📚 Year 11 CAIE Geography: Quick Reference Handbook of Formulas and Theorems | Year 11 CAIE地理：公式定理速查手册
For Year 11 CAIE IGCSE Geography students, mastering the essential formulas and conceptual models is key to success in both Paper 1 and Paper 2. This quick reference handbook brings together all the critical quantitative skills, demographic equations, river and coastal measurements, and the most important geographical theorems you will encounter in your exams. Keep this guide handy while revising skills-based questions and case study analyses.
对于Year 11 CAIE IGCSE地理的学生来说，掌握基本公式和概念模型是在Paper 1和Paper 2中取得成功的关键。这本速查手册汇集了你在考试中会遇到的所有关键定量技能、人口统计方程、河流与海岸测量以及最重要的地理定理。在复习技能类题目和案例分析时，请随身携带这份指南。
1. Map Scales and Distance Conversions | 地图比例尺与距离换算
Map scale expresses the relationship between a distance on the map and the corresponding distance on the ground. It can be given as a ratio (e.g. 1:50 000), a word statement (e.g. ‘2 cm to 1 km’) or a linear scale bar.
地图比例尺表示地图上的距离与地面实际距离之间的关系。它可以表示为比率（如1:50 000）、文字说明（如”图上2厘米代表实地1千米”）或直线比例尺。
The basic formula for converting map distance to real-world distance is: Real Distance = Map Distance × Scale Factor. Always ensure both distances are in the same units before converting. For a 1:25 000 map, a 4 cm measurement represents 4 cm × 25 000 = 100 000 cm = 1 km.
将地图距离转换为实地距离的基本公式为：实地距离 = 地图距离 × 比例尺因子。换算前务必确保单位一致。对于1:25 000的地图，图上4厘米的测量值表示实地4 cm × 25 000 = 100 000 cm = 1 km。
When using a linear scale, place a piece of paper along the route, mark the distance, and then align the paper against the scale bar. For winding roads or rivers, use a string to follow the curves before measuring its length.
使用直线比例尺时，可将纸条沿路线放置，标出距离，然后将纸条与比例尺对齐读数。对于弯曲的道路或河流，可用细线沿曲线摆放，再测量细线长度。
2. Area Calculation on Maps | 地图面积计算
To calculate the area of a regular shape (e.g. a rectangular field) on a topographic map, first measure its length and width in centimetres, convert each to real ground distances using the scale, and then multiply: Area = Length × Width.
计算地形图上规则形状（如矩形田地）的面积时，先以厘米为单位测量其长度和宽度，利用比例尺分别转换为实地距离，然后相乘：面积 = 长 × 宽。
For irregular shapes, the grid square method is used. Count the number of full grid squares inside the feature and estimate the number of partial squares. On a 1:50 000 map, one grid square (2 cm × 2 cm) represents 1 km² on the ground.
对于不规则形状，则使用格网法。统计该要素内部完整格网方格的数量，并估算部分方格的数量。在1:50 000地图上，一个方格（2厘米×2厘米）代表实地1平方千米。
Formula for area using grid squares: Area = (Number of full squares + ½ × Number of partial squares) × Area per grid square. Always check the map’s grid interval to determine the area represented by each square.
使用格网方格计算面积的公式为：面积 = （完整方格数 + ½ × 部分方格数）× 每个方格的实地面积。务必检查地图的格网间距，以确定每个方格所代表的实际面积。
3. Gradient and Slope | 梯度与坡度
Gradient is a measure of how steep a slope is, often expressed as a ratio 1 : R. The formula is: Gradient = Vertical Interval (VI) ÷ Horizontal Equivalent (HE). VI is the difference in height between two points; HE is the horizontal ground distance between them.
梯度是衡量坡度陡缓的指标，通常表示为1 : R的比值。公式为：梯度 = 垂直间距（VI）÷ 水平距离（HE）。VI是两点之间的高差；HE是它们之间的水平地面距离。
Both VI and HE must be in the same unit (e.g. metres). To express the gradient as 1 : R, divide HE by VI after ensuring VI is made equal to 1 unit. For example, VI = 20 m, HE = 400 m gives gradient = 20/400, which simplifies to 1 : 20.
