Volume of a Cylinder | 圆柱的体积

📚 Volume of a Cylinder | 圆柱的体积

This article explains how to calculate the volume of a cylinder, a key topic in the Cambridge KS3 mathematics syllabus, directly building on the content from p255 of your textbook. We will derive the formula, walk through worked examples, and highlight common pitfalls to help you master this essential skill.

本文讲解如何计算圆柱的体积，这是剑桥 KS3 数学大纲中的重要主题，内容直接基于教材第 255 页。我们将推导公式，逐步讲解示例，并指出常见错误，帮助你掌握这一关键技能。

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1. Understanding a Cylinder | 认识圆柱

A cylinder is a three‑dimensional shape with two identical circular faces (the bases) and a curved surface connecting them. The two bases lie in parallel planes, and the axis joining the centres of the bases is perpendicular to the bases in a right cylinder – the type we always use at KS3. The key measurements are the radius r of the circular base and the vertical height h between the bases.

圆柱是一种三维图形，有两个完全相同的圆形面（底面）和一个连接底面的曲面。两个底面位于平行平面内，连接底面圆心的轴线垂直于底面——这就是 KS3 阶段始终使用的直圆柱。关键尺寸是圆形底面的半径 r 和两个底面之间的垂直高度 h。

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2. The Volume Formula | 体积公式

The volume V of a cylinder is found by multiplying the area of its circular base by its height. Because the area of a circle is πr², the formula is:

圆柱的体积 V 等于底面积乘以高。因为圆的面积是 πr²，所以公式为：

V = πr²h

Here r is the radius of the base and h is the perpendicular height of the cylinder. This formula works for all right cylinders when the same units are used for r and h.

这里 r 是底面半径，h 是圆柱的垂直高度。当 r 和 h 采用相同单位时，该公式适用于所有直圆柱。

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3. Deriving the Formula – From Prism to Cylinder | 公式推导——从棱柱到圆柱

Think of a cylinder as a circular prism. The volume of any prism is Area of cross-section × length. For a cylinder standing upright, the cross-section is a circle (area = πr²) and the ‘length’ is the height h. So V = πr² × h.

可以把圆柱看作圆形棱柱。任何棱柱的体积都等于横截面积 × 长度。对于直立的圆柱，横截面是圆（面积 = πr²），‘长度’就是高 h。因此 V = πr² × h。

This reasoning links the cylinder formula to the more general volume idea you already know for cuboids and prisms. It also explains why the formula stays the same whether the cylinder is tall or short.

这一推理将圆柱公式与你已经熟悉的长方体和棱柱的体积概念联系起来。这也解释了为什么无论圆柱是高是矮，公式都保持不变。

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4. Units of Volume | 体积单位

Volume is always measured in cubic units. If the radius and height are in centimetres, the volume will be in cubic centimetres (cm³). If the measurements are in metres, the volume is in cubic metres (m³). Other common units are mm³, and for larger capacities we sometimes convert cubic centimetres to litres (1 litre = 1000 cm³).

体积总是以立方单位来度量。如果半径和高以厘米为单位，体积就是立方厘米 (cm³)。如果测量值以米为单位，体积则是立方米 (m³)。其他常见单位还有 mm³，对于较大的容量，我们有时会将立方厘米换算为升（1 升 = 1000 cm³）。

Remember that the unit of volume is the cube of the unit of length. Using the same unit for r and h is essential; if r is given in mm and h in cm, you must first convert to the same unit.

请记住，体积单位是长度单位的立方。r 和 h 必须使用相同的单位；如果 r 用 mm 而 h 用 cm，你必须先统一单位。

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5. Worked Example 1 – Find Volume Given r and h | 示例 1——已知半径和高求体积

Question: A cylinder has base radius 5 cm and height 12 cm. Calculate its volume. Give your answer in terms of π and to 3 significant figures.

