📚 Perimeter, Area and Volume | 周长、面积与体积
Understanding how to calculate perimeter, area and volume is essential for solving real-world problems and forms a key part of the Cambridge Lower Secondary Mathematics curriculum. This revision guide covers the main formulas and techniques you need to master.
掌握如何计算周长、面积和体积对于解决实际问题至关重要，也是剑桥初中数学课程的核心部分。本复习指南涵盖你需要掌握的主要公式和方法。
1. Perimeter of Rectangles and Squares | 矩形和正方形的周长
The perimeter of any polygon is the total length of its sides. For a rectangle, you add the lengths of all four sides, which is the same as P = 2 x (length + width) or P = 2l + 2w, where l stands for length and w for width.
任何多边形的周长是其所有边长的总和。对于矩形，将四条边的长度相加，即 P = 2 x (长 + 宽) 或 P = 2l + 2w，其中 l 表示长，w 表示宽。
For a square, all sides are equal, so P = 4 x side length, usually written as P = 4s.
对于正方形，四条边相等，因此 P = 4 x 边长，通常写作 P = 4s。
Always include the correct unit, such as cm or m, when giving a perimeter.
给出周长时务必带上正确的单位，如 cm 或 m。
Example: A rectangle has length 9 cm and width 6 cm. Its perimeter = 2(9 + 6) = 2 x 15 = 30 cm.
举例：一个矩形长 9 cm，宽 6 cm，周长 = 2(9 + 6) = 2 x 15 = 30 cm。
2. Area of Rectangles and Squares | 矩形和正方形的面积
The area of a rectangle is found by multiplying its length by its width. The formula is A = l x w or A = length x width.
矩形的面积等于长乘宽。公式为 A = l x w 或 A = 长 x 宽。
For a square, since length equals width, the area is simply A = s x s = s2, where s is the side length.
对于正方形，因为长等于宽，面积即 A = s x s = s2，其中 s 是边长。
Area is measured in square units, for example cm2 or m2. This shows that two dimensions are multiplied.
面积以平方单位度量，例如 cm2 或 m2。这表示进行了两个维度的乘法运算。
Example: A square with side 7 m has area 7 m x 7 m = 49 m2.
举例：边长为 7 m 的正方形，面积为 7 m x 7 m = 49 m2。
3. Area of Triangles | 三角形的面积
The area of a triangle is half the area of a rectangle with the same base and height. The formula is A = ½ x base x height, often written as A = (b x h) / 2.
三角形的面积是同底同高的矩形面积的一半。公式为 A = ½ x 底 x 高，通常写作 A = (b x h) / 2。
The height must be perpendicular to the base – this is the vertical height, not the slanted side.
高必须垂直于底——这是垂直高度，而不是斜边长度。
Different types of triangles (scalene, isosceles, right-angled) all follow this same rule, as long as you use the perpendicular height.
不同类型的三角形（不等边、等腰、直角三角形）都遵循这一规则，只要使用垂直高度即可。
Example: A triangle with base 10 cm and vertical height 4 cm has area = ½ x 10 x 4 = 20 cm2.
举例：一个底 10 cm、垂直高 4 cm 的三角形，面积 = ½ x 10 x 4 = 20 cm2。
4. Area of Parallelograms | 平行四边形的面积
A parallelogram looks like a slanted rectangle, but its area is still calculated using base x perpendicular height. The formula is A = b x h, where h is the perpendicular distance between the two parallel bases.
平行四边形看起来像一个倾斜的矩形，但其面积仍用底 x 垂直高计算。公式为 A = b x h，其中 h 是两条平行底边之间的垂直距离。
You cannot use the slanted side length as the height; always draw or identify the perpendicular height.
不能使用斜边长度作为高；始终画出或找出垂直高度。
Example: A parallelogram has base 12 cm and perpendicular height 5 cm. Its area = 12 x 5 = 60 cm2.
举例：一个平行四边形底长 12 cm，垂直高 5 cm，其面积 = 12 x 5 = 60 cm2。
5. Area of Trapeziums | 梯形的面积
A trapezium (or trapezoid) has one pair of parallel sides. Its area is found by averaging the lengths of the parallel sides and then multiplying by the perpendicular height.
