📚 Year 11 CIE Geography: Formula & Theorem Quick Reference | Year 11 CIE 地理：公式定理速查手册
Mastering quantitative skills is essential for success in CIE IGCSE Geography. This quick reference guide brings together the most important formulas, indices, and conceptual 'theorems' you need to interpret data, analyse geographical patterns, and answer calculation-based questions with confidence. From population density to river discharge, each entry is presented clearly with both the formula and a worked example where appropriate.
掌握定量技能对于在 CIE IGCSE 地理考试中取得成功至关重要。这份速查手册汇集了最重要的公式、指标和概念性“定理”，帮助你解读数据、分析地理格局并自信地回答计算类问题。从人口密度到河流流量，每个条目都清晰地列出了公式以及适当的计算示例。
1. Population Density and Distribution | 人口密度与分布
Population density measures the number of people living per unit area, usually expressed as persons per square kilometre (km²). It is calculated by dividing the total population of a region by its total land area. This simple formula helps geographers compare how crowded different places are and can reveal patterns of settlement and resource pressure.
人口密度衡量单位面积上居住的人口数量，通常以每平方公里人数表示。计算方法是用一个地区的总人口除以其总土地面积。这个简单的公式有助于地理学家比较不同地方的拥挤程度，并揭示聚落格局和资源压力。
Population Density = Total Population ÷ Total Land Area (km²)
sons per km². Remember that high density does not always mean overpopulation; it depends on the carrying capacity of the land.
示例：如果一个城市有200万人分布在500平方公里的土地上，其人口密度为 2,000,000 ÷ 500 = 4,000 人/km²。请记住，高密度并不总是意味着人口过剩；这取决于土地的承载能力。
2. Population Change: Natural Increase and Migration | 人口变化：自然增长与迁移
The natural increase rate (NIR) is the difference between the crude birth rate (CBR) and crude death rate (CDR), typically expressed per thousand of the population. A positive NIR indicates a growing population from natural causes alone, while a negative NIR points to natural decrease. To understand overall population change, net migration must also be considered.
自然增长率（NIR）是粗出生率（CBR）与粗死亡率（CDR）之差，通常以每千人口表示。正自然增长率表示仅因自然原因造成的人口增长，而负自然增长率则表示人口自然减少。要了解总体人口变化，还必须考虑净迁移。
Natural Increase Rate (NIR) = (CBR − CDR) ÷ 10
Often NIR is given as a percentage; you can convert from per thousand: 20 per 1000 = 2% ssed per 1000. The total population growth rate is approximately NIR + Net Migration Rate.
自然增长率通常以百分比表示；你可以从千分比进行换算：千分之20 = 2%。净迁移率同样以千分比表示。总人口增长率约等于自然增长率加净迁移率。
3. Dependency Ratio | 抚养比
The dependency ratio is a measure of the economic pressure on the productive population. It compares the number of dependents (aged 0–14 and 65+) to the working-age population (15–64). A high dependency ratio may strain healthcare, education, and social services, especially in countries with youthful or ageing populations.
抚养比衡量的是生产性人口所承受的经济压力。它比较受抚养人口（0-14岁和65岁以上）与劳动年龄人口（15-64岁）的数量。高抚养比可能给医疗、教育和社会服务带来压力，尤其是在人口年轻化或老龄化的国家。
Dependency Ratio (%) = (Number of Dependents ÷ Working-Age Population) × 100
ependents and 50 million working-age adults has a dependency ratio of (30 ÷ 50) × 100 = 60%, meaning there are 60 dependents for every 100 workers.
示例：一个拥有3000万受抚养人口和5000万劳动年龄成年人的国家，其抚养比为 (30 ÷ 50) × 100 = 60%，意味着每100名劳动者需抚养60名受抚养者。
4. Population Pyramids and Demographic Transition Model | 人口金字塔与人口转变模型
A population pyramid is a graphical representation of the age-sex structure of a population. While not a formula in the numerical sense, interpreting pyramids involves using percentages and comparing cohort sizes. The Demographic Transition Model (DTM) is a conceptual 'theorem' that describes how birth and death rates change as a country develops, passing through five stages from high fluctuating to low fluctuating rates.
人口金字塔是人口年龄-性别结构的图形表示。虽然它不是数字意义上的公式，但解读金字塔需要使用百分比并比较各年龄段人群的规模。人口转变模型（DTM）是一个概念性“定理”，描述了一个国家随着发展出生率和死亡率如何变化，经历从高波动到低波动率的五个阶段。
Key checks: When reading a pyramid, identify the percentage of population in each age group, usually from census data. The shape indicates the stage of the DTM – a wide base suggests Stage 1 or 2, while a rectangular shape suggests Stage 4.
