📚 IGCSE CCEA Geography: Formula & Theorem Quick Reference Handbook | IGCSE CCEA 地理：公式定理速查手册
This quick reference handbook compiles the essential formulas, ratios and decision rules needed for CCEA IGCSE Geography. It covers mapwork, population, settlement, rivers, coasts, weather and development, and it is designed to be used alongside past papers and fieldwork tasks.
本速查手册汇编了 CCEA IGCSE 地理所需的核心公式、比率与判读规则，涵盖地图技能、人口、聚落、河流、海岸、天气与发展，适合配合历年真题与野外考察任务使用。
Every formula below is written in exam-ready form: state the formula, substitute the values with units, then give the answer with the correct unit and a sensible level of rounding. Accuracy marks in CCEA papers are usually awarded for the correct unit as well as the correct number.
以下每个公式都按考场可直接使用的形式给出：先写公式，再代入带单位的数值，最后给出单位正确、精度合理的结果。CCEA 试卷中的准确性分数通常同时考察单位与数值。
1. Map Scale and Distance | 地图比例尺与距离
Map scale links a measurement on the map to the real distance on the ground. A scale can be written as a representative fraction (1:50,000), as a statement (1 cm to 500 m), or drawn as a linear bar. Always express distance in the same unit before converting.
地图比例尺将地图上的测量长度与实地距离联系起来。比例尺可写成数值比例尺（1:50,000）、文字说明（1 厘米代表 500 米），或画成直线比例尺。换算前必须先把单位统一。
t centimetres to kilometres, divide by 100,000. Remember that 1 km = 1,000 m = 100,000 cm.
对于 1:50,000 这样的数值比例尺，地图上每 1 厘米代表实地 50,000 厘米。要把厘米换算成公里，需除以 100,000。记住 1 公里 = 1,000 米 = 100,000 厘米。
Distance (km) = Map distance (cm) × Scale denominator ÷ 100,000
Map distance (cm) = Real distance (km) × 100,000 ÷ Scale denominator
n a 1:25,000 map, a road is 6.4 cm long. Real length = 6.4 × 25,000 = 160,000 cm = 1.6 km.
示例：在 1:25,000 地图上，一条道路长 6.4 厘米。实地长度 = 6.4 × 25,000 = 160,000 厘米 = 1.6 公里。
| Scale | 1 cm represents | 1 km on map |
|---|---|---|
| 1:10,000 | 100 m | 10 cm |
| 1:25,000 | 250 m | 4 cm |
| 1:50,000 | 500 m | 2 cm |
| 1:100,000 | 1 km | 1 cm |
For curved distances such as a river or a winding road, place a piece of string or the straight edge of paper along the route, mark the start and finish, then measure the marked length against the linear scale. A large scale (1:10,000) shows more detail; a small scale (1:100,000) shows a larger area with less detail.
测量河流、蜿蜒道路等曲线距离时，可用细绳或纸边沿线路贴合，标出起点与终点，再放到直线比例尺上读数。大比例尺（1:10,000）显示的细节更多；小比例尺（1:100,000）覆盖范围更大但细节更少。
2. Area Calculation on Maps | 地图面积计算
Map area calculations require you to convert map length and width into ground distances first. For regular features such as rectangular fields, multiply ground length by ground width. Never multiply the two map measurements together and then apply a single conversion — the scale factor must be squared.
地图面积计算要求先把地图上的长和宽分别换算成实地距离。对于矩形农田等规则地物，将实地长度乘以实地宽度。切勿先把地图上的两个测量值相乘再做一次换算——比例尺因子必须平方。
Ground area (km²) = Ground length (km) × Ground width (km)
Ground area (km²) = Map area (cm²) × (Scale denominator)² ÷ 10¹⁰
For irregular shapes such as a lake or a woodland, count full grid squares and then add the area of partial squares. Each partial square is usually counted as 0.5 if more than half covered and 0 if less than half covered, so an estimate always carries a small margin of error.