问题：一个圆柱的底面半径为 5 cm，高为 12 cm。计算它的体积，分别用含 π 和保留三位有效数字的形式给出答案。

Solution: Using V = πr²h, substitute r = 5 and h = 12.

解答：使用公式 V = πr²h，代入 r = 5, h = 12。

V = π × 5² × 12 = π × 25 × 12 = 300π cm³

The exact volume is 300π cm³. Using π ≈ 3.14 or a calculator, 300π ≈ 300 × 3.14159 = 942.477… ≈ 942 cm³ (3 s.f.).

体积的精确值为 300π cm³。使用 π ≈ 3.14 或计算器，300π ≈ 300 × 3.14159 = 942.477… ≈ 942 cm³（保留三位有效数字）。

Notice that squaring the radius gives cm², and multiplying by the height in cm gives cm³, confirming the unit is correct.

注意，半径平方后得到 cm²，再乘以以 cm 为单位的高便得到 cm³，这验证了单位是正确的。

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6. Worked Example 2 – Find Height Given Volume and r | 示例 2——已知体积和半径求高

Question: A cylinder holds 500π cm³ of water. The base has a radius of 5 cm. What is the depth of the water?

问题：一个圆柱装有 500π cm³ 的水，其底面半径为 5 cm。水的深度是多少？

Solution: The volume of water is the volume of a cylinder with radius 5 cm and unknown height h. Set up the equation:

解答：水的体积等于一个底面半径为 5 cm、高未知的圆柱的体积。列出方程：

π × 5² × h = 500π

Divide both sides by π: 25h = 500, so h = 500 ÷ 25 = 20 cm. The depth is 20 cm.

两边同时除以 π：25h = 500，因此 h = 500 ÷ 25 = 20 cm。水深为 20 cm。

This type of rearrangement is common in both KS3 and later IGCSE papers. Always check that the final unit makes sense.

这种公式变形在 KS3 和后来的 IGCSE 试卷中都很常见。务必检查最终单位是否合理。

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7. Worked Example 3 – Find Radius Given Volume and h | 示例 3——已知体积和高求半径

Question: A cylindrical candle has a volume of 900π cm³ and a height of 25 cm. Find its radius.

问题：一支圆柱形蜡烛的体积为 900π cm³，高 25 cm。求它的半径。

Solution: V = πr²h, so 900π = π × r² × 25. Cancel π: 900 = 25r². So r² = 900 ÷ 25 = 36, hence r = √36 = 6 cm. Only the positive square root is taken because radius is a length.

解答：V = πr²h，所以 900π = π × r² × 25。约去 π：900 = 25r²。因此 r² = 900 ÷ 25 = 36，于是 r = √36 = 6 cm。只取正的平方根，因为半径是长度。

When solving for r, remember that the radius must be positive. Writing the answer with the correct unit is just as important as the number.

当要求解 r 时，记住半径必须为正数。用正确单位写出答案与数值本身同样重要。

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8. Volume of a Hollow Cylinder (Extension) | 空心圆柱的体积（拓展）

A hollow cylinder, such as a pipe, has an outer radius R and an inner radius r. The volume of the material is the difference between the volume of the outer cylinder and the inner cylinder:

空心圆柱（例如水管）有外半径 R 和内半径 r。材料的体积等于外圆柱体积与内圆柱体积之差：

V = πR²h – πr²h = πh(R² – r²)

This is a useful extension that sometimes appears in challenge questions. You can also think of it as πh(R – r)(R + r), but the left-hand form is simpler for calculation.

这是一个有用的拓展，偶尔会出现在挑战题中。你也可以把它写成 πh(R – r)(R + r)，但左边的形式对计算更简单。

At KS3, this is not always required, but it helps to understand how volume formulas can be combined.

在 KS3 阶段这不总是必考内容，但有助于理解如何组合体积公式。

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9. Real-life Applications | 实际应用

Cylindrical shapes appear everywhere: cans, pipes, candles, silos, and even some batteries. Knowing how to calculate the volume helps manufacturers determine how much liquid a can will hold, or how much concrete is needed for a cylindrical pillar.

