📚 Year 11 CAIE Geography: Formula & Theorem Quick Reference Handbook | Year 11 CAIE 地理：公式定理速查手册
Master the essential formulae, models, and theories required for CAIE IGCSE Geography. This handbook collects every calculation, key diagram, and geographical theorem you will need to interpret data, analyse patterns, and score top marks on Paper 2 and Paper 4. Use it alongside case studies to strengthen your geographical skills.
掌握 CAIE IGCSE 地理所必需的核心公式、模型与理论。本手册汇集了你解读数据、分析模式并在 Paper 2 和 Paper 4 中取得高分所需的每一种计算、关键图表和地理定理。将其与案例研究配合使用，可夯实你的地理技能。
1. Population Formulas | 人口计算公式
Natural Increase Rate = (Birth Rate – Death Rate) / 10 (gives the annual percentage change, with birth and death rates expressed per 1000 people). A positive result means natural growth; a negative result means natural decrease.
自然增长率 = (出生率 – 死亡率) / 10（得出年百分比变化，出生率和死亡率均以每千人为单位）。正值表示自然增长，负值表示自然减少。
Population Density = Total Population / Total Land Area (km²). This measures how crowded a country or region is. High densities are found in urban areas and fertile river valleys.
人口密度 = 总人口 / 土地总面积 (km²)。衡量一个国家或地区的拥挤程度。城区和肥沃的河谷地带人口密度较高。
Dependency Ratio = [(Population aged 0–14 + Population aged 65+) / Population aged 15–64] × 100. The result is the number of dependents per 100 working-age people.
抚养比 = [(0–14 岁人口 + 65 岁以上人口) / 15–64 岁人口] × 100。结果为每 100 名劳动适龄人口对应的受抚养人数。
ant Mortality Rate = (Number of deaths of infants under 1 year / Total live births) × 1000. It reflects healthcare quality and is often used alongside life expectancy to assess development.
婴儿死亡率 = (1 岁以下婴儿死亡数 / 活产总数) × 1000。反映医疗保健质量，常与预期寿命一起用于衡量发展水平。
2. Demographic Transition Model | 人口转变模型
The DTM describes how a country's birth rate, death rate and total population change over time as it develops economically. Stage 1: High fluctuating – both rates are high and variable, so population remains low and stable. Stage 2: Early expanding – death rate falls sharply due to improved medicine and sanitation, while birth rate stays high, causing rapid population growth. Stage 3: Late expanding – birth rate begins to decline as contraception spreads and children become less of an economic asset; growth slows. Stage 4: Low fluctuating – both rates are low and balanced, leading to a stable or slowly growing population. Stage 5: Declining – birth rate drops below death rate, resulting in natural decrease and an ageing population.
人口转变模型描述了随着经济发展，一个国家的出生率、死亡率和总人口如何随时间变化。阶段 1：高位波动 – 两项比率均居高且多变，人口保持低位稳定。阶段 2：早期扩张 – 由于医疗和卫生改善，死亡率急剧下降，而出生率居高不下，导致人口迅速增长。阶段 3：晚期扩张 – 随着避孕普及和儿童经济价值降低，出生率开始下降，增长放缓。阶段 4：低位波动 – 两项比率均低且均衡，人口稳定或缓慢增长。阶段 5：衰退 – 出生率降至死亡率以下，导致自然减少和人口老龄化。
3. Population Pyramids | 人口金字塔
Expansive pyramids (wide base, concave sides) indicate high birth rates, high death rates, and a young population – typical of Stage 1 or early Stage 2 countries. Examples: Niger, Uganda. Constrictive pyramids (narrow base, wider middle) show low birth rates and an ageing population – typical of Stage 4 or 5 countries. Examples: Japan, Italy. Stationary pyramids (relatively straight sides) reflect low but stable birth and death rates. The shape reveals historical events like wars or baby booms, and helps governments plan for schools, healthcare, and pensions.
扩张型金字塔（底部宽、侧面凹）表明高出生率、高死亡率和年轻型人口 – 常见于阶段 1 或阶段 2 早期的国家，如尼日尔、乌干达。收缩型金字塔（底部窄、中部宽）显示低出生率和老龄人口 – 常见于阶段 4 或阶段 5 的国家，如日本、意大利。静止型金字塔（侧边近乎垂直）反映低而稳定的出生率和死亡率。金字塔轮廓能揭示战争或婴儿潮等历史事件，并帮助政府规划学校、医疗和养老金。
4. Settlement and Urban Land Use Models | 聚落与城市土地利用模型
Burgess Concentric Zone Model (1925): A city grows outward from a central business district (CBD) in concentric rings. Zone 1 CBD, Zone 2 transition zone (industry and old housing), Zone 3 low-class residential, Zone 4 middle-class residential, Zone 5 commuter zone. Criticised for being too simplistic and based solely on Chicago.
