📚 Year 11 Cambridge Geography: Formula & Theorem Quick Reference Handbook | 剑桥地理：公式定理速查手册
Mastering the quantitative side of geography is essential for Cambridge IGCSE success. This handbook compiles every key formula, ratio and theorem you need – from population dynamics to river hydraulics – presented in a clear, exam-ready format. Each entry is accompanied by a worked example to reinforce your understanding and boost your confidence in calculations and data-response questions.
掌握地理学科中的定量分析方法是剑桥 IGCSE 考试取得成功的关键。本手册汇集了你需要掌握的所有重要公式、比率和定理——从人口动态到河流水力学——并以清晰、紧扣考点的形式呈现。每一个条目都配有详细的示例解析，帮助你巩固理解，提升在计算题和数据分析题中的信心。
1. Population Density | 人口密度
Population density measures how crowded an area is. It is calculated by dividing the total population by the total land area. The formula is:
人口密度衡量一个地区的拥挤程度。它用总人口除以总面积计算，公式为：
PD = P ÷ A
make comparisons fair. For example, if a region has 2,400,000 people and an area of 800 km², then PD = 2,400,000 ÷ 800 = 3,000 persons/km².
其中 PD 代表人口密度（人/平方公里），P 代表总人口，A 代表土地面积（平方公里）。为公平比较，应使用土地面积，排除大面积水域。例如，某地区人口为 240 万，土地面积为 800 平方公里，则人口密度 = 2400000 ÷ 800 = 3000 人/平方公里。
2. Crude Birth Rate and Crude Death Rate | 粗出生率与粗死亡率
The Crude Birth Rate (CBR) is the number of live births per 1,000 people in a population per year. The Crude Death Rate (CDR) is the number of deaths per 1,000 people per year. These rates allow simple comparison between countries of different sizes.
粗出生率 (CBR) 是指某地区一年内每千人中活产婴儿的数量。粗死亡率 (CDR) 是指一年内每千人中的死亡人数。这些比率便于在不同规模的国家之间进行简单比较。
CBR = (Number of births ÷ Total population) × 1,000
CDR = (Number of deaths ÷ Total population) × 1,000
Both rates are expressed as per mille (‰). For example, a country with 50,000 births in a year and a mid‑ear population of 2,000,000 has CBR = (50,000 ÷ 2,000,000) × 1,000 = 25 per 1,000. The mid‑year population is used to account for seasonal fluctuations.
两者都以千分比 (‰) 表示。例如，某国一年内出生 5 万人，年中人口为 200 万，则 CBR = (50000 ÷ 2000000) × 1000 = 25‰。使用年中人口是为了消除季节性波动的影响。
3. Natural Increase Rate | 自然增长率
The Natural Increase Rate (NIR) shows how fast a population is growing from births and deaths alone, ignoring migration. It is usually expressed as a percentage.
自然增长率 (NIR) 表示仅考虑出生和死亡（忽略迁移）时人口的增长速度，通常用百分比表示。
NIR (%) = (CBR – CDR) ÷ 10
First calculate CBR and CDR per 1,000, then subtract and divide by 10. If CBR = 32‰ and CDR = 8‰, NIR = (32 – 8) ÷ 10 = 2.4%. This means the population grows by 2.4% each year before migration. A negative NIR indicates natural decrease.
先计算出 CBR 和 CDR（‰），然后相减并除以 10。若 CBR = 32‰，CDR = 8‰，则 NIR = (32 – 8) ÷ 10 = 2.4%。这意味着在考虑迁移之前，人口每年增长 2.4%。若 NIR 为负值，则表示人口自然减少。
4. Net Migration Rate | 净迁移率
Net migration rate is the difference between the number of immigrants and emigrants per 1,000 population. It complements natural increase to give total population growth.