对于湖泊、林地等不规则图形，先统计完整方格，再加部分方格的面积。覆盖超过一半的部分方格通常按 0.5 计算，不足一半的按 0 计算，因此估算结果总会存在一定误差。
| Scale | 1 cm² on map represents | 10 cm² on map represents |
|---|---|---|
| 1:10,000 | 0.01 km² | 0.1 km² |
| 1:25,000 | 0.0625 km² | 0.625 km² |
| 1:50,000 | 0.25 km² | 2.5 km² |
| 1:100,000 | 1 km² | 10 km² |
100 m on the ground, so a 3 cm by 2 cm field covers 300 m × 200 m = 60,000 m² = 0.06 km².
如果题目要求用米作答而比例尺是 1:10,000，记住地图上 1 厘米等于实地 100 米，因此一块 3 厘米 × 2 厘米的农田面积为 300 米 × 200 米 = 60,000 平方米 = 0.06 平方公里。
3. Gradient, Cross-sections and Contours | 坡度、地形剖面与等高线
Gradient measures how steeply the land rises or falls. It is the vertical interval divided by the horizontal distance, with both values in the same unit. Gradient can be shown as a ratio (1:12), a percentage, or an angle in degrees.
坡度衡量地表上升或下降的陡缓程度，等于垂直高差除以水平距离，两者必须使用相同单位。坡度可表示为比例（1:12）、百分比或角度。
Gradient = Vertical interval (m) ÷ Horizontal distance (m)
Gradient (%) = (Vertical rise ÷ Horizontal distance) × 100
r a horizontal distance of 2 km. Gradient = 120 ÷ 2,000 = 0.06, which is 1:16.7 or about 6%.
示例：一座山丘在 2 公里的水平距离内升高 120 米。坡度 = 120 ÷ 2,000 = 0.06，即 1:16.7，约为 6%。
Vertical exaggeration shows how many times a cross-section is stretched vertically compared with the map scale:
垂直夸大倍数表示地形剖面在垂直方向上相对于地图比例尺被拉伸的倍数：
Vertical exaggeration = Horizontal scale denominator ÷ Vertical scale denominator
nes that are close together indicate a steep slope; widely spaced contours indicate gentle slopes; contours crossing a valley form a V pointing upstream.
若地图比例尺为 1:50,000，垂直比例尺为 1 厘米代表 100 米（分母 10,000），则垂直夸大倍数为 50,000 ÷ 10,000 = 5 倍。等高线密集表示坡陡，稀疏表示坡缓；等高线穿过河谷时形成指向上游的 V 形。
4. Bearings, Direction and Aspect | 方位角、方向与坡向
A bearing is always measured clockwise from north and is written with three figures, from 000° to 360°. To take a bearing on a map, draw a north line through the starting point, join the two points, and measure the angle clockwise.
方位角始终从正北方向顺时针测量，写成三位数字，范围从 000° 到 360°。在地图上量角时，先过起点画一条正北线，连接两点，再顺时针量取角度。
Back bearing = Forward bearing ± 180°
If the forward bearing is less than 180°, add 180°; if it is greater than 180°, subtract 180°. For example, the back bearing of 065° is 245°, and the back bearing of 310° is 130°.
若前方位角小于 180°，加 180°；若大于 180°，减 180°。例如 065° 的反方位角是 245°，310° 的反方位角是 130°。
Direction can also be described in compass points (N, NE, ESE) or as an aspect, which is the compass direction that a slope faces. In the northern hemisphere, south-facing slopes receive more insolation, so they are warmer and often used differently from north-facing slopes.
方向也可用罗盘方位（北、东北、东南偏东）描述，或用坡向表示，即坡面所朝向的罗盘方向。在北半球，朝南坡接受的太阳辐射更多，因此更温暖，土地利用方式往往与朝北坡不同。
rue bearing = Magnetic bearing + Magnetic declination (east) − Magnetic declination (west)
Check whether the declination is east or west before adding or subtracting; a wrong sign is a very common error in mapwork questions.
加或减之前要先判断磁偏角是东偏还是西偏；符号搞错是地图技能题中最常见的失分点。
5. Population Rates and Density | 人口比率与人口密度
Population data questions normally ask for a rate per 1,000 people, a percentage, or a density. Always identify what the rate is measured against before substituting numbers.