VI和HE必须使用相同单位（如米）。要以1 : R的形式表示梯度，先将VI定为1单位，然后用HE除以VI。例如，VI = 20 m, HE = 400 m，梯度 = 20/400，化简为1 : 20。
Slope angle can also be calculated using trigonometry: tan(θ) = VI ÷ HE, then find θ = tan⁻¹(VI/HE). Remember that a gradient of 1 : 5 indicates a much steeper slope than 1 : 50.
坡度角也可用三角学计算：tan(θ) = VI ÷ HE，然后求θ = tan⁻¹(VI/HE)。记住，梯度1 : 5表示比1 : 50陡得多。
4. Population Density | 人口密度
Population density is the average number of people per unit area, usually expressed as persons per km². The formula is: Population Density = Total Population ÷ Land Area (km²).
人口密度是单位面积上的平均人口数量，通常表示为人/平方千米。公式为：人口密度 = 总人口 ÷ 土地面积（平方千米）。
Be careful with units: if the area is given in hectares, convert to km² (1 km² = 100 hectares). Population density figures can mask variations within a country; it is an average, so internal disparities (e.g. dense urban cores vs. sparse rural areas) are not shown.
注意单位：如果面积以公顷给出，则转换为平方千米（1 km² = 100公顷）。人口密度数据可能掩盖一个国家内部的差异；它是一个平均值，因此无法显示内部差异（如稠密的城市核心与稀疏的农村地区）。
High population density can indicate pressure on resources and infrastructure, while low density might suggest underused potential or physical constraints such as harsh climate. Always interpret density figures in context with other geographical indicators.
高人口密度可能表明资源与基础设施面临压力，而低密度可能意味着潜力未充分利用或自然条件限制（如恶劣气候）。务必结合其他地理指标解读密度数据。
5. Birth Rate, Death Rate and Natural Increase | 出生率、死亡率和自然增长率
The crude birth rate (CBR) is the number of live births per 1000 people per year: CBR = (Total live births ÷ Total population) × 1000. The crude death rate (CDR) is similarly: CDR = (Total deaths ÷ Total population) × 1000.
粗出生率（CBR）是每年每千人中的活产婴儿数：CBR = （活产婴儿总数 ÷ 总人口）× 1000。粗死亡率（CDR）类似：CDR = （死亡总人数 ÷ 总人口）× 1000。
The natural increase rate (NIR) is expressed as a percentage or per 1000: Natural Increase = (CBR – CDR) / 10 (to get a percentage). For example, if CBR = 25 per 1000 and CDR = 10 per 1000, NIR = (25 – 10) / 10 = 1.5%.
自然增长率（NIR）以百分比或千分比表示：自然增长率 = (CBR – CDR) / 10（得百分比）。例如，若CBR为每千人25，CDR为每千人10，则NIR = (25 – 10) / 10 = 1.5%。
High natural increase is typical of Stage 2 of the Demographic Transition Model (high births, falling deaths). Low or negative natural increase is seen in Stage 4 or 5 countries. CAIE often asks candidates to calculate or interpret these rates.
高自然增长率常见于人口转变模型的第二阶段（高出生率、死亡率下降）。低增长或负增长见于第四或第五阶段国家。CAIE常要求考生计算或解读这些比率。
6. Population Growth Rate | 人口增长率
The overall population growth rate includes migration as well as natural change: Growth Rate (%) = ((Births – Deaths) + (Immigration – Emigration)) ÷ Total Population × 100. This captures net migration’s impact.
总人口增长率包括迁移和自然变化：增长率（%）= （（出生 – 死亡） + （迁入 – 迁出）） ÷ 总人口 × 100。这反映了净迁移的影响。
When using data over several years, the annual growth rate can be approximated by the formula: r = (1/t) × ln(P(t)/P(0)), but for IGCSE, simple arithmetic using the above is sufficient. Focus on comparing growth rates between countries or time periods.
