | 棱柱表面积 |
| Volume of a cylinder | V = πr²h | 圆柱体积 |
| Surface area of a cylinder | SA = 2πrh + 2πr² | 圆柱表面积 |
Final checklist: identify the solid, choose the formula, unify the units
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