人口数据题通常要求计算每 1,000 人的比率、百分比或人口密度。代入数字前，先弄清比率的计算基数是什么。
Population density = Total population ÷ Total area (km²)
Birth rate = (Births in a year ÷ Total population) × 1,000
Death rate = (Deaths in a year ÷ Total population) × 1,000
Natural increase is the difference between the birth rate and the death rate. When the answer is per 1,000, divide by 10 to express it as a percentage.
自然增长是出生率与死亡率之差。若结果以每千人计，除以 10 即可换算成百分比。
Natural increase rate (per 1,000) = Birth rate − Death rate
Natural increase (%) = (Birth rate − Death rate) ÷ 10
Population change = Natural increase + Net migration
Net migration = Immigration − Emigration
The rule of 70 gives a quick estimate of how long a population takes to double at a given growth rate.
70 法则可快速估算在给定增长率下人口翻倍所需的时间。
Doubling time (years) ≈ 70 ÷ Annual growth rate (%)
So a country growing at 2% per year doubles in about 35 years. Migration can reverse this: an area with natural decrease may still grow if net migration is strongly positive.
因此每年增长 2% 的国家约 35 年人口翻倍。迁移可能改变趋势：自然增长为负的地区若净迁入很强，人口仍会增加。
6. Population Pyramids and Dependency Ratio | 人口金字塔与抚养比
A population pyramid shows the age and sex structure of a population. A broad base indicates a high birth rate, a narrow top indicates shorter life expectancy, and a bulge in the middle may record an earlier baby boom or a period of in-migration.
人口金字塔显示人口的年龄与性别结构。底部宽表示出生率高，顶部窄表示预期寿命较短，中部的凸起可能反映此前的婴儿潮或迁入高峰。
dency ratio = ((Population aged 0-14 + Population aged 65+) ÷ Population aged 15-64) × 100
A dependency ratio of 60 means that every 100 people of working age support 60 people who are economically dependent. A high ratio increases pressure on health care, schools and pensions.
抚养比为 60 意味着每 100 名劳动年龄人口需负担 60 名经济受抚养人口。抚养比高会对医疗、学校与养老金造成更大压力。
Sex ratio = (Number of males ÷ Number of females) × 100
Percentage of population in an age group = (Group total ÷ Total population) × 100
Pyramids are classified as expansive (stage 2 of the demographic transition model, wide base), stationary (stage 4, straight sides) or constrictive (stage 5, narrower base). Link each shape to a named country and to the demographic transition model for full marks.
金字塔可分为扩张型（人口转变模型第 2 阶段，底部宽）、稳定型（第 4 阶段，两侧近乎垂直）与收缩型（第 5 阶段，底部更窄）。作答时把每种形状与具体国家和人口转变模型联系起来，才能拿到满分。
7. Settlement: Nearest Neighbour Index and Sphere of Influence | 聚落：最近邻指数与影响范围
The nearest neighbour index measures whether settlements or other point features are clustered, random or evenly spaced. It compares the observed mean distance between each point and its nearest neighbour with the distance expected under a random pattern.
最近邻指数用于判断聚落或其他点状要素是聚集、随机还是均匀分布。它把每个点到最近邻点的实际平均距离，与随机分布下应有的平均距离进行比较。
Expected mean distance (random) = 1 ÷ (2 × √N ÷ A)
earest neighbour index = Observed mean nearest neighbour distance ÷ Expected mean distance
In these formulas N is the number of points and A is the area of the study region in km², so the density N ÷ A is measured in points per km². The index runs from 0 (perfectly clustered) through 1 (random) to about 2.15 (perfectly regular or dispersed).
公式中 N 为点的数量，A 为研究区域面积（平方公里），因此密度 N ÷ A 的单位是每平方公里的点数。指数从 0（完全聚集）经 1（随机）到约 2.15（完全规则或均匀分布）。
Sphere of influence describes the area from which a settlement or service draws its customers. Larger settlements offer higher-order goods with a larger threshold population and a wider range. Reilly's break-point formula estimates the boundary between two competing centres.
影响范围指一个聚落或服务设施吸引顾客的区域。规模较大的聚落提供高级商品，门槛人口更高、服务范围更广。Reilly 断点公式可估算两个竞争中心之间的分界。
Break point = d ÷ (1 + √PB ÷ PA)
Here d is the distance between town A and town B, and PA and PB are their populations. The rank-size rule provides a further check on the settlement hierarchy.