梯形有一组对边平行。其面积计算方法是：取两平行边长度的平均值，再乘以垂直高。
The formula is A = ½ x (a + b) x h, where a and b are the lengths of the parallel sides and h is the perpendicular height between them.
公式为 A = ½ x (a + b) x h，其中 a 和 b 是两条平行边的长度，h 是它们之间的垂直高度。
It helps to remember this as ‘half the sum of the parallel sides times the height’.
可以记住这个口诀：‘上底加下底之和的一半乘高’。
Example: Parallel sides are 8 cm and 6 cm, height is 5 cm. Area = ½ x (8 + 6) x 5 = ½ x 14 x 5 = 35 cm2.
举例：平行的两边分别为 8 cm 和 6 cm，高为 5 cm。面积 = ½ x (8 + 6) x 5 = ½ x 14 x 5 = 35 cm2。
6. Circumference of a Circle | 圆的周长
The perimeter of a circle is called the circumference. You need to know the value of π (pi), approximately 3.14 or the fraction 22/7.
圆的周长叫做圆周。你需要知道 π（圆周率）的值，约为 3.14 或分数 22/7。
If you know the radius (r), use the formula C = 2πr. If you know the diameter (d), use C = πd.
如果已知半径 (r)，使用公式 C = 2πr。如果已知直径 (d)，则使用 C = πd。
The diameter is twice the radius: d = 2r. So both formulas give the same result.
直径是半径的两倍：d = 2r。因此两个公式会给出相同结果。
Example: A circle has radius 7 cm. Circumference = 2 x π x 7 ≈ 2 x 3.14 x 7 = 43.96 cm (or use 2 x 22/7 x 7 = 44 cm).
举例：一个半径为 7 cm 的圆，周长 = 2 x π x 7 ≈ 2 x 3.14 x 7 = 43.96 cm（或用 2 x 22/7 x 7 = 44 cm）。
7. Area of a Circle | 圆的面积
The area enclosed by a circle is A = πr2, meaning pi times the radius squared. Remember you square only the radius, not the whole product.
圆所围成的面积是 A = πr2，即圆周率乘以半径的平方。记住只是半径被平方，而非整个乘积。
If you are given the diameter, divide it by 2 to find the radius first, then apply the formula.
如果给定直径，先除以 2 得到半径，再套用公式。
The area of a circle is always in square units, and you should leave your answer in terms of π when an exact value is required.
圆的面积单位总是平方单位，当要求精确值时，答案可以保留 π。
Example: Radius = 5 cm, area = π x 52 = 25π cm2 or about 78.5 cm2 (using 3.14).
举例：半径 = 5 cm，面积 = π x 52 = 25π cm2 或约为 78.5 cm2（用 3.14 计算）。
Summary of area formulas so far:
到目前面积公式总结：
| Shape | Formula |
| Rectangle | A = l x w |
| Square | A = s2 |
| Triangle | A = ½ x b x h |
| Parallelogram | A = b x h |
| Trapezium | A = ½ x (a + b) x h |
| Circle | A = πr2 |
8. Composite Shapes | 复合图形
A composite (or compound) shape is made up of two or more simple shapes. To find its area, split it into rectangles, triangles, circles or other known figures.
复合（组合）图形由两个或多个简单图形组成。求其面积时，将其拆分为矩形、三角形、圆或其他已知图形。
Calculate the area of each part separately and then add them together. Sometimes you may need to subtract areas, for example when finding the area of a shaded region.
分别计算每个部分的面积，然后相加。有时需要减去某些面积，例如求阴影部分面积时。
With perimeter, you still add the outer side lengths, but be careful not to include internal lines where shapes join.
对于周长，仍然是将外部边长相加，但要小心不要把图形连接处的内部线段也包括进去。
Example: An L-shaped figure can be divided into two rectangles 6 by 2 and 4 by 3. Area = 6×2 + 4×3 = 12 + 12 = 24 cm2.
