📚 Mastering IGCSE Mensuration: Perimeter, Area, Surface Area and Volume | 掌握 IGCSE 测量：周长、面积、表面积与体积
Mensuration is one of the most practical and widely tested topics in IGCSE Mathematics. It connects measurement, units, algebra, geometry and spatial reasoning. This article covers the key formulas, methods and exam strategies you need for perimeter, area, surface area and volume.
测量是 IGCSE 数学中最实用、考查最广泛的专题之一。它把测量、单位、代数、几何和空间推理联系在一起。本文涵盖周长、面积、表面积和体积所需的关键公式、方法与应试策略。
1. Units, Prefixes and Conversions | 单位、前缀与换算
Before using any mensuration formula, check the units in the question. Length is measured in mm, cm, m or km. Area is given in square units such as cm² and m², while volume is given in cubic units such as cm³ and m³. Converting area and volume units is not the same as converting length units.
在使用任何测量公式之前，都要先检查题目中的单位。长度以 mm、cm、m 或 km 为单位。面积以 cm²、m² 等平方单位表示，体积以 cm³、m³ 等立方单位表示。面积和体积的单位换算与长度单位换算不同。
ituting into a formula, express all lengths in the same unit first. If the question asks for an answer in a particular unit, convert at the end or at the start carefully.
例如，1 m = 100 cm，但 1 m² = 10 000 cm²，1 m³ = 1 000 000 cm³。代入公式时，应先将所有长度用同一单位表示。如果题目要求答案使用特定单位，应在开始或最后仔细换算。
2. Perimeter of 2D Shapes | 二维图形的周长
Perimeter is the total distance around the outside of a two-dimensional shape. For a rectangle with length L and width W, the perimeter is 2L + 2W. For a triangle, simply add the three side lengths. For a circle, the perimeter is called the circumference.
周长是围绕二维图形外部的总距离。对于长为 L、宽为 W 的矩形，周长为 2L + 2W。对于三角形，直接将三条边相加即可。对于圆，周长称为圆周。
Circumference = 2πr or πd
In a semicircle, the perimeter includes the curved arc plus the straight diameter. If the radius is r, the perimeter of a semicircle is πr + 2r. Many students forget the diameter, so always include all outer edges.
在半圆中，周长包括弯曲的弧长加上直线直径。如果半径为 r，则半圆的周长为 πr + 2r。许多学生忘记加直径，因此一定要包含所有外边缘。
3. Area of Triangles and Quadrilaterals | 三角形与四边形的面积
Area measures the amount of surface inside a two-dimensional boundary. The most common area formulas are for rectangles, triangles, parallelograms and trapeziums. In IGCSE exams, you must choose the correct formula and use the perpendicular height, not the slant height.
面积测量二维边界内的表面大小。最常见的面积公式适用于矩形、三角形、平行四边形和梯形。在 IGCSE 考试中，你必须选择正确公式，并使用垂直高度，而不是斜高。
- A rectangle has area = length × width.
- 矩形的面积 = 长 × 宽。
- A triangle has area = ½ × base × perpendicular height.
- 三角形的面积 = ½ × 底 × 垂直高。
- A parallelogram has area = base × perpendicular height.
- 平行四边形的面积 = 底 × 垂直高。
- A trapezium has area = ½ × (a + b) × h, where a and b are the parallel sides.
- 梯形的面积 = ½ × (a + b) × h，其中 a 和 b 是平行边。
When a shape is rotated or drawn on a grid, the perpendicular height may need to be found from a diagram rather than measured along the side. Always identify the base and its corresponding perpendicular height.
当图形被旋转或画在网格上时，垂直高度可能需要从图中求出，而不是沿边测量。一定要确定底边及其对应的垂直高。
4. Circles: Area, Arc Length and Sectors | 圆：面积、弧长与扇形
Circle calculations appear frequently in IGCSE paper 2 and paper 4. The two key full-circle formulas are the circumference and the area. For parts of a circle, use the fraction of the angle at the centre over 360°.
圆的计算经常出现在 IGCSE 试卷二和试卷四中。两个基本的全圆公式是周长和面积。对于圆的一部分，使用圆心角占 360° 的分数。
Circumference = 2πr and Area = πr²
². This works because θ/360 is the fraction of the whole circle represented by the sector.
如果扇形角为 θ 度，则弧长为 θ/360 × 2πr，扇形面积为 θ/360 × πr²。这是因为 θ/360 是该扇形占整个圆的比例。
Be careful when a question gives the diameter instead of the radius. Divide the diameter by 2 before applying the circle formulas.
当题目给出直径而不是半径时要小心。在使用圆公式前，应先将直径除以 2。
5. Composite Shapes and Shaded Regions | 复合图形与阴影区域
A composite shape is made by combining simpler shapes such as rectangles, triangles and semicircles. To find its area, split the shape into parts, calculate each part separately, then add or subtract the results.
复合图形由矩形、三角形和半圆等更简单的图形组合而成。要求其面积，应将图形拆分成若干部分，分别计算每一部分，然后相加或相减。
For shaded regions, a common method is to subtract the area of a smaller shape from a larger one. For example, the shaded area between a rectangle and an inscribed circle is rectangle area minus circle area.
对于阴影区域，一个常用方法是用大图形的面积减去小图形的面积。例如，矩形与其内切圆之间的阴影面积等于矩形面积减去圆的面积。
Always write down the parts you are adding or subtracting. This helps you earn method marks even if a numerical slip occurs later. Labelling the diagram can also reduce confusion.
始终写下你正在相加或相减的各个部分。这样即使之后出现数值失误，也能帮助你获得方法分。在图上标注也能减少混淆。
6. Surface Area of Prisms and Cylinders | 棱柱与圆柱的表面积
Surface area is the total area of all faces of a three-dimensional solid. A cuboid with length L, width W and height H has three pairs of identical faces, so its surface area is 2(LW + LH + WH).
表面积是三维立体所有面的总面积。长为 L、宽为 W、高为 H 的长方体有三对相同的面，因此其表面积为 2(LW + LH + WH)。
A cylinder has two circular ends and a curved surface. If the radius is r and the height is h, the curved surface area is 2πrh and the total surface area is 2πrh + 2πr².
圆柱有两个圆形底面和一个曲面。如果半径为 r、高为 h，则曲面面积为 2πrh，总表面积为 2πrh + 2πr²。
Drawing a net can help you identify every face and avoid missing hidden surfaces in composite solids. For open containers, remember to exclude the face that is not present.
画展开图有助于识别每个面，避免在复合立体中遗漏隐藏的面。对于开口容器，记得排除不存在的那个面。
7. Volume of Prisms and Cylinders | 棱柱与圆柱的体积
Volume measures the space inside a three-dimensional solid. For any prism, the volume is found by multiplying the cross-sectional area by the length. A cylinder is a prism with a circular cross-section.
体积测量三维立体内部的空间。对于任何棱柱，体积等于横截面积乘以长度。圆柱是一个横截面为圆形的棱柱。
Volume of prism = area of cross-section × length
a cylinder, it becomes πr²h. Always identify the cross-section first, especially when a prism is shown on its side or with an irregular end face.
对于
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图