IGCSE Maths Mensuration: Cones and Spheres | IGCSE 数学测量：圆锥与球

📚 IGCSE Maths Mensuration: Cones and Spheres | IGCSE 数学测量：圆锥与球

Cones and spheres appear throughout the IGCSE course, from straightforward volume calculations to harder composite-solid and ratio problems. This article brings together the essential formulas, worked examples and exam tips you need.

圆锥与球贯穿 IGCSE 课程的始终，从简单的体积计算到较难的组合体与比例问题都会涉及。本文汇总了你需要掌握的基本公式、例题和考试技巧。

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1. The essential formulae | 必背公式

Before attempting any question, write down the relevant formula from memory. IGCSE papers expect you to know these without a formula sheet.

在开始做题之前，请先凭记忆写出相关公式。IGCSE 试卷要求你在没有公式表的情况下记住这些公式。

Shape	Volume	Surface area
Cone	V = ⅓πr²h	A = πr² + πrl
Sphere	V = 4/3πr³	A = 4πr²
Hemisphere	V = 2/3πr³	Total A = 3πr²

The cone volume formula uses the perpendicular height h, while the cone surface area formula uses the slant height l.

圆锥体积公式使用垂直高度 h，而圆锥表面积公式使用斜高 l。

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2. Cone vocabulary: r, h and l | 圆锥的三个量：r、h 和 l

A cone has three important lengths: the radius r of the circular base, the perpendicular height h from the apex to the centre of the base, and the slant height l along the sloping side.

圆锥有三个重要的长度：底面圆的半径 r、从顶点到底面中心的垂直高度 h，以及沿侧面的斜高 l。

These three quantities are connected by Pythagoras' theorem because the radius and perpendicular height form a right-angled triangle with the slant height.

这三个量由勾股定理联系起来，因为半径、垂直高度和斜高构成直角三角形。

l² = r² + h²

You will often be given two of these three values and must calculate the third before finding the surface area.

考试中常常给出这三个量中的两个，你需要先求出第三个，再计算表面积。

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3. Volume of a cone | 圆锥的体积

The volume of a cone is one third of the volume of a cylinder with the same base radius and perpendicular height.

圆锥的体积等于同底同高的圆柱体积的三分之一。

Always use the perpendicular height h in this formula, not the slant height l.

代入公式时一定要使用垂直高度 h，而不是斜高 l。

V = ⅓πr²h

Example: for r = 3 cm and h = 4 cm, V = ⅓ × π × 3² × 4 = 12π cm³.

例如：当 r = 3 cm、h = 4 cm 时，V = ⅓ × π × 3² × 4 = 12π cm³。

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4. Surface area of a cone | 圆锥的表面积

A cone's total surface area is the curved surface area plus the circular base area.

圆锥的总表面积等于侧面曲面面积加上底面圆面积。

The curved surface area is πrl, where l is the slant height. The base area is πr².

侧面曲面面积为 πrl，其中 l 是斜高；底面积为 πr²。

A = πr² + πrl = πr(r + l)

If the question asks for the curved surface area only, omit the πr² term.

如果题目只要求侧面积，就去掉 πr² 这一项。

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5. Volume of a sphere | 球的体积

The volume of a sphere depends only on its radius r.

球的体积只取决于它的半径 r。

V = 4/3πr³

For a sphere of radius 6 cm, V = 4/3 × π × 6³ = 288π cm³.

对于半径为 6 cm 的球，V = 4/3 × π × 6³ = 288π cm³。

Be careful to cube the radius before multiplying by 4/3 and π.

计算时注意先对半径求立方，再乘以 4/3 和 π。

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6. Surface area of a sphere | 球的表面积

The surface area formula is surprisingly simple and is worth memorising exactly.

球的表面积公式简单得出奇，值得准确记忆。

A = 4πr²

For r = 6 cm, A = 4 × π × 6² = 144π cm².

当 r = 6 cm 时，A = 4 × π × 6² =

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