举例：一个 L 形图形可以分割为两个矩形，分别为 6×2 和 4×3。面积 = 6×2 + 4×3 = 12 + 12 = 24 cm2。
9. Surface Area of Prisms | 棱柱的表面积
A prism is a 3D shape with a constant cross-section, such as a cuboid, triangular prism or cylinder (which is a circular prism). The surface area is the total area of all its faces.
棱柱是一种具有恒定横截面的三维图形，例如长方体、三棱柱或圆柱（即圆形棱柱）。表面积是所有面的总面积。
For a cuboid with length l, width w and height h, the surface area = 2(lw + lh + wh). Each pair of opposite faces contributes two identical areas.
对于长为 l、宽为 w、高为 h 的长方体，表面积 = 2(lw + lh + wh)。每对相对的面贡献两个相同的面积。
To find the surface area of any prism: work out the area of the two end faces (the cross-section) plus the areas of the rectangular faces that wrap around the sides.
求任何棱柱的表面积：计算出两个端面（横截面）的面积，再加上所有侧面的矩形面积。
Example: A triangular prism with cross-section area 12 cm2 and perimeter 16 cm, length 10 cm. Surface area = 2×12 + 16×10 = 24 + 160 = 184 cm2.
举例：一个三棱柱横截面面积 12 cm2，横截面周长 16 cm，柱长 10 cm。表面积 = 2×12 + 16×10 = 24 + 160 = 184 cm2。
10. Volume of Prisms | 棱柱的体积
The volume of any prism is given by V = area of cross-section x length (or height). This works for cuboids, cylinders, triangular prisms and all other uniform prisms.
任何棱柱的体积公式为 V = 横截面积 x 长度（或高）。这适用于长方体、圆柱、三棱柱和其他所有均匀棱柱。
For a cuboid, this becomes V = l x w x h. For a cylinder, V = πr2h, where r is the radius of the circular base and h is the height.
对于长方体，这是 V = l x w x h。对于圆柱，V = πr2h，其中 r 是圆形底面的半径，h 是高。
Volume is measured in cubic units: cm3, m3 and so on. It tells you how much space an object occupies.
体积以立方单位度量：cm3、m3 等。它表示物体占据的空间大小。
Important: When calculating volume, make sure all dimensions are in the same unit before multiplying.
重要：计算体积时，确保所有尺寸单位一致后再相乘。
Example: A cylinder with radius 3 cm and height 10 cm has volume = π x 32 x 10 = 90π cm3 ≈ 282.6 cm3.
举例：一个半径 3 cm、高 10 cm 的圆柱，体积 = π x 32 x 10 = 90π cm3 ≈ 282.6 cm3。
11. Units and Conversions | 单位与换算
Perimeter units are always linear (cm, m, km). Area units are squared because two lengths are multiplied. Volume units are cubed because three lengths are multiplied.
周长单位始终是长度单位（cm、m、km）。面积单位带平方，因为由两个长度相乘。体积单位带立方，因为由三个长度相乘。
Key conversions to remember:
需记住的关键换算：
-
Length: 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m
长度：1 cm = 10 mm，1 m = 100 cm，1 km = 1000 m
-
Area: 1 m2 = 10 000 cm2 (because 100 cm x 100 cm = 10 000 cm2)
面积：1 m2 = 10 000 cm2（因为 100 cm x 100 cm = 10 000 cm2）
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Volume: 1 m3 = 1 000 000 cm3 (100 cm x 100 cm x 100 cm)
体积：1 m3 = 1 000 000 cm3（100 cm x 100 cm x 100 cm）
When converting between units of area or volume, be careful not to use the same factor as for length. Square the linear conversion factor for area, and cube it for volume.
在进行面积或体积单位换算时，注意不要直接使用长度单位的换算系数。面积需使用长度换算系数的平方，体积使用立方。
Example: Convert 2 m2 to cm2: 2 x (100)2 = 2 x 10 000 = 20 000 cm2.
举例：将 2 m2 转换为 cm2：2 x (100)2 = 2 x 10 000 = 20 000 cm2。
These conversions are common in KS3 questions and help link geometry with measurement skills.
这些换算在 KS3 题目中很常见，有助于将几何与测量技能联系起来。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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