📚 IGCSE Maths Mensuration: Area, Volume and Surface Area | IGCSE 数学：测量——面积、体积与表面积
Mensuration is one of the most practical and formula-driven topics in IGCSE Mathematics. It deals with measuring lengths, areas, surface areas and volumes of 2D and 3D shapes, and it appears regularly on both Core and Extended papers.
测量是 IGCSE 数学中非常实用且以公式为核心的主题之一。它涉及二维和三维图形的长度、面积、表面积和体积的计算，在 Core 和 Extended 试卷中都经常出现。
1. Key Units and Conversions | 关键单位与换算
Before using any formula, check that all lengths are in the same unit. Length is usually measured in mm, cm or m; area is measured in mm², cm² or m²; volume is measured in mm³, cm³ or m³.
在使用任何公式之前，请检查所有长度是否使用同一单位。长度通常以 mm、cm 或 m 为单位；面积以 mm²、cm² 或 m² 为单位；体积以 mm³、cm³ 或 m³ 为单位。
For volume conversions, cube the length conversion factor, so 1 m³ = 100³ = 1 000 000 cm³.
换算面积单位时，将长度换算系数平方。因为 1 m = 100 cm，所以 1 m² = 100² = 10 000 cm²。换算体积单位时，将长度换算系数立方，因此 1 m³ = 100³ = 1 000 000 cm³。
1 m² = 10 000 cm²
1 m³ = 1 000 000 cm³
2. Area of Rectangles, Triangles and Parallelograms | 矩形、三角形与平行四边形的面积
The area of a rectangle is found by multiplying its length by its width. For a parallelogram, the area is base multiplied by perpendicular height, not the sloping side.
矩形的面积等于长乘以宽。对于平行四边形，面积等于底边乘以垂直高度，而不是乘以斜边。
Area of rectangle = l × w
矩形面积 = 长 × 宽
The area of a triangle is half of the base multiplied by the perpendicular height. The base and height must meet at a right angle.
三角形的面积等于底边乘以垂直高度的一半。底边与高必须成直角。
A = ½ × b × h
3. Area of Trapeziums | 梯形的面积
A trapezium has one pair of parallel sides. Its area is calculated by taking the average of the two parallel sides and multiplying by the perpendicular distance between them.
梯形有一组平行边。其面积等于两条平行边的平均值乘以它们之间的垂直距离。
A = ½ × (a + b) × h
Here a and b are the lengths of the parallel sides, and h is the perpendicular height. Do not use the length of a non-parallel side as the height.
这里 a 和 b 是两条平行边的长度，h 是垂直高度。不要用非平行边的长度作为高。
4. Circles: Circumference and Area | 圆：周长与面积
calculated using the diameter or the radius. The diameter is twice the radius, so d = 2r.
圆的周长可以用直径或半径来计算。直径是半径的两倍，所以 d = 2r。
C = πd = 2πr
The area of a circle is π multiplied by the square of the radius. Always use the radius, not the diameter, when applying this formula.
圆的面积等于 π 乘以半径的平方。使用该公式时始终使用半径，而不是直径。
A = πr²
5. Arc Length and Sector Area | 弧长与扇形面积
A sector is a fraction of a circle, like a slice of pizza. The fraction is determined by the sector angle θ divided by 360°.
扇形是圆的一部分，就像一块披萨。这个比例由扇形角 θ 除以 360° 确定。
For a sector with radius r and angle θ measured in degrees, the arc length is:
对于半径为 r、圆心角为 θ 度（以度为单位）的扇形，弧长为：
Arc length = θ/360 × 2πr
The area of the same sector is:
同一扇形的面积为：
Sector area = θ/360 × πr²
If θ is given in radians, the formulas become rθ and ½r²θ, but IGCSE questions usually use degrees.
如果 θ 以弧度给出，公式变为 rθ 和 ½r²θ，但 IGCSE 题目通常使用度。
6. Surface Area of Prisms and Cylinders | 棱柱与圆柱的表面积
A prism has a constant cross-section. Its total surface area is twice the area of the cross-section plus the perimeter of the cross-section multiplied by the length of the prism.
