📚 IGCSE Mensuration: Area, Surface Area and Volume | IGCSE 测量学：面积、表面积与体积
Mensuration is one of the most practical and heavily examined topics in IGCSE Mathematics. It covers perimeter, area, surface area and volume of common 2D and 3D shapes, together with unit conversions, density and working backwards from known values.
测量学是 IGCSE 数学中最实用、考查频率最高的主题之一。它涵盖常见二维和三维图形的周长、面积、表面积与体积，还涉及单位换算、密度以及根据已知条件反向求解。
This revision guide takes you through the key formulas, worked examples and common pitfalls for the IGCSE syllabus.
本复习指南将带你梳理 IGCSE 大纲中的关键公式、典型例题与常见失分点。
1. Units and Conversions | 单位与换算
Always convert all lengths to the same unit before substituting into a formula. In IGCSE mensuration, common length units include mm, cm, m and km.
在代入公式前，先把所有长度换算成相同单位。IGCSE 测量学常见长度单位包括 mm、cm、m 和 km。
1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m
For area and volume, the conversion factor must be squared or cubed. This is because area is two-dimensional and volume is three-dimensional.
面积和体积的单位换算需要将换算倍数平方或立方。这是因为面积是二维量，体积是三维量。
1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³
A common error is to convert 5 m² into 500 cm². The correct conversion is 5 × 10 000 = 50 000 cm².
常见的错误是把 5 m² 直接写成 500 cm²。正确的换算应为 5 × 10000 = 50000 cm²。
2. Perimeter and Area of Rectangles, Triangles and Parallelograms | 矩形、三角形与平行四边形的周长和面积
The perimeter of a rectangle with length l and width w is found by adding all four sides. Its area is length multiplied by width.
长为 l、宽为 w 的矩形，其周长通过四条边相加得到，面积等于长乘宽。
Rectangle: P = 2l + 2w, A = lw
For a triangle, the area is half the base times the perpendicular height. The height must meet the base at a right angle, not along a slanted side.
三角形的面积等于底边与垂直高度乘积的一半。高必须与底边成直角，而不是沿着斜边。
Triangle: A = ½ × b × h
A parallelogram has the same area formula as a rectangle when you use the perpendicular height, not the slanted side length.
平行四边形使用与矩形相同的面积公式，但必须使用垂直高度，而不是斜边长度。
Parallelogram: A = b × h
For example, a rectangle measuring 8 cm by 5 cm has perimeter 26 cm and area 40 cm². A triangle with base 10 cm and perpendicular height 6 cm has area 30 cm².
例如，一个长 8 cm、宽 5 cm 的矩形，周长为 26 cm，面积为 40 cm²。底为 10 cm、垂直高度为 6 cm 的三角形面积为 30 cm²。
3. Area of Trapezium, Rhombus and Kite | 梯形、菱形与风筝形的面积
A trapezium has one pair of parallel sides. Its area is the average of the two parallel sides multiplied by the perpendicular distance between them.
梯形有一对平行边。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。
Trapezium: A = ½ × (a + b) × h
Here a and b are the parallel sides and h is the perpendicular height. For a trapezium with parallel sides 7 cm and 13 cm and height 4 cm, the area is ½ × (7 + 13) × 4 = 40 cm².
其中 a 和 b 是平行边，h 是垂直高度。若梯形的平行边分别为 7 cm 和 13 cm，高为 4 cm，则面积为 ½ × (7 + 13) × 4 = 40 cm²。
For a rhombus or a kite, the area can be found from the lengths of the diagonals. The two diagonals must be measured at right angles to each other.
菱形或风筝形的面积可由对角线长度求得。两条对角线必须互相垂直。
Rhombus or kite: A = ½ × d₁ × d₂
Do not confuse the diagonal of a rhombus with its side length. The side length is not used directly in this formula unless the question gives the diagonals.
不要把菱形的对角线与边长混淆。边长不能直接用于该公式，除非题目给出对角线长度。
4. Circumference and Area of a Circle | 圆的周长与面积
The circumference of a circle is the distance around it. The area is the space inside it. Both formulas use the radius r, while the circumference can also use the diameter d.
圆的周长是围绕圆形一圈的距离，面积是圆内部的区域大小。两个公式都使用半径 r，而周长也可以用直径 d 表示。
Circumference: C = 2πr = πd
Area: A = πr²
When a question asks for an exact answer, leave the result in terms of π. Otherwise use the π button on your calculator or take π as 3.142, then round as instructed.
如果题目要求精确值，请将结果保留为 π 的形式。否则使用计算器上的 π 键或取 π = 3.142，然后按要求四舍五入。
For example, a circle of radius 5 cm has circumference 10π ≈ 31.4 cm and area 25π ≈ 78.5 cm² to 3 significant figures.
例如，半径为 5 cm 的圆，周长为 10π ≈ 31.4 cm，面积为 25π ≈ 78.5 cm²（保留三位有效数字）。
5. Arc Length and Sector Area | 弧长与扇形面积
A sector is a portion of a circle bounded by two radii and an arc. The fraction of the circle used is the central angle θ divided by 360°.
扇形是由两条半径和一段弧围成的圆的一部分。它占整个圆的比例等于圆心角 θ 除以 360°。
Arc length: L = θ/360 × 2πr
Sector area: A = θ/360 × πr²
These formulas only work when θ is in degrees, which is the usual case in IGCSE questions. Always check the unit of the angle before substituting.
