📚 Surface Area & Volume of Common Solids: Formula Summary | 常见几何体的表面积与体积公式汇总
In IGCSE and A-Level mathematics, calculating the surface area and volume of three-dimensional solids is a core skill tested regularly in both pure and applied papers. A clear command of the standard formulas, their derivations, and their units will help you solve problems quickly and accurately.
在 IGCSE 和 A-Level 数学考试中，计算三维几何体的表面积与体积是高频考点。熟练掌握标准公式、理解其推导过程并注意单位换算，能够帮助你在各类题型中快速而准确地得分。
1. Why Surface Area and Volume Matter | 表面积与体积为何重要
Surface area measures the total external area of a solid and is expressed in square units such as cm² or m². Volume measures the amount of space enclosed by a solid and is expressed in cubic units such as cm³ or m³.
表面积衡量的是几何体外部所有面的总面积，单位为平方单位（如 cm²、m²）；体积衡量的是几何体内部所占空间的大小，单位为立方单位（如 cm³、m³）。
Real-world applications include packaging design, container capacity, paint coverage, and material costing. Exam questions often combine these formulas with ratios, similar figures, or optimisation problems.
实际应用包括包装设计、容器容量、涂料用量以及材料成本估算。考试题常将这些公式与比例、相似图形或最优化问题结合考查。
2. Cuboid and Cube | 长方体与正方体
For a cuboid with length l, width w and height h, the volume is found by multiplying the three dimensions.
对于长、宽、高分别为 l、w、h 的长方体，体积等于三个维度的乘积。
V = l × w × h
The total surface area is the sum of the areas of six rectangular faces.
总表面积等于六个矩形面的面积之和。
S = 2(lw + wh + lh)
edges are equal, so if its edge length is s, the formulas simplify to V = s³ and S = 6s².
正方体是长、宽、高都相等的特殊长方体。若棱长为 s，则体积为 V = s³，表面积为 S = 6s²。
3. Right Prisms | 直棱柱
A prism has a uniform cross-section; for a right prism, the length or height is perpendicular to the base. The volume equals the area of the base multiplied by the height.
棱柱具有统一的横截面；对于直棱柱，高垂直于底面。体积等于底面积乘以高。
V = Abase × h
The surface area of a prism is the sum of the areas of the two identical bases and the lateral faces. The lateral area equals the perimeter of the base multiplied by the height.
棱柱的表面积等于两个相同底面与所有侧面面积之和；侧面积等于底面周长乘以高。
S = 2A_base + Pbase × h
This general rule works for triangular prisms, hexagonal prisms and any other prism you may meet. Always identify the correct base shape before substituting values.
这一通用规则适用于三棱柱、六棱柱等任何棱柱。代入数值前务必先正确识别底面形状。
4. Cylinders | 圆柱
A cylinder is a prism-like solid with circular bases. If its radius is r and its height is h, then its volume is:
圆柱是具有圆形底面的棱柱型几何体。若底面半径为 r，高为 h，则体积为：
V = πr²h
The lateral surface of a cylinder, when unwrapped, forms a rectangle of width h and length equal to the circumference 2πr. Adding the two circular bases gives the total surface area:
圆柱的侧面展开后是一个宽为 h、长为底面周长 2πr 的矩形。加上两个圆形底面后得到总表面积：
S = 2πr² + 2πrh = 2πr(r + h)
For an open cylinder (e.g. a tank without a lid), subtract the missing base: Sopen = πr² + 2πrh.
对于无盖圆柱（如无盖的水箱），需减去缺少的一个底面：Sopen = πr² + 2πrh。
5. Pyramids | 棱锥
A pyramid has a polygonal base and triangular lateral faces that meet at an apex. Its volume is exactly one third of the product of the base area and the perpendicular height.
棱锥有一个多边形底面，侧面为交于顶点的三角形。其体积等于底面积与垂直高乘积的三分之一。
V = (1/3) Abase × h
For surface area, add the base area to the sum of the areas of all triangular faces. Note that the perpendicular height h differs from the sloping (slant) height used in lateral calculations.
求表面积时，将底面积与所有三角形侧面积相加即可。注意：垂直高 h 不同于用于侧面计算的斜高。
6. Cones | 圆锥
A cone is a pyramid-like solid with a circular base. The formula for its volume mirrors that of a pyramid, with the base area πr².
圆锥是具有圆形底面的棱锥型几何体，其体积公式与棱锥类似，底面面积为 πr²。
V = (1/3) πr²h
If l is the slant height, the curved (lateral) surface area is πrl, and the total surface area is:
若斜高为 l，则侧面积为 πrl，总表面积为：
S = πr² + πrl = πr(r + l)
Remember the relationship between r, h and l given by Pythagoras' theorem: l = √r² + h².
切记 r、h、l 之间满足勾股定理：l = √r² + h²。
7. Spheres | 球体
A sphere is perfectly round. Given its radius r, the volume and surface area are given by the following formulas.
球体是完全对称的圆形立体。已知半径 r 时，体积和表面积由以下公式给出。
V = (4/3) πr³
S = 4πr²
/dr = 4πr² = S. This useful relationship occasionally appears in calculus-based questions.
注意：体积对 r 的导数恰好等于表面积，即 dV/dr = 4πr² = S。这一有趣关系偶尔出现在与微积分结合的题目中。
8. Hemispheres | 半球
A hemisphere is half a sphere. Its curved surface area is half that of a sphere, and the flat circular base contributes an extra πr².
半球是球体的一半。其曲面部分面积为球体表面积的一半，再加上一个半径为 r 的圆形底面 πr²。
V = (2/3) πr³
S = 2πr² + πr² = 3πr²
If the hemisphere is hollow or open, only the curved area 2πr² is used. Always read the question twice to check whether the base is included.
若半球为空心或敞口，则只计算曲面面积 2πr²。做题时务必读清题目，确认底面是否计入。
9. Frustums | 圆台与棱台
A frustum is the portion of a cone or pyramid remaining after its top is cut off by a plane parallel to the base. The volume of a conical frustum with lower radius R, upper radius r and height h is:
圆台（或棱台）是用平行于底面的平面截去锥体顶部后剩余的部分。若圆台下底面半径为 R、上底面半径为 r、高为 h，则体积为：
V = (1/3) πh (R² + Rr + r²)
The curved surface area is given by π(R + r)l, where l is the slant height. For a pyramidal frustum, subtract the small pyramid from the large one to find volume.
圆台侧面积为 π(R + r)l，其中 l 为斜高。对于棱台，可用大棱锥体积减去小棱锥体积的方法求解。
When a frustum appears in a question, painting or filling it often requires only the curved surface or the volume, so identify exactly which part is asked for.
遇到台体问题时，常只需求侧面积或体积，请先确定题目要求的究竟是哪个部分。
10. Units and Dimensional Consistency | 单位与量纲一致
Length is measured in mm, cm, m or km. Squaring a length gives area units (mm², cm², m², km²), and cubing gives volume units (mm³, cm³, m³).
长度单位为 mm、cm、m 或 km；长度平方得到面积单位（mm²、cm²、m²、km²），立方则得到体积单位（mm³、cm³、m³）。
1 mL, 1000 cm³ = 1 L, and 1 m³ = 1000 L. Also 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³.
重要换算关系包括：1 cm³ = 1 mL，1000 cm³ = 1 L，1 m³ = 1000 L；同时 1 m² = 10,000 cm²，1 m³ = 1,000,000 cm³。
Always keep all dimensions in the same unit before applying a formula; mixing metres
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