净迁移率 是每千人中迁入与迁出人数的差值，它与自然增长一起决定了总人口增长。
NMR = [(Immigrants – Emigrants) ÷ Total population] × 1,000
15,000 people move in and 5,000 move out of a population of 2,000,000, then NMR = (15,000 – 5,000) ÷ 2,000,000 × 1,000 = +5 per 1,000. A positive value means net in‑migration; a negative value means net out‑migration. To find the total growth rate, add natural increase (as a % converted to per thousand) to NMR.
例如，某地区人口 200 万，迁入 1.5 万人，迁出 0.5 万人，则 NMR = (15000 – 5000) ÷ 2000000 × 1000 = +5‰。正值表示净迁入，负值表示净迁出。计算总增长率时，可将自然增长率（‰）与净迁移率相加。
5. Dependency Ratio | 抚养比
The dependency ratio expresses the proportion of the economically inactive population (young 0–14 and old 65+) relative to the working‑age population (15–64). It indicates the pressure on the productive population.
抚养比 表示非经济活动人口（0–14 岁少儿和 65 岁以上老年人）与劳动适龄人口（15–64 岁）的比例，反映了劳动人口所承受的供养压力。
Dependency Ratio (%) = [(Population 0–14 + Population 65+) ÷ Population 15–64] × 100
If a country has 30 million children, 10 million elderly, and 80 million working‑age adults, the ratio = (30 m + 10 m) ÷ 80 m × 100 = 50% in developing countries (many children) and in ageing developed countries (many elderly).
若某国有 3000 万儿童、1000 万老人和 8000 万劳动适龄人口，抚养比 = (3000 万 + 1000 万) ÷ 8000 万 × 100 = 50%。这意味着每 100 名劳动者需供养 50 名非劳动者。高抚养比常见于发展中国家（儿童占比高）以及老龄化发达国家（老年占比高）。
6. River Discharge | 河流流量
Discharge is the volume of water flowing past a cross‑section of a river per unit time, measured in cubic metres per second (cumecs). It is a fundamental concept in fluvial hydrology.
流量 是指单位时间内通过河道某一横截面的水体体积，单位为立方米每秒（cumecs）。它是河流水文学中的基本概念。
Q = A × V
where Q = discharge (m³/s), A = cross‑ ertical section, and sum them. Mean velocity is measured with a flowmeter or float method.
其中 Q 代表流量（m³/s），A 代表横截面积（m²），V 代表平均流速（m/s）。计算 A 时，需在河道不同位置测量宽度和深度，算出各垂直截面的面积并求和。平均流速可使用流速仪或浮标法测定。
| Symbol | Meaning | Typical unit |
|---|---|---|
| Q | Discharge | m³/s (cumecs) |
| A | Cross-sectional area | m² |
| V | Mean velocity | m/s |
7. Hydraulic Radius | 水力半径
The hydraulic radius describes the efficiency of a river channel’s cross‑section in conveying water. A larger hydraulic radius means less friction and faster flow.
水力半径 描述了河流横截面的输水效率。水力半径越大，摩擦力越小，水流越快。
R = A ÷ P
where R = hydraulic radius (m), A = cross‑sectional area (m²), P = wetted perimeter (m) – ontact with water. For a rectangular channel of width w and depth d, P = w + 2d. In a semi‑circular channel, R is large because the wetted perimeter is small relative to area.
其中 R 代表水力半径（m），A 代表横截面积（m²），P 代表湿周（m）——即河床和河岸与水接触的长度。对于宽度为 w、深度为 d 的矩形渠道，P = w + 2d。在半圆形河道中，由于湿周相对于面积较小，R 值较大。
ue is used in the Manning equation to estimate velocity, but for IGCSE you only need to compute R itself.
例如，某河段 A = 12 m²，P = 10 m，则 R = 12 ÷ 10 = 1.2 m。此数值可用于曼宁公式估算流速，但 IGCSE 阶段只需会计算 R 本身。
8. River Gradient | 河流坡度
Gradient is the steepness of a river bed, usually expressed as a dimensionless ratio or as a fall in metres per kilometre. It influences flow velocity and erosive power.