伯吉斯同心圆模型 (1925)：城市从中央商务区 (CBD) 向外以同心环状扩展。区域 1 CBD，区域 2 过渡带（工业与旧住宅），区域 3 低阶层住宅，区域 4 中阶层住宅，区域 5 通勤区。该模型被认为过于简单且仅基于芝加哥。
Hoyt Sector Model (1939): Land use develops in sectors or wedges radiating from the CBD along transport routes. Industry follows railways or rivers; high-class residential develops along desirable corridors. Explains why certain sectors remain industrial or wealthy over time.
霍伊特扇形模型 (1939)：土地利用沿交通线从 CBD 向外呈扇形或楔形发展。工业沿铁路或河流布局，高阶层住宅则沿优质廊道发展。解释了为何某些扇区长期保持工业化或富裕。
Harris & Ullman Multiple Nuclei Model (1945): Cities develop around multiple centres (nuclei) rather than a single CBD, influenced by the rise of car ownership and decentralisation. Examples include edge cities and specialised districts such as university zones or airports.
哈里斯和厄尔曼多核心模型 (1945)：城市围绕多个中心（核心）而非单一 CBD 发展，受汽车普及和去中心化影响。例证包括边缘城市以及大学区、机场等专门化区域。
5. River and Coastal Formulas | 河流与海岸计算公式
Gradient = Vertical Fall (m) / Horizontal Distance (m). Expressed as a ratio, e.g. 1:50 meaning a 1 m drop for every 50 m horizontally. Used to compare upper, middle and lower courses of a river.
坡度 = 垂直落差 (m) / 水平距离 (m)。以比例如 1:50 表示，即每 50 m 水平距离下降 1 m。用于比较河流上、中、下游的坡度。
Drainage Density = Total Length of Streams (km) / Drainage Basin Area (km²). High drainage density indicates rapid runoff, impermeable rock, and a flashy flood hydrograph. Low densities suggest permeable rock and slower runoff.
河网密度 = 河流总长度 (km) / 流域面积 (km²)。高河网密度表明快速地表径流、不透水岩石及暴涨暴落型洪水过程线。低密度则意味着透水岩石和较慢的径流。
Discharge (Q) = Cross-sectional Area (A) × Velocity (V). Measured in cubic metres per second (m³/s). Increases downstream as tributaries join. Fieldwork often uses a flowmeter and tape measure to calculate Q.
流量 (Q) = 横截面积 (A) × 流速 (V)。单位为立方米 / 秒 (m³/s)。因支流汇入，下游流量增大。野外调查常使用流速仪和卷尺来计算 Q。
6. Map Skills Formulas | 地图技能公式
Scale Conversion: Real Distance = Map Distance × Scale Factor. On a 1:50,000 map, 1 cm represents 50,000 cm (0.5 km). Always convert to metres or kilometres as required.
比例尺换算：实际距离 = 图上距离 × 比例因子。在 1:50,000 地图上，1 cm 代表 50,000 cm (0.5 km)。始终按题目要求转换为米或千米。
Gradient from a Map = Height Difference (m) / Horizontal Distance (m). Use contour lines to find the vertical interval, then measure the straight-line distance between the two points. Express as a ratio in its simplest form.
根据地图计算坡度 = 高差 (m) / 水平距离 (m)。利用等高线找出垂直间距，然后测量两点间的直线距离。化为最简整数比。
Area Estimation Using Grid Squares uares. Multiply by the area represented by one square (e.g. 1 km² if grid is 1 km × 1 km).
使用网格方块估算面积：数出图形覆盖的完整方格，再加半个上述数量的部分方格。乘以一个方格代表的实际面积（若方格为 1 km × 1 km，则为 1 km²）。
7. Weather and Climate Formulas | 天气与气候计算公式
Temperature Range = Maximum Temperature – Minimum Temperature. Use daily, monthly or annual figures. Large annual ranges indicate a continental climate, whereas coastal locations have smaller ranges.
气温较差 = 最高气温 – 最低气温。可使用日、月或年数据。年较差大表明大陆性气候，沿海地区则温差较小。
Mean Daily Temperature = (Max + Min) / 2. To calculate mean monthly temperature, sum all daily means and divide by the number of days in the month.
日平均气温 = (最高 + 最低) / 2。计算月平均气温时，将所有日平均值相加，再除以当月的天数。
Total Annual Precipitation = Sum of all monthly rainfall amounts (mm). Climate graphs often combine bar charts for rainfall and line graphs for temperature. Linking rainfall totals to seasonal patterns helps explain ecosystems and river regimes.
年降水总量 = 各月降水量总和 (mm)。气候图表常将降水柱状图与气温曲线图结合。将降水总量与季节模式联系起来，有助于解释生态系统和河流情势。
8. Development Indicators | 发展指标
GNI per Capita = Total Gross National Income / Total Population. This gives an average income per person in US dollars, enabling comparisons between countries. However, it disguises internal inequalities and does not measure non-economic aspects of development.