关键检查点：阅读金字塔时，确定每个年龄组的人口百分比，通常来自人口普查数据。金字塔的形状表明DTM的阶段——底部宽大表示第1或第2阶段，而矩形的形状表示第4阶段。
5. Urbanisation: Rate and Level | 城市化：速度与水平
The level of urbanisation is the proportion of a country's total population living in urban areas. The rate of urbanisation is the speed at which this proportion is increasing, usually expressed as an annual percentage change. Both are needed to understand urban growth patterns, especially in developing countries where rapid rural-to-urban migration occurs.
城市化水平是指一个国家总人口中居住在城市地区的比例。城市化速度是该比例增加的速度，通常以年度百分比变化来表示。理解城市增长模式需要两者兼顾，特别是在发生快速城乡迁移的发展中国家。
Level of Urbanisation (%) = (Urban Population ÷ Total Population) × 100
Rate of urbanisation is calculated by the difference in urban percentage between two years divided by the number of years, or more commonly from census data expressed as annual growth rate of urban population.
城市化速度的计算方法是，两个年份之间城市人口百分比的差除以年数，或者更常见的是根据人口普查数据，以城市人口的年增长率来表示。
6. River Discharge, Velocity and Cross-Sectional Area | 河流流量、流速与横截面积
River discharge (Q) is the volume of water passing a given point in the channel per second, measured in cubic metres per second (cumec, m³/s). It is a fundamental measure in river studies, combining the channel's cross-sectional area and the mean velocity of flow. Discharge changes with rainfall, catchment size and human modifications.
河流流量（Q）是每秒钟通过河床某一给定点的水量，以立方米/秒（cumec, m³/s）计量。它是河流研究中的一项基本指标，结合了河床的横截面积和平均流速。流量随降雨、流域大小和人类改造而变化。
Discharge (Q) = Cross-Sectional Area (A) × Mean Velocity (V)
To calculate cross-sectional area, divide the channel into vertical sections, measure width and depth at each, then sum up the areas of the trapezoids. Mean velocity can be measured with a flowmeter at 0.6 depth from the surface.
要计算横截面积，需将河道分成若干垂直截面，测量每个截面的宽度和深度，然后将各梯形面积相加。平均流速可用流速计在距水面0.6倍深度处测量。
7. Drainage Density and Stream Order | 河网密度与水系等级
Drainage density is the total length of all streams and rivers in a drainage basin divided by the total area of the basin. It provides insight into how well a basin is drained, the geology, rainfall intensity, and vegetation cover. Stream order, following the Strahler or Shreve methods, is a 'theorem' of stream hierarchy used to compare basin characteristics.
河网密度是流域内所有河流总长度除以流域总面积。它反映出流域的排水效率、地质条件、降雨强度和植被覆盖情况。依据斯特拉勒或什里夫方法确定的水系等级，是一种用于比较流域特征的水系层级“定理”。
Drainage Density (km/km²) = Total Stream Length (km) ÷ Basin Area (km²)
In Strahler ordering, fingertip tributaries are order 1; where two order-1 streams join, they form an order-2 stream, and so on. High drainage density is typical of impermeable rock or sparse vegetation.
在斯特拉勒分级法中，末梢支流为1级；两条1级河流汇合形成2级河流，以此类推。高河网密度常见于不透水岩石或植被稀疏的地区。
8. Measuring Development: HDI and Other Indices | 发展衡量：人类发展指数及其他指标
The Human Development Index (HDI) is a composite index combining life expectancy, education (mean and expected years of schooling), and Gross National Income (GNI) per capita. Each component is normalised on a scale from 0 to 1, and the HDI is the geometric mean of the three indices. Unlike simple GDP per capita, HDI captures broader well-being.
人类发展指数（HDI）是一个综合指标，纳入了预期寿命、教育（平均受教育年限和预期受教育年限）以及人均国民总收入（GNI）。每个组成部分均在0到1的尺度上进行归一化，HDI是这三个指数的几何平均数。与简单的人均GDP不同，HDI反映了更广泛的福祉。
HDI = (Ilife × I_education × Iincome)1/3
Each index is calculated as (actual value – minimum) ÷ (maximum – minimum). CIE expects you to interpret HDI values and trends, not compute them from raw data, but understanding the structure helps analyse development gaps.
