📚 Year 11 CIE Geography: Formula and Theorem Quick Reference Guide | Year 11 CIE 地理：公式定理速查手册
This concise reference guide is designed for Year 11 CIE Geography students who need quick access to essential formulas, key theorems, and models. All entries are presented in paired English–Chinese paragraphs to reinforce understanding and are aligned with the Cambridge IGCSE syllabus. Use this handbook to master population calculations, hydrological measures, map skills, and important theoretical frameworks.
本速查手册为 Year 11 CIE 地理学生设计，旨在快速查阅核心公式、关键定理与模型。全部内容采用中英双语对照段落呈现，以加深理解，并贴合剑桥 IGCSE 大纲。利用本手册掌握人口计算、水文测量、地图技能及重要理论框架。
1. Population Density | 人口密度
Population density measures how crowded an area is. It is expressed as the number of people per square kilometre (km²). The basic formula divides the total population by the land area. Urban areas usually show high density, while rural or mountainous zones show low density.
人口密度衡量一个地区的拥挤程度，用每平方公里人数表示。基本公式为总人口除以土地面积。城区通常密度高，而农村或山区密度低。
Population Density = Total Population ÷ Area (km²)
If a city has 800,000 people living in 400 km², the population density is 2,000 persons per km². When interpreting density, remember that built‑up areas may have much higher net densities than the administrative area suggests.
若某城市有80万人居住在400 km²范围内，人口密度为每平方公里2000人。解读密度时要注意，建成区的净密度可能远高于行政区面积所显示的数值。
2. Crude Birth Rate and Crude Death Rate | 粗出生率与粗死亡率
The crude birth rate (CBR) and crude death rate (CDR) are fundamental demographic indicators. They are expressed per 1,000 people per year and ignore the population’s age structure, hence ‘crude’. They allow comparisons across countries of different sizes.
粗出生率（CBR）和粗死亡率（CDR）是基本的人口指标，以每年每千人为单位，忽略了年龄结构，因此称为“粗”。它们便于不同规模国家间比较。
CBR (‰) = (Number of live births in a year ÷ Total population) × 1000
CDR (‰) = (Number of deaths in a year ÷ Total population) × 1000
A country with 500,000 people and 10,000 births records a CBR of 20 per 1,000. High CBRs are typical of agricultural societies; low CDRs often reflect improved healthcare. The gap between CBR and CDR drives natural change.
一个拥有50万人口和1万新生儿的国家，CBR 为20‰。高出生率常见于农业社会；低死亡率往往反映医疗改善。出生率与死亡率之差驱动自然变动。
3. Natural Increase Rate | 自然增长率
Natural increase is the difference between the number of live births and deaths in a population, usually expressed as a percentage. It excludes migration. The rate indicates whether a population is growing naturally without considering people moving in or out.
自然增长是一国人口出生与死亡人数之差，通常用百分比表示，不含迁移影响。该比率显示排除迁入迁出后，人口是否自然增长。
Natural Increase Rate (%) = [(CBR − CDR) / 1000] × 100
If CBR is 35 ‰ and CDR is 10 ‰, the rate of natural increase is (25/1000)×100 = 2.5% urces, while a negative rate implies natural decrease, as seen in some ageing populations.
若出生率为35‰，死亡率为10‰，则自然增长率为(25/1000)×100 = 2.5%每年。快速自然增长可能使资源吃紧，而负值意味着自然减少，如某些老龄化社会。
4. Total Fertility Rate and Infant Mortality Rate | 总和生育率和婴儿死亡率
Total fertility rate (TFR) is the average number of children a woman would have during her reproductive years. Infant mortality rate (IMR) measures deaths of infants under one year old per 1,000 live births. Both influence population structure and health policy.
总和生育率（TFR）是妇女育龄期内预期生育子女的平均数。婴儿死亡率（IMR）衡量每千名活产婴儿中一岁前死亡的数量。两者共同影响人口结构和卫生政策。
Infant Mortality Rate (‰) = (Deaths of infants <1 year ÷ Total live births) × 1000
TFR is derived from age‑specific fertility data, not a simple arithmetic formula. A TFR of 2.1 is considered replacement level. High IMR often correlates with low TFR later, as families have more children to ensure survival.
