📚 IGCSE CAIE Geography: Formulae & Theorems Quick Reference | IGCSE CAIE 地理：公式定理速查手册
In IGCSE CAIE Geography, a small set of formulae and key conceptual models can earn many marks in Paper 1, Paper 2 and the alternative to coursework. This quick reference handbook condenses the essential population, settlement, map work, weather, river and development calculations into one place.
在 IGCSE CAIE 地理中，少量公式和重要概念模型可以在 Paper 1、Paper 2 和课程作业替代试卷中帮你拿到很多分数。本速查手册将核心的人口、聚落、地图、天气、河流和经济发展计算浓缩在一起。
1. Key Demographic Formulae | 核心人口公式
Demographic formulae are nearly always presented per 1,000 people, except population density and dependency ratio. In exam questions, write the formula first, substitute values, and then give the answer to one decimal place unless told otherwise.
人口公式通常以每 1000 人表示，但人口密度和依赖比率除外。考试时，先写出公式，再代入数值，最后按题目要求保留小数位（通常保留至一位小数）。
| Formula | Use / 用途 |
|---|---|
| Birth rate = (live births ÷ total population) × 1000 | 每千人出生人数 / births per 1,000 people |
| Death rate = (deaths ÷ total population) × 1000 | 每千人死亡人数 / deaths per 1,000 people |
| Natural increase = births − deaths | 自然增长人数 / absolute natural change |
| Natural increase rate (%) = (natural increase ÷ total population) × 100 | 自然增长率 / natural increase as percentage |
| Population density = total population ÷ land area | 每平方千米人口数 / people per km² |
| Dependency ratio = ((population aged 0–14 + population aged 65+) ÷ working-age population aged 15–64) × 100 | 抚养比率 / dependants per 100 working-age people |
| Infant mortality rate = (deaths under 1 year ÷ total live births) × 1000 | 每千名活产婴儿死亡率 / infant deaths per 1,000 live births |
| Total fertility rate = number of live births per woman | 每名妇女平均生育子女数 / births per woman |
Always check whether the question asks for a rate per 1000 or a percentage. Some candidates lose marks by multiplying by 100 instead of 1000.
务必检查题目要求的是千分率还是百分率。很多考生因为该乘 100 时乘了 1000 而丢分。
2. Migration and Population Change | 迁移与人口变化
Net migration is the difference between immigration and emigration. Natural increase plus net migration gives total population change. This is often summarised as the population equation.
净迁移是移入与移出之差。自然增长加上净迁移等于总人口变化。这通常被概括为人口方程。
Net migration = immigrants − emigrants
Net migration rate = (net migration ÷ total population) × 1000
Population change = (births − deaths) + (immigrants − emigrants)
Population growth rate (%) = ((final population − initial population) ÷ initial population) × 100
In a statement of population change, births and immigrants are usually increases, while deaths and emigrants are decreases. Show the signs clearly when you present your working.
在人口变化计算中，出生和迁入通常是增加项，死亡和迁出是减少项。列式计算时要清楚地写出正负号。
3. Settlement and Urbanisation | 聚落与城市化
Urbanisation is measured as the percentage of people living in urban areas. It can increase through rural-urban migration or natural increase in towns and cities.
城市化是用居住在城市地区的人口比例来衡量。农村向城市迁移或城镇人口自然增长都会使城市化水平上升。
Urban population percentage = (urban population ÷ total population) × 100
Urban growth rate (%) = ((current urban population − previous urban population) ÷ previous urban population) × 100
Rate of rural-urban migration = (number of rural-urban migrants ÷ total population) × 1000
Settlement hierarchy questions often refer to the concepts of threshold population and range. Threshold population is the minimum number of people needed to support a service; range is the maximum distance people are willing to travel to use it.
聚落等级体系题目经常涉及门槛人口和商品服务范围。门槛人口是支撑一项服务所需的最低人口数量；服务范围是人们愿意为使用该服务而移动的最大距离。
4. Map Skills: Scale, Distance and Area | 地图技能：比例尺、距离与面积
Map work questions require accurate use of the linear scale or representative fraction. Distance is calculated from map distance multiplied by the scale factor; area is usually found by converting grid-square measurements to ground distances.
地图技能题目要求正确使用线段比例尺或数字比例尺。距离等于图上距离乘以比例因子；面积通常通过将网格正方形的测量值换算为实地距离来求出。
| Scale | Conversion |
|---|---|
| 1:50,000 | 1 cm = 0.5 km |
| 1:25,000 | 1 cm = 0.25 km |
| 1:100,000 | 1 cm = 1 km |
| 1:10,000 | 1 cm = 0.1 km |
Ground distance = map distance × scale factor
Ground distance in km = (map distance in cm × scale denominator) ÷ 100,000
Ground area = map area × (scale factor)²
On a 1:50,000 map, a 2 cm × 2 cm grid square represents 1 km². On a 1:25,000 map, a 4 cm × 4 cm grid square represents 1 km². Use the grid-square method whenever possible because it is faster and reduces calculation errors.