圆柱形无处不在：罐头、水管、蜡烛、粮仓，甚至某些电池。掌握体积计算有助于制造商确定罐子能装多少液体，或者建造一根圆柱形柱子需要多少混凝土。

When dealing with real objects, always check that the given dimensions are in consistent units before applying the formula. For example, if a can’s radius is given in cm and its capacity is required in litres, convert cm³ to litres at the end.

在处理实际物体时，应用公式前一定要检查所给尺寸的单位是否一致。例如，如果罐子的半径以 cm 给出，而容量要求以升表示，那么应在最后将 cm³ 换算为升。

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10. Common Mistakes to Avoid | 常见错误与避免方法

Mistake 1: Confusing radius and diameter. If the problem gives the diameter, always divide by 2 to find the radius before using V = πr²h.

错误 1：混淆半径与直径。如果题目给出的是直径，在使用 V = πr²h 之前，务必先除以 2 求得半径。

Mistake 2: Using the slant height. For a cylinder, the height is the perpendicular distance between the bases, not the length of the curved side.

错误 2：用了斜高。对于圆柱，高是两底面间的垂直距离，不是侧面的倾斜长度。

Mistake 3: Forgetting to square the radius. The formula contains r², not r. Squaring a number makes a big difference – 5² = 25, not 10.

错误 3：忘记将半径平方。公式中的是 r²，而不是 r。平方一个数会产生很大差异——5² = 25，而不是 10。

Mistake 4: Writing the volume in square units. Volume is always a cubic unit. Check that your final answer ends with ³.

错误 4：用平方单位来表示体积。体积总是立方单位。检查你的最终答案是否带有³。

Taking a moment to check these points after each calculation can save many marks.

每次计算后花一点时间检查这些要点，可以避免丢失很多分数。

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11. Practice Questions with Hints | 练习题与提示

Try these questions to test your understanding. Full solutions are best done on paper before reading the hints.

尝试以下题目来检验你的理解。最好先在纸上完成完整解答，再看提示。

• Q1: A cylinder has radius 7 cm and height 10 cm. Find its volume, leaving your answer in terms of π.
  Hint: V = π × 7² × 10.
  Q1：一个圆柱的半径为 7 cm，高为 10 cm。求它的体积，答案用含 π 的形式表示。
  提示：V = π × 7² × 10。
• Q2: The volume of a cylinder is 200π cm³ and its height is 8 cm. Calculate the radius.
  Hint: 200π = πr² × 8 → r² = ?
  Q2：一个圆柱的体积为 200π cm³，高为 8 cm。计算其半径。
  提示：200π = πr² × 8 → r² = ?
• Q3: A cylindrical water tank has diameter 1.4 m and height 2 m. Find its capacity in litres.
  Hint: radius = 0.7 m, volume in m³, then 1 m³ = 1000 litres.
  Q3：一个圆柱形水箱的直径为 1.4 m，高为 2 m。求它的容量（单位：升）。
  提示：半径 = 0.7 m，体积以 m³ 表示，然后 1 m³ = 1000 升。

Regular practice with these types of problems will make the formula second nature.

经常练习这类题目将使你对公式的运用变得得心应手。

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12. Summary | 总结

The volume of a cylinder is given by V = πr²h. It comes from multiplying the area of the circular base by the perpendicular height. Always use consistent units, remember that r is the radius (not the diameter), and give your final answer in cubic units. Whether you are solving for V, r, or h, setting up the equation clearly and cancelling π when it appears on both sides will lead you to the correct answer.

圆柱的体积公式为 V = πr²h。它来源于圆形底面积乘以垂直高度。务必使用一致的单位，记住 r 是半径（不是直径），并以立方单位给出最终答案。无论你是求 V、r 还是 h，清晰地列出方程，并在两边都出现 π 时将其约去，都能引导你得到正确的答案。

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