每个指数的计算为（实际值 – 最小值）÷（最大值 – 最小值）。CIE期望你能解释HDI值和趋势，而不是根据原始数据进行计算，但理解其结构有助于分析发展差距。
9. Climate Graphs: Calculating Temperature Range and Precipitation | 气候图：计算气温范围和降水量
A climate graph (or climograph) combines monthly temperature (line) and precipitation (bar) data to reveal seasonal patterns. Two key quantities you may be asked to calculate are the annual temperature range and the total annual precipitation. The temperature range is simply the difference between the highest and lowest mean monthly temperatures.
气候图（或气候图表）综合了每月气温（折线）和降水量（柱状）数据，以揭示季节规律。你可能需要计算的两个关键量是年气温范围和年总降水量。气温范围即最高与最低月平均气温之间的差值。
Annual Temperature Range (°C) = Hottest Month Mean − Coldest Month Mean
Total Annual Precipitation (mm) = Sum of 12 Monthly Precipitations
Example: If the hottest month averages 28°C and the coldest −2°C, the range is 30°C. Large ranges often indicate continental interiors, while small ranges are typical of coastal or equatorial climates.
示例：如果最热月平均28°C，最冷月平均−2°C，则气温范围为30°C。大的气温范围通常表示大陆内部，小的范围则常见于沿海或赤道气候。
10. Settlement Hierarchy and Threshold Population | 聚落层级与门槛人口
Settlement hierarchy describes the ranking of settlements by size, range of services, and sphere of influence. The concept of threshold population – the minimum number of people needed to support a service – behaves like a 'theorem' in central place theory. High-order services (e.g., hospital, department store) require a larger threshold than low-order services (e.g., corner shop).
聚落层级描述了根据规模、服务范围和影响范围对聚落进行的分级。门槛人口的概念——即支撑某项服务所需的最低人口数量——类似于中心地理论中的一个“定理”。高级服务（如医院、百货商店）需要的门槛人口比低级服务（如街角小店）更大。
While exact formulas aren't normally calculated at IGCSE, you may be given data to analyse: determine whether a settlement has sufficient population to sustain a new service by comparing the threshold provided. The sphere of influence is the area from which a settlement draws its customers.
虽然IGCSE通常不要求精确计算公式，但你可能会被提供数据进行分析：通过比较给定的门槛人口，判断一个聚落是否有足够的人口来维持一项新服务。影响范围即聚落吸引顾客的区域。
11. Coastal Processes: Wave Energy and Longshore Drift | 海岸过程：波浪能量与沿岸漂移
In physical geography, quantitative aspects of coasts include wave frequency, fetch length, and the rate of longshore drift. Longshore drift (littoral drift) is the movement of sediment along the coast driven by waves approaching at an angle. The volume of transported material can be estimated using sediment traps and is expressed in cubic metres per day in fieldwork.
在自然地理学中，海岸的定量方面包括波浪频率、风区长度和沿岸漂移的速率。沿岸漂移（滨海漂移）是由波浪以一定角度接近海岸所引起的沉积物沿海岸线的运动。运移的物质体积可通过沉积物捕集器估算，在实地考察中以立方米/天表示。
The fetch is the uninterrupted distance of open water over which the wind blows; a longer fetch generates more powerful waves. Empirical relationships link fetch, wind speed, and significant wave height, but at IGCSE level you simply need to identify fetch on a map and explain its influence.
风区是风吹过的无阻碍开阔水域的距离；风区越长，产生的波浪能量越大。经验关系联系着风区长度、风速和有效波高，但在IGCSE级别，你只需在地图上识别风区并解释其影响。
12. Earthquakes: Magnitude, Epicentre and the Richter Scale | 地震：震级、震中与里氏震级
The Richter scale is a logarithmic scale that measures the magnitude of an earthquake based on seismic wave amplitude. Each whole number increase represents a tenfold increase in wave amplitude and roughly 31.6 times more energy release. While the moment magnitude scale (Mw) is now preferred, IGCSE questions sometimes still refer to the Richter scale for historical events.
里氏震级是一种对数尺度，根据地震波的振幅测量地震震级。每增加一个整数等级，表示波幅增加十倍，释放的能量大约增加31.6倍。虽然矩震级（Mw）目前更受青睐，但在IGCSE问题中，对于历史事件有时仍会提及里氏震级。
Key relationships: The epicentre is the point on the Earth's surface directly above the focus (hypocentre). Depth of focus influences surface impact – shallow-focus earthquakes cause more damage. There is no simple equation to learn, but you should understand the logarithmic nature of magnitude.
关键关系：震中是地球表面上直接位于震源（焦点）上方的点。震源深度影响地表破坏——浅源地震造成更大破坏。虽然没有简单的方程式需要记忆，但你应该理解震级是对数性质的。
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