坡度 指河床的陡峭程度，通常用无量纲比值或每公里的落差（米/公里）来表示。它影响流速和侵蚀能力。
Gradient = Vertical drop (m) ÷ Horizontal distance (m)
tretch: Gradient = 50 ÷ 5,000 = 0.01, or 10 m/km. Steep gradients are typical in upper courses; gentle gradients downstream reduce velocity (although discharge increases and efficiency may rise).
通常转换为米/公里：坡度 (m/km) = (落差 ÷ 河道长度) × 1000。例如，某河流在 5 公里河段内下降了 50 米，则坡度 = 50 ÷ 5000 = 0.01，即 10 米/公里。上游坡度陡，下游坡度缓，虽然流速可能因水力效率提高而不减反增。
9. Fieldwork: Measuring Average Velocity | 实地考察：平均流速测量
In river fieldwork, average velocity is often determined using a float. A buoyant object (e.g., an orange) is timed over a known distance. The surface velocity is then multiplied by a correction factor to estimate mean velocity.
在河流实地考察中，平均流速常通过浮标法测量。让一个漂浮物（如橙子）漂过已知距离并计时，然后将表面流速乘以校正系数以估算平均流速。
Velocity (m/s) = Distance (m) ÷ Time (s)
Because friction with the bed means the surface flows faster, a correction factor of 0.8– m/s; mean velocity ≈ 2 × 0.8 = 1.6 m/s. Always take multiple readings to improve accuracy.
由于河床摩擦，水面流速大于平均流速，因此对于粗糙河床，校正系数常取 0.8–0.85：平均流速 ≈ 表面流速 × 0.8。若漂浮物通过 10 米距离耗时 5 秒，表面流速 = 10 ÷ 5 = 2 m/s，则平均流速 ≈ 2 × 0.8 = 1.6 m/s。务必多次测量以提高准确性。
10. Human Development Index (Overview) | 人类发展指数概览
The Human Development Index (HDI) is not a single formula but a composite index combining three dimensions: health (life expectancy at birth), education (mean years of schooling and expected years of schooling), and standard of living (GNI per capita, PPP). Each dimension is normalised to an index between 0 and 1.
人类发展指数 (HDI) 并非单一公式，而是一个综合指数，涵盖三个维度：健康（出生时预期寿命）、教育（平均受教育年限和预期受教育年限）以及生活水平（人均国民总收入，按购买力平价计算）。每个维度被标准化为 0 到 1 之间的指数。
HDI = ∛(Health Index × Education Index × Income Index)
The indices are calculated using the minimum and maximum values (goalposts) set by the UN. For example, life expectancy goalposts are 20 and 85 years; GNI per capita goalposts are $100 and $75,000 (PPP). Although you will not be asked to compute HDI in the exam, understanding its structure and the geometric mean (cube root) helps you explain why a country with very low income cannot fully be compensated by high health and education.
各指数通过联合国设定的最小值和最大值（目标值）计算。例如，预期寿命的目标值为 20 岁和 85 岁；人均 GNI 的目标值为 100 美元和 75000 美元（购买力平价）。虽然考试不要求计算 HDI，但理解其构成以及几何平均数（立方根）的作用，有助于解释为何一个收入极低的国家无法完全通过较高的健康和教育来弥补。
- Health Index = (Actual life expectancy – 20) ÷ (85 – 20) | 健康指数 =（实际预期寿命 – 20）÷（85 – 20）
- Education Index = (Mean years index + Expected years index) ÷ 2, each calculated similarly | 教育指数 =（平均受教育年限指数 + 预期受教育年限指数）÷ 2，各自按类似方法计算
- Income Index = [ln(actual GNIpc) – ln(100)] ÷ [ln(75,000) – ln(100)] | 收入指数 = [ln（实际人均 GNI） – ln(100)] ÷ [ln(75000) – ln(100)]
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