Volume and Surface Area of Cones and Spheres | 圆锥与球的体积和表面积

📚 Volume and Surface Area of Cones and Spheres | 圆锥与球的体积和表面积

In this lesson, we explore two elegant and essential 3D shapes: cones and spheres. You will learn the formulas for their volumes and surface areas, apply them to worked examples, and avoid common exam pitfalls.

本节课我们将探索两个优美且重要的三维图形：圆锥和球。你将学习它们的体积与表面积公式，通过例题加以运用，并避开常见的考试陷阱。

1. What Are Cones and Spheres? | 什么是圆锥和球？

A cone is a three-dimensional shape with a circular base and a single vertex (apex) located directly above the centre of the base. A sphere is a perfectly round 3D shape in which every point on its surface is the same distance from its centre.

圆锥是一种具有圆形底面和唯一顶点的三维图形，该顶点位于底面圆心的正上方。球则是一个完美对称的三维图形，其表面上每一点到球心的距离都相等。

You must understand three key measurements for a cone:

对于圆锥，你必须理解三个关键度量：

• Radius (r) | 半径（r）: the distance from the centre of the circular base to its edge. | 从底面圆心到边缘的距离。
• Perpendicular height (h) | 垂直高（h）: the vertical distance from the apex to the centre of the base. | 从顶点到底面圆心的垂直距离。
• Slant height (l) | 母线长（l）: the distance along the sloping surface from the apex to the edge of the base. | 沿圆锥侧面从顶点到底面边缘的距离。

For a sphere, only the radius r is needed, since a sphere’s height and width are simply twice its radius (the diameter).

对于球，只需要半径 r 就够了，因为球的高和宽都等于直径（半径的两倍）。

2. Volume of a Cone | 圆锥的体积

The volume of a cone is exactly one third of the volume of a cylinder with the same base and height. This is the key relationship to remember.

圆锥的体积恰好是同底等高圆柱体积的三分之一。这是需要牢记的关键关系。

The formula is:

其公式为：

V = ⅓ × π × r² × h

Where V represents volume, r is the

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