在 1:50,000 地图上，一个 2 厘米 × 2 厘米的网格正方形代表 1 平方千米。在 1:25,000 地图上，一个 4 厘米 × 4 厘米的网格正方形代表 1 平方千米。尽量使用网格法，因为更快且能减少计算错误。
5. Map Skills: Gradient and Cross Section | 地图技能：坡度与剖面
Gradient compares vertical rise to horizontal distance. It is usually expressed as a ratio in the form 1:n, where n is calculated by dividing the horizontal equivalent by the vertical interval.
坡度是比较垂直上升量与水平距离的关系。它通常表示为 1:n 的比率形式，其中 n 等于水平距离除以垂直高差。
Gradient = vertical interval ÷ horizontal equivalent
Gradient as 1:n = 1 ÷ (horizontal equivalent ÷ vertical interval)
Example: if the vertical interval is 100 m and the horizontal equivalent is 2,000 m, the gradient is 100 ÷ 2,000 = 1 in 20. Both measurements must use the same unit before dividing.
例如：如果垂直高差为 100 米，水平距离为 2000 米，则坡度为 100 ÷ 2000 = 1:20。两个测量值在相除前必须使用相同单位。
6. Weather and Climate Calculations | 天气与气候计算
Climate statistics are based on temperature and precipitation records. Temperature range, mean daily temperature and annual total rainfall are common calculations.
气候统计数据以温度和降水记录为基础。温度范围、日平均温度和年总降水量是常见计算。
| Calculation | Formula |
|---|---|
| Temperature range | Maximum temperature − minimum temperature |
| Mean daily temperature | (daily maximum + daily minimum) ÷ 2 |
| Mean monthly temperature | sum of daily mean temperatures ÷ number of days |
| Mean annual temperature | sum of monthly mean temperatures ÷ 12 |
| Total annual precipitation | sum of all monthly rainfall totals |
| Percentage of annual rainfall in a month | (monthly rainfall ÷ annual rainfall) × 100 |
When reading a climate graph, read the temperature scale and precipitation scale separately, since they usually have different units and scales. Temperature range is a simple subtraction, but many candidates accidentally add the signs incorrectly when negative values appear.
阅读气候图时，要分别读取温度标尺和降水标尺，因为两者的单位和刻度通常不同。温度范围是简单减法，但当出现负值时，很多考生会不小心把正负号弄错。
7. Hydrology and Fluvial Calculations | 水文与河流计算
River discharge is the volume of water passing a point in a river per second. It is calculated by multiplying the cross-sectional area of the channel by the average velocity of the water.
河流流量是每秒通过河流某一点的水量。它由河道横截面积乘以平均流速得出。
Discharge (Q) = cross-sectional area (A) × average velocity (V)
Cross-sectional area (m²) = average width (m) × average depth (m)
Drainage density = total stream length ÷ drainage basin area
Discharge is usually expressed in cubic metres per second (m³/s). In fieldwork, velocity is usually measured in metres per second, so ensure the area is in square metres before you multiply.
流量通常以立方米每秒（m³/s）表示。在实地调查中，流速通常以米每秒测量，因此在相乘前要确保横截面积的单位为平方米。
8. Economic Development Indicators | 经济发展指标
Economic development indicators help compare countries. GDP per capita, GNI per capita and employment structure percentages are the most common calculations in the IGCSE CAIE Geography specification.
经济发展指标有助于比较不同国家。人均 GDP、人均 GNI 和就业结构百分比是 IGCSE CAIE 地理大纲中最常见的计算。
| Indicator | Formula |
|---|---|
| GDP per capita | GDP ÷ total population |
| GNI per capita | GNI ÷ total population |
| Percentage employed in a sector | (number employed in sector ÷ total workforce) × 100 |
| Percentage change | ((new value − old value) ÷ old value) × 100 |
| Trade balance | value of exports − value of imports |
When comparing countries, use the same indicator and the same year. A percentage change can be positive or negative; state whether the change is an increase or a decrease to make your answer clear.
比较国家时，要使用同一指标和同一年份。百分比变化可以是正数或负数；答题时明确说明是增加还是减少，使答案更清晰。
9. Agriculture, Industry and Energy Ratios | 农业、工业与能源比率
Agricultural and industrial productivity calculations measure efficiency. Yield, labour productivity and energy consumption per capita are useful in food, industry and resource topics.
农业和工业生产率计算用于衡量效率。产量、劳动生产率和人均能源消费在粮食、工业和资源专题中非常有用。
Crop yield = total output ÷ cultivated area (tonnes per hectare)
Agricultural labour productivity = total output ÷ number of workers
Industrial labour productivity = total output ÷ number of workers
Energy consumption per capita = total energy consumption ÷ total population
These ratios are often used to explain why some countries have high food security or industrial output despite a small workforce. Show the units in your final answer, such as hectares, tonnes or kWh
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