Volume and Surface Area of Spheres | 球的体积与表面积

📚 Volume and Surface Area of Spheres | 球的体积与表面积

In IGCSE Mathematics, you must be confident when calculating the volume and surface area of spheres, hemispheres and composite solids. These topics appear regularly on both Core and Extended papers, often mixed with cones or cylinders.

在 IGCSE 数学中，你必须熟练掌握球体、半球和组合体的体积与表面积计算。此类题目在核心卷和拓展卷中经常出现，并常与圆锥或圆柱结合考查。

1. The Key Formula for a Sphere | 球体的核心公式

A sphere is the set of all points in three-dimensional space at a fixed distance r from the centre. For a sphere of radius r, the two key formulas are:

球体是三维空间中到定点距离为 r 的所有点组成的图形。对于半径为 r 的球体，两个核心公式如下：

V = 4/3 × π × r³

A = 4 × π × r²

Here V stands for volume and A stands for surface area. You should memorise both formulas precisely, as they are not always given on the formula sheet.

这里 V 表示体积，A 表示表面积。你应当准确记住这两个公式，因为它们在公式表中不一定提供。

2. Understanding the Units | 理解单位

Volume is measured in cubic units such as cm³, m³ or mm³. Surface area is measured in square units such as cm², m² or mm². Always check whether the question asks for volume or area; mixing units is a common mistake.

体积以立方单位（如 cm³、m³ 或 mm³）计量，表面积以平方单位（如 cm²、m² 或 mm²）计量。始终检查题目要求的是体积还是表面积；混淆单位是常见错误。

When a question gives measurements in centimetres, the volume will be in cm³ and the surface area in cm². If you convert lengths to metres first, the volume will be in m³ and the area in m².

当题目给出的长度单位是厘米时，体积单位就是 cm³，表面积单位就是 cm²。如果你先把长度换算成米，那么体积单位就是 m³，面积单位就是 m²。

3. Working with Radius and Diameter | 半径与直径的使用

The formulas use radius r. If a question gives diameter d, divide by 2 first: r = d ÷ 2. For example, a diameter of 12 cm gives r = 6 cm.

公式中使用半径 r。如果题目给出直径 d，要先除以 2：r = d ÷ 2。例如，直径为 12 cm，则 r = 6 cm。

A very common error is to substitute the diameter directly into the formula. Always pause and ask: is this value the radius or the diameter?

一个非常常见的错误是把直径直接代入公式。始终停下来问一问：这个数值是半径还是直径？

4. Finding Volume: Step-by-Step | 体积求解：逐步方法

The safest method is to follow a clear sequence when finding the volume of a sphere.

求球体体积时，最稳妥的方法是按照清晰的步骤进行。

Write the formula: V = 4/3 × π × r³. 中文：写出公式：V = 4/3 × π × r³。
Substitute the value of r. 中文：代入半径 r 的值。
Calculate r³ first. 中文：先计算 r³。
Multiply by 4, divide by 3, and multiply by π. 中文：乘 4，除以 3，再乘 π。
Add the correct units, such as cm³. 中文：加上正确单位，如 cm³。

For example, a sphere with radius 3 cm has volume V = 4/3 × π × 3³ = 4/3 × π × 27 = 36π cm³, which is approximately 113.1 cm³.

例如，半径为 3 cm 的球体体积为 V = 4/3 × π × 3³ = 4/3 × π × 27 = 36π cm³，约等于 113.1 cm³。

5. Finding Surface Area: Step-by-Step | 表面积求解：逐步方法

Use the same careful approach for surface area.

表面积的计算也应采用同样谨慎的方法。

Write the formula: A = 4 × π × r². 中文：写出公式：A = 4 × π × r²。
Substitute the value of r. 中文：代入半径 r 的值。
Square r first. 中文：先计算 r²。
Multiply by 4 and π. 中文：乘 4 和 π。
Add the correct units, such as cm². 中文：加上正确单位，如 cm²。

For the same sphere with radius 3 cm, the surface area is A = 4 × π × 3² = 4 × π × 9 = 36π cm², which is approximately 113.1 cm².

对于同一个半径为 3 cm 的球体，表面积为 A = 4 × π × 3² = 4 × π × 9 = 36π cm²，约等于 113.1 cm²。

6. Hemispheres and Composite Solids | 半球与组合体

A hemisphere is half a sphere. Its volume is half the volume of a full sphere, so V = 2/3 × π × r³.

半球是半个球体。它的体积是整个球体体积的一半，因此 V = 2/3 × π × r³。

For surface area, the curved part is half the sphere surface area, which is 2πr². If the hemisphere is closed, you must also add the circular base πr². Therefore the total surface area of a closed hemisphere is:

对于表面积，曲面部分是球表面积的一半，即 2πr²。如果半球是封闭的，还必须加上圆形底面 πr²。因此封闭半球的总表面积为：

A = 2πr² + πr² = 3πr²

If the question asks for the curved surface area only, use 2πr². Always read the wording carefully.

如果题目只要求曲面面积，则使用 2πr²。始终仔细阅读题干表述。

7. Exact Answers in Terms of π | 用 π 表示精确答案

Many IGCSE questions ask for an exact answer in terms of π. Do not replace π with 3.14 unless the question asks for a decimal approximation. For example, the exact volume of a sphere with radius 5 cm is 500π/3 cm³.

许多 IGCSE 题目要求用 π 表示精确答案。除非题目明确要求小数近似，否则不要用 3.14 代替 π。例如，半径为 5 cm 的球的精确体积为 500π/3 cm³。

Leaving your answer in terms of π also avoids rounding errors. You can then round at the very end if a decimal value is required.

将答案用 π 表示也能避免舍入误差。如果题目需要小数值，你可以在最后一步再舍入。

8. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Forgetting to cube the radius in the volume formula. 中文：在体积公式中忘记将半径立方。
Using diameter instead of radius. 中文：误用直径而不是半径。
Confusing volume formulas with surface area formulas. 中文：混淆体积公式与表面积公式。
Omitting the base circle when finding the total surface area of a closed hemisphere. 中文：求封闭半球总表面积时遗漏底面圆。
Writing incorrect units, such as cm² for volume or cm³ for area. 中文：写出错误单位，例如体积写成 cm²，面积写成 cm³。

To avoid these errors, write down the formula first, identify r clearly, and check the units at the end.

为了避免这些错误，请先写下公式，明确 r 的值，并在最后检查单位。

9. Exam-Style Worked Example 1 | 考试型例题 1

A solid sphere has a diameter of 10 cm. Find the exact volume and surface area in terms of π.

一个实心球体的直径为 10 cm。求用 π 表示的精确体积和表面积。

Step 1: find the radius. r = d ÷ 2 = 10 ÷ 2 = 5 cm.

步骤 1：求半径。r = d ÷ 2 = 10 ÷ 2 = 5 cm。

Step 2: calculate volume. V = 4/3 × π × 5³ = 4/3 × π × 125 = 500π/3 cm³.

步骤 2：计算体积。V = 4/3 × π × 5³ = 4/3 × π × 125 = 500π/3 cm³。

Step 3: calculate surface area. A = 4 × π × 5² = 4 × π × 25 = 100π cm².

步骤 3：计算表面积。A = 4 × π × 5² = 4 × π × 25 = 100π cm²。

10. Exam-Style Worked Example 2 | 考试型例题 2

A closed hemisphere has radius 4 cm. Find its total surface area and volume. Give exact answers in terms of π.

一个封闭半球的半径为 4 cm。求其总表面积

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Volume and Surface Area of Spheres | 球的体积与表面积