TFR 来自年龄别生育率数据，并非简单算式。总和生育率2.1被视为更替水平。高婴儿死亡率后期常与生育率下降相关，因为家庭为确保存活而多生孩子。
| Indicator | Typical High Value | Typical Low Value |
|---|---|---|
| TFR | 5–7 (Niger) | 1.2–1.5 (South Korea) |
| IMR (‰) | >60 (Sierra Leone) | <5 (Japan, Singapore) |
5. Dependency Ratio | 抚养比
The dependency ratio compares the economically dependent population (young and old) with the working‑age population. A high ratio indicates greater pressure on the productive group to support dependants. It is widely used in planning schools, pensions and healthcare.
抚养比将经济依赖人口（少儿和老年）与劳动年龄人口进行比较。高抚养比表明劳动人口抚养负担更重，广泛用于教育、养老金和医疗规划。
Dependency Ratio (%) = [(Population 0–14 + Population 65+) ÷ Population 15–64] × 100
Some definitions use 0–19 and 60+, but the IGCSE standard uses 0– . For example, if a country has 2 million young, 0.5 million elderly and 5 million working‑age, the dependency ratio is ((2+0.5)/5)×100 = 50%. Youth and old‑age dependency can be separated to identify specific pressures.
部分定义使用0–19与60+，但IGCSE标准采用0–14和65+。例如，某国有200万少儿、50万老年人和500万劳动人口，抚养比为((2+0.5)/5)×100 = 50%。可分拆少儿抚养比和老年抚养比来识别特定压力。
6. Net Migration Rate | 净迁移率
Net migration measures the difference between immigrants and emigrants in a given year, expressed per 1,000 population. It reflects both push and pull factors and provides insight into regional population change beyond natural increase.
净迁移衡量一年内移民迁入与迁出的差额，以每千人为单位。它体现推拉因素，并提供自然增长之外的人口变化见解。
Net Migration Rate (‰) = [(Immigrants − Emigrants) ÷ Total population] × 1000
A positive net migration rate may boost labour supply and cultural diversity, while negative net migration can lead to brain drain. In many high‑income countries, net migration offsets low natural increase.
正的净迁移率可增加劳动力和文化多样性，负净迁移则可能导致人才流失。在许多高收入国家，净迁移弥补了低自然增长。
7. Urbanisation Level | 城市化水平
Urbanisation level, often called the urban population percentage, is the proportion of a country’s population living in areas classified as urban. Criteria for ‘urban’ vary between countries, so comparisons require caution.
城市化水平，常指城市人口百分比，即一个国家居住在城市区域的人口比例。各国对“城市”的定义不同，比较时需谨慎。
Urbanisation Level (%) = (Urban population ÷ Total population) × 100
If a country’s total population is 40 million and 28 million live in urban settlements, the urbanisation level is 70%. High‑income countries generally exceed 80%, while many low‑income countries remain below 40%. This rate is central to studying land‑use models and urban sprawl.
若某国总人口4000万，其中2800万居住在城市，城市化水平为70%。高收入国家通常超过80%，而许多低收入国家仍低于40%。该比率对研究土地利用模型和城市扩张至关重要。
8. River Discharge and Rainfall Intensity | 河流流量与降雨强度
River discharge is the volume of water flowing through a cross‑section of a river per unit time, typically in cubic metres per second (cumecs). Rainfall intensity helps hydrologists predict flood peaks and design drainage systems.
河流流量是单位时间内通过河道横断面的水量，单位通常为立方米每秒（cumecs）。降雨强度帮助水文学家预测洪峰并设计排水系统。
Discharge (Q) = Cross‑sectional area (A) × Mean velocity (V)
Rainfall Intensity (mm/hr) = Total rainfall (mm) ÷ Duration (hours)
If a river channel has a cross‑sectional area of 15 m² and water flows at 2 m/s, discharge is 30 m³/s. Rainfall intensity of 60 mm over 3 hours gives 20 mm/hr. These formulas are used with flood hydrographs to interpret lag time and peak discharge.
若河床横断面积为15 m²，流速为2 m/s，则流量为30 m³/s。降雨强度60毫米持续3小时，即为20毫米/小时。这些公式配合洪水过程线，用于解释滞时和洪峰流量。
9. Gradient and Vertical Exaggeration | 坡度和垂直放大率
Gradient describes the steepness of a slope, often expressed as a ratio or fraction. Vertical exaggeration is the ratio of the vertical scale to the horizontal scale on a cross‑section, used to emphasize relief.