人均国民总收入 (GNI) = 国民总收入 / 总人口。得出以美元计的人均收入，便于国别比较。但它掩盖了国内不平等，且不衡量发展的非经济层面。
Human Development Index (HDI) combines three dimensions: health (life expectancy at birth), education (mean years of schooling and expected years), and standard of living (GNI per capita adjusted for PPP). Each dimension is normalised and the geometric mean is taken. HDI gives a more holistic picture than income alone.
人类发展指数 (HDI) 结合三个维度：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（按购买力平价调整的人均 GNI）。各维度经标准化后取几何平均值。HDI 比仅看收入更能全面反映发展。
Dependency Ratio (see Section 1) is also a key development indicator; high ratios indicate a heavy burden on the working population, common in both youthful and ageing populations.
抚养比（见第 1 节）也是关键的发展指标；高抚养比表明劳动人口负担沉重，常见于年轻型或老龄型人口。
9. Agricultural Location Theory (von Thünen) | 农业区位理论（冯·杜能模型）
von Thünen's model (1826) explains agricultural land use around a central market city. Four concentric rings form: Market gardening and dairy (perishable, high transport cost) nearest to the market; forest for fuel (in 19th-century context); extensive field crops like grain; and ranching and livestock grazing on the outer fringe. Land rent decreases with distance, and the cost of transport dictates which product is profitable at each zone. The model assumes uniform relief, soil fertility, and a single market – assumptions rarely met in reality, but the principle remains useful for understanding rural land use patterns.
冯·杜能模型 (1826) 解释了围绕中心市场城市的农业土地利用。形成四个同心环：市场园艺和乳业（易腐烂、运输成本高）最靠近市场；森林提供燃料（19 世纪背景）；大宗作物如谷物；最外缘是放牧业。地租随距离递减，运输成本决定各圈层何种产品能够盈利。模型假设地形、土壤均一且只有一个市场 – 现实中极少满足，但这一原理仍有助于理解乡村土地利用格局。
10. Industrial Location Theory (Weber) | 工业区位理论（韦伯）
Alfred Weber's Least Cost Theory (1909) states that industries locate where production costs are minimised. The three key factors are: transport costs (moving raw materials and finished goods), labour costs (cheap, skilled workforce), and agglomeration (benefits of clustering with other firms). When raw materials are heavier or bulkier than the finished product, the factory locates near the raw material; when the opposite is true, it locates near the market. A 'locational triangle' is used to visualise the pull of these factors.
阿尔弗雷德·韦伯的最小成本理论 (1909) 指出，工业布局在使生产成本最低的地方。三大关键因素为：运输成本（运输原料和成品）、劳动力成本（廉价熟练工）及集聚效应（与其它企业聚集带来的收益）。当原料比成品更重或体积更大时，工厂靠近原料地；反之则靠近市场。使用“区位三角形”可直观显示这些因素的拉力。
11. Natural Hazards Measurement | 自然灾害度量
The Richter scale measures the magnitude (energy released) of an earthquake logarithmically: each whole number increase represents a tenfold increase in wave amplitude and roughly 32 times more energy. The Mercalli scale measures observed intensity (effects on people and structures) on a I–XII scale. The Saffir–Simpson scale rates hurricanes from Category 1 to 5 based on sustained wind speed and potential damage. None of these are simple algebraic formulas, but understanding their scales is fundamental to analysing natural hazard impacts.
里氏震级 以对数方式度量地震的大小（释放的能量）：每增加一个整级，地震波振幅增大 10 倍，能量约增大约 32 倍。麦加利烈度 按 I–XII 级度量观测到的强度（对人和建筑物的影响）。萨菲尔–辛普森等级 根据持续风速和潜在损害将飓风分为 1 至 5 级。这些并非简单的代数公式，但理解其量度标准是分析自然灾害影响的基础。
12. General Geographical Skills | 一般地理技能
Percentage Change = [(New Value – Old Value) / Old Value] × 100. Use this to describe trends in population, GDP, or trade over time. Always state whether the change is an increase or decrease.
百分比变化 = [(新值 – 旧值) / 旧值] × 100。用于描述人口、GDP 或贸易随时间变化的趋势。务必说明变化是增加还是减少。
Line of Best Fit on a Scattergraph: Not a formula but a skill. Draw a straight line passing through as many points as possible, with equal numbers above and below. The line suggests a relationship (positive, negative, or no correlation) between two variables, such as GNI per capita and life expectancy.
散点图上的最佳拟合线：虽非公式却是一项技能。画一条尽可能穿过最多点、且两侧点数大致相等的直线。该线可以揭示两个变量间的关系（正相关、负相关或无相关），如人均 GNI 与预期寿命。
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