当使用多年数据时，年增长率可用公式r = (1/t) × ln(P(t)/P(0))近似计算，但IGCSE阶段用上述简单算术即可。重点在于比较不同国家或时期的增长率。
A positive growth rate indicates a rising population, while a negative rate suggests population decline. Such trends have implications for service provision, economic planning and environmental sustainability.
正增长率表明人口在增加，负增长率则意味着人口减少。这种趋势对服务供给、经济规划和环境可持续性都有影响。
7. Dependency Ratio | 抚养比率
The dependency ratio indicates the economic burden on the working-age population. Formula: Dependency Ratio = ((Population aged 0–14) + (Population aged 65+)) ÷ (Population aged 15–64) × 100.
抚养比率反映了劳动年龄人口所承受的经济负担。公式：抚养比率 = （（0–14岁人口） + （65岁以上人口）） ÷ （15–64岁人口） × 100。
The result shows the number of dependents per 100 working-age people. A high ratio, say 85, means 85 dependents for every 100 workers, placing pressure on healthcare, education and pensions. Countries in early stages of the demographic transition tend to have high youth dependency; ageing populations in developed countries increase old-age dependency.
结果显示每100名劳动年龄人口对应的受抚养人数。若比率较高，如85，意味着每100名劳动者需抚养85人，给医疗、教育和养老金带来压力。人口转变早期阶段的国家往往有较高的少儿抚养负担；发达国家的人口老龄化则增加了老年抚养负担。
When analyzing a population pyramid, you can estimate the dependency ratio by comparing the widths of the base and top to the middle. This supports explanation of socio-economic challenges.
分析人口金字塔时，可通过比较底部和顶部与中部的宽度来估算抚养比率。这有助于深入解释社会经济挑战。
8. River Discharge and Drainage Density | 河流流量与河网密度
River discharge (Q) is the volume of water flowing through a river cross-section per unit time. Formula: Discharge (m³/s) = Cross-sectional Area (A) × Velocity (V). Area is measured in m² by multiplying mean width by mean depth.
河流流量（Q）是单位时间内通过河流横截面的水体体积。公式：流量（立方米/秒）= 横截面积（A）× 流速（V）。面积通过平均宽度乘以平均深度得出，单位为平方米。
Drainage density is a measure of how closely spaced the river channels are in a basin: Drainage Density = Total length of all streams (km) ÷ Basin area (km²). High drainage density often indicates impermeable rock, steep slopes or high rainfall, leading to rapid runoff.
河网密度衡量流域内河道间距的疏密：河网密度 = 所有河道总长度（千米） ÷ 流域面积（平方千米）。高河网密度通常表示不透水岩石、陡坡或高降雨量，导致快速径流。
These calculations are essential in flood risk analysis and understanding basin hydrology. CAIE paper 2 skills questions may require reading values from a cross-section graph and calculating discharge.
这些计算对于洪水风险分析和理解流域水文至关重要。CAIE Paper 2技能题可能要求从横截面图中读取数值并计算流量。
9. The Bradshaw Model (River Variables) | 布拉德肖模型（河流变量变化）
The Bradshaw Model is a theoretical model that describes how key river characteristics change from source to mouth. Discharge, occupied channel width, mean depth and average velocity all increase downstream, while load particle size and channel bed roughness decrease.
布拉德肖模型是一个理论模型，描述了从源头到河口，河流关键特征如何变化。流量、水面宽度、平均深度和平均流速均向下游增加，而泥沙粒径和河床糙率则减小。
You may be asked to draw or interpret a Bradshaw diagram, with arrows indicating the direction of change. This model is tested through fieldwork data comparison; deviations from the model can occur due to human interference (e.g. dams) or tributary inputs.
你可能会被要求绘制或解释布拉德肖图，图中用箭头指示变化方向。该模型通过实地数据比较进行检验；由于人为干预（如水坝）或支流汇入，可能会出现偏离模型的情况。
Remember the expected trends: cross-sectional area, hydraulic radius and efficiency increase with distance downstream. In exam answers, support commentary with specific data or sketch graphs based on fieldwork.