坡度描述坡面陡峭程度，常用比率或分数表示。垂直放大率是剖面图中垂直比例尺与水平比例尺的比值，用于突出地势起伏。
Gradient = Vertical interval (rise) ÷ Horizontal equivalent (run)
Vertical Exaggeration = Vertical scale ÷ Horizontal scale
m over a horizontal distance of 2 km (2000 m), the gradient is 100/2000 = 1 in 20. A cross‑ xaggeration of (1/10,000) ÷ (1/100,000) = 10 times. Always state scales in the same units.
若道路水平距离2 km内上升100 m，则坡度为100/2000 = 1:20。若剖面图垂直比例尺1 cm代表100 m，水平比例尺1 cm代表1 km，垂直放大率为(1/10,000) ÷ (1/100,000) = 10倍。计算时须统一单位。
10. Map Scale and Area Calculation | 地图比例尺和面积计算
res or the formula length × width after converting map distances to real ground distances.
地图比例尺表示图上距离与对应实地距离的比值。图上面积计算常用方格法，或先转换距离再用长×宽计算。
Map Scale = Map distance ÷ Ground distance (in same units)
onvert each to real metres, then multiply: Area (m²) = real length × real width. For irregular shapes, use dot grids or tracing squares to count partial and full squares.
比例尺1:50,000表示图上1 cm等于实地500 m。计算地图上矩形森林面积时，量取长宽以厘米为单位，分别换算为实际米数，再相乘：面积(m²) = 实地长 × 实地宽。不规则形状可使用网点格或透明方格纸统计完整和部分方格。
11. Demographic Transition Model | 人口过渡模型
The Demographic Transition Model (DTM) is a theorem describing how birth and death rates change as a country develops through five stages. It is a foundational concept for explaining population growth and age structure shifts over time.
人口过渡模型（DTM）描述随着国家发展，出生率和死亡率经历五个阶段的定理。它是解释人口增长和年龄结构变化的基础概念。
- Stage 1: High fluctuating birth and death rates – low natural increase.
- Stage 2: Death rates drop, birth rates remain high → rapid population growth.
- Stage 3: Birth rates start to fall, slowing growth.
- Stage 4: Low birth and death rates – population stabilises.
- Stage 5: Birth rate falls below death rate, leading to natural decrease.
- 阶段1：高出生率和高死亡率波动——低自然增长。
- 阶段2：死亡率下降，出生率保持高位→人口快速增长。
- 阶段3：出生率开始下降，增长放缓。
- 阶段4：低出生率和低死亡率——人口稳定。
- 阶段5：出生率低于死亡率，引起自然减少。
No country remains in Stage 1 today; most low‑income countries are in Stage 2 or early Stage 3, while many European nations are in Stage 4 or 5. The model is a simplification and does not account for migration or government policy, but it remains a powerful comparative tool for IGCSE case studies.
当今已无国家停留在阶段1；多数低收入国家处于阶段2或早期阶段3，而许多欧洲国家处于阶段4或5。该模型为简化模型，未考虑迁移或政策，但仍是IGCSE案例研究的有效比较工具。
12. Distance Decay and Gravity Model | 距离衰减与重力模型
Distance decay is the principle that interaction between two places decreases as the distance between them increases. The gravity model quantifies this, stating that interaction is proportional to the product of the populations and inversely proportional to the square of the distance.
距离衰减指两地间互动随距离增加而减弱的原理。重力模型将其量化，认为互动量与人口乘积成正比，与距离平方成反比。
Interaction I = k × (P₁ × P₂) ÷ D²
ment areas, and the distribution of services. Geographers also apply it to traffic flow: flow between two settlements = (Pop₁ × Pop₂) ÷ distance².
其中P₁和P₂为两城市人口，D为距离，k为常数。该模型有助于解释迁移流、零售商圈和服务分布。地理学家也将其用于交通流：两聚落间交通流量 = (人口₁ × 人口₂) ÷ 距离²。
0,000 (ignoring k). A larger city further away might still generate stronger flow due to mass. This model underlies the sphere of influence in settlement hierarchies.
例如，两镇人口分别为8万和5万，相距10公里，理论互动量与(80,000 × 50,000) ÷ 100 = 40,000,000成正比（忽略k）。距离更远的大城市因规模大仍可产生更强流动。该模型是聚落等级体系影响范围的基础。
Distance decay also applies to the spread of innovations and disease – the farther from a source, the slower the adoption or infection rate. Learning these theorems alongside the formulas equips you to interpret both physical and human geography patterns effectively.
距离衰减同样适用于创新扩散和疾病传播——离源头越远，采纳或感染率越低。掌握这些定理与公式，可有效解释自然和人文地理格局。
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