棱柱的横截面保持不变。其总表面积等于横截面积的两倍加上横截面周长乘以棱柱的长度。
SA = 2 × cross-sectional area + perimeter × length
For a cylinder with radius r and height h, the curved surface area is the circumference of the circular base multiplied by the height, which gives 2πrh.
对于半径为 r、高为 h 的圆柱，侧面积等于圆形底面的周长乘以高，即 2πrh。
The total surface area of a closed cylinder includes two circular ends and the curved surface:
封闭圆柱的总表面积包括两个圆形底面和侧面：
Total SA = 2πr² + 2πrh
If the cylinder is open at one end, add only one circle to the curved surface area.
如果圆柱一端开口，则只需将侧面积加上一个圆的面积。
7. Volume of Prisms and Cylinders | 棱柱与圆柱的体积
The volume of any prism is found by multiplying the area of its cross-section by its length. This works for triangular prisms, cuboids and any other shape with a constant cross-section.
任何棱柱的体积都可以通过横截面积乘以长度来求出。这适用于三棱柱、长方体以及任何横截面不变的形状。
V = area of cross-section × length
For a cylinder, the cross-section is a circle of area πr², so the volume is:
对于圆柱，横截面是面积为 πr² 的圆，因此体积为：
V = πr²h
special prism with a rectangular cross-section, so its volume is length × width × height.
长方体是一种特殊的棱柱，其横截面为矩形，因此其体积为 长 × 宽 × 高。
8. Pyramids, Cones and Spheres | 棱锥、圆锥与球体
A pyramid tapers to a point, so its volume is one third of the base area multiplied by the vertical height. The vertical height must be perpendicular to the base.
棱锥逐渐收尖至一点，因此其体积等于底面积乘以垂直高度的三分之一。垂直高度必须垂直于底面。
V = ⅓ × base area × height
A cone is a circular-based pyramid. Its volume is one third of πr²h, and its curved surface area is πrl, where l is the slant height.
圆锥是以圆为底的锥体。其体积为 πr²h 的三分之一，侧面积为 πrl，其中 l 是母线长。
V = ⅓ × πr²h
Curved SA of cone = πrl
For a sphere of radius r, the volume and surface area are:
对于半径为 r 的球体，其体积和表面积为：
V = 4/3 × πr³
SA = 4πr²
9. Compound Shapes and Real-Life Problems | 复合图形与实际应用题
Many IGCSE mensuration questions involve compound shapes made from rectangles, triangles, circles and sectors. Break the shape into simple parts, calculate each part separately, then add or subtract as needed.
许多 IGCSE 测量题涉及由矩形、三角形、圆和扇形组成的复合图形。将图形拆分成简单部分，分别计算每个部分，然后根据需要相加或相减。
For example, the area of a garden path around a circular pond can be found by subtracting the area of the inner circle from the area of the outer circle.
例如，圆形池塘周围花园小径的面积可以通过外圆面积减去内圆面积来求出。
In real-life problems, look for hidden right-angled triangles in packaging, roofs and 3D solids. Pythagoras' theorem is often needed to find a missing height or slant length.
在实际问题中，要注意包装、屋顶和三维几何体中隐藏的直角三角形。通常需要使用勾股定理来求出缺失的高或斜边。
10. Common Mistakes | 常见错误
question gives the diameter, divide it by 2 before substituting into A = πr² or V = πr²h.
在圆面积和体积公式中使用直径而不是半径是一种非常常见的错误。如果题目给出的是直径，在代入 A = πr² 或 V = πr²h 之前要先除以 2。
Another common mistake is forgetting to square or cube conversion factors when changing area or volume units. For example, 1 cm² is not 10 mm²; it is 100 mm².
另一个常见错误是在换算面积或体积单位时忘记平方或立方换算系数。例如，1 cm² 不是 10 mm²，而是 100 mm²。
Confusing vertical height with slant height in cones and pyramids also loses marks. The slant height is used for surface area of a cone, while the vertical height is used for volume.
混淆圆锥和棱锥的垂直高度与
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