这些公式仅当 θ 以度为单位时适用，这也是 IGCSE 题目中的常见情况。代入前务必检查角的单位。
For a sector with radius 12 cm and centre angle 60°, the arc length is 60/360 × 2π × 12 = 4π ≈ 12.6 cm, and the sector area is 60/360 × π × 12² = 24π ≈ 75.4 cm².
对于半径为 12 cm、圆心角为 60° 的扇形，弧长为 60/360 × 2π × 12 = 4π ≈ 12.6 cm，扇形面积为 60/360 × π × 12² = 24π ≈ 75.4 cm²。
6. Surface Area of Prisms and Cylinders | 棱柱与圆柱的表面积
A prism has a constant cross-section along its length. Its surface area is the total area of all faces: two identical ends plus the side faces.
棱柱沿长度方向具有相同的横截面。其表面积是所有面的总面积：两个相同的端面加上侧面。
Prism surface area = 2 × cross-sectional area + perimeter of cross-section × length
For a cylinder, the two ends are circles and the curved surface can be unrolled into a rectangle. The rectangle has length equal to the circumference 2πr and height h.
对于圆柱体，两个端面是圆形，曲面可以展开成一个矩形。该矩形的长等于圆的周长 2πr，宽等于圆柱的高 h。
Cylinder surface area = 2πr² + 2πrh
For example, a cylinder with radius 3 cm and height 10 cm has surface area 2π × 3² + 2π × 3 × 10 = 18π + 60π = 78π ≈ 245 cm².
例如，半径为 3 cm、高为 10 cm 的圆柱，表面积为 2π × 3² + 2π × 3 × 10 = 18π + 60π = 78π ≈ 245 cm²。
7. Volume of Prisms, Cylinders, Pyramids, Cones and Spheres | 棱柱、圆柱、棱锥、圆锥与球的体积
The volume of any prism or cylinder is found by multiplying the cross-sectional area by the length or height.
任何棱柱或圆柱的体积都等于横截面积乘以长度或高度。
Prism: V = cross-sectional area × length
Cylinder: V = πr²h
Pyramids and cones take up one third of the space of the corresponding prism or cylinder with the same base and height.
棱锥和圆锥的体积等于同底等高棱柱或圆柱体积的三分之一。
Pyramid or cone: V = ⅓ × base area × height
Cone: V = ⅓πr²h
For a sphere, the volume and surface area depend only on the radius r.
球体的体积和表面积只取决于半径 r。
Sphere: V = (4/3)πr³, Surface area = 4πr²
For example, a cone with radius 5 cm and vertical height 12 cm has volume ⅓ × π × 5² × 12 = 100π ≈ 314 cm³. A sphere with radius 6 cm has volume (4/3) × π × 6³ = 288π ≈ 905 cm³.
例如，半径为 5 cm、垂直高度为 12 cm 的圆锥体积为 ⅓ × π × 5² × 12 = 100π ≈ 314 cm³。半径为 6 cm 的球体积为 (4/3) × π × 6³ = 288π ≈ 905 cm³。
8. Compound Shapes and Frustums | 复合图形与平截头体
For compound 2D shapes, split the shape into rectangles, triangles, semicircles and trapeziums. Find the area of each part, then add or subtract as needed.
对于复合二维图形，可将图形拆分为矩形、三角形、半圆和梯形。分别计算各部分的面积，再按需要相加或相减。
A frustum is the remaining solid when a smaller pyramid or cone is cut off the top of a larger one. Its volume is the large volume minus the small removed volume.
平截头体是从较大的棱锥或圆锥顶部切去一个较小的相似体后剩余的部分。它的体积等于大体积减去被切去的小体积。
Frustum volume = V large − V small
To find missing heights or radii in a frustum, use similar triangles. If the linear scale factor is k, then the area scale factor is k² and the volume scale factor is k³.
要求出平截头体中缺失的高度或半径，可使用相似三角形。若线性比例因子为 k，则面积比例因子为 k²，体积比例因子为 k³。
For example, if a cone of height 15 cm is cut into a top cone of height 5 cm, the linear scale factor from the small cone to the large cone is 3, so the large volume is 3³ = 27 times the small volume.
例如，一个高 15 cm 的圆锥被切出一个高 5 cm 的顶部小圆锥，小圆锥到大圆锥的线性比例因子为 3，因此大圆锥体积是小圆锥的 3³ = 27 倍。
9. Density and Working Backwards | 密度与反向求解
Density links mass and volume. The formula is often used in mensuration problems involving solids made from a given material.
密度将质量与体积联系起来。涉及由特定材料制成的固体的测量学问题中经常会用到该公式。
Density: ρ = mass / volume
Rearranged forms are mass = density × volume and volume = mass / density. Always check that the units are consistent, such as g/cm³ with g and cm³, or kg/m³ with kg and m³.
变形后可得：质量 = 密度 × 体积，体积 = 质量 / 密度。务必检查单位是否一致，例如 g/cm³ 对应 g 和 cm³，kg/m³ 对应 kg 和 m³。
Working backwards is also common when the volume or surface area is given and you need to find an unknown radius or height. Substitute the known values and solve the resulting equation.
已知体积或表面积求未知半径或高度的反向求解也很常见。代入已知数值并解方程即可。Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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