记住预期趋势：横截面积、水力半径和效率随下游距离增加而增大。在考试作答时，应结合实地考察中的具体数据或简图来支持评述。
10. The Demographic Transition Model | 人口转变模型
The Demographic Transition Model (DTM) describes how birth rates and death rates change over time as a country develops, leading to five stages. Stage 1: high fluctuating; Stage 2: early expanding; Stage 3: late expanding; Stage 4: low fluctuating; Stage 5: declining.
人口转变模型（DTM）描述了随着国家的发展，出生率和死亡率如何随时间变化，形成五个阶段。第一阶段：高位静止；第二阶段：早期扩张；第三阶段：晚期扩张；第四阶段：低位静止；第五阶段：人口负增长。
Although not a numerical formula, the DTM is a key geographical theorem. Exam questions often ask you to place case study countries in their appropriate stage and justify with data on birth/death rates, GNI, or industrial structure.
虽然DTM不是一个数值公式，但它是一项关键的地理定理。考试题目常要求将案例研究国家归入合适的阶段，并用出生/死亡率、国民总收入或产业结构等数据进行论证。
Remember that the model is Eurocentric and not all countries follow the exact sequence; some, such as NICs, may pass through stages more rapidly. Criticisms include its failure to account for migration or government policies like China’s one-child policy.
记住，该模型以欧洲为中心，并非所有国家都严格遵循这一顺序；一些新兴工业化国家（NIC）可能更快地经历各个阶段。其局限性包括未能考虑迁移因素或如中国独生子女政策这样的政府干预。
11. The Burgess Model (Urban Land Use) | 伯吉斯模型（城市土地利用）
Burgess’s concentric zone model is a classic urban structure theorem. It divides a city into five concentric rings: the Central Business District (CBD), transition zone (industry and old housing), low-class residential, medium-class residential, and commuter zone/suburbs.
伯吉斯的同心圆模型是一个经典的城市结构定理。它将城市划分为五个同心圆环带：中央商务区（CBD）、过渡带（工业与旧住宅）、低阶层住宅区、中阶层住宅区和通勤区/郊区。
As distance from the CBD increases, land value generally decreases, and housing density lowers while green space increases. The model is based on 1920s Chicago and assumes flat land, equal accessibility, and a radial transport network.
随着离CBD距离增加，土地价值普遍下降，住房密度降低，而绿地增加。该模型基于20世纪20年代的芝加哥，假设地势平坦、通达性均等以及放射状交通网络。
In CAIE exams, you may compare this with the Hoyt sector model or the Harris and Ullman multiple nuclei model. Criticisms of Burgess include its ignorance of physical barriers, changing commuter patterns, and the growth of edge cities.
在CAIE考试中，你可能需要将此模型与霍伊特扇形模型或哈里斯-厄尔曼多核心模型进行比较。对伯吉斯模型的批评包括：它忽略了自然障碍、通勤模式的变化以及边缘城市的发展。
12. The Hydrological Cycle and Water Balance | 水文循环与水量平衡
The water balance equation sums up the hydrological cycle for a drainage basin: Precipitation (P) = Streamflow (Q) + Evapotranspiration (E) ± Change in Storage (ΔS). It is a fundamental theorem used in physical geography.
水量平衡方程总结了流域内的水文循环：降水（P） = 径流量（Q） + 蒸发散（E） ± 蓄水量变化（ΔS）。这是自然地理中使用的一项基本定理。
For exam purposes, you may need to calculate one component given the others. For instance, if P = 1200 mm, Q = 500 mm and E = 400 mm, then ΔS = 1200 – 500 – 400 = +300 mm (surplus recharge). A negative ΔS indicates net depletion of soil and groundwater stores.
考试中，你可能需要根据其他已知分量计算某一分量。例如，若P = 1200 mm，Q = 500 mm，E = 400 mm，则ΔS = 1200 – 500 – 400 = +300 mm（盈余补给）。负的ΔS表示土壤和地下水储量的净消耗。
Understanding the balance helps explain river regimes, drought and flood risk. This concept links to climate graphs and storm hydrographs, which are essential skills in Paper 2.
理解水量平衡有助于解释河流情势、干旱和洪水风险。这一概念与气候图和暴雨流量过程线相关，这些都是Paper 2中的基本技能。
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