A-Level CAIE Geography: Formula & Theorem Quick Reference | A-Level CAIE 地理：公式定理速查手册

📚 A-Level CAIE Geography: Formula & Theorem Quick Reference | A-Level CAIE 地理：公式定理速查手册

This handbook compiles the most frequently tested formulae, calculations and statistical techniques in CAIE A-Level Geography. It is designed for quick revision across physical, human and environmental geography topics, allowing you to recognise the formula type, apply it correctly and interpret the result within an exam context.

本手册汇总了 CAIE A-Level 地理中最常考测的公式、计算方法与统计技术，覆盖自然地理、人文地理与环境地理等主题，帮助你在考试中快速识别公式类型、正确代入、并合理解释计算结果。

1. Map Skills & Scale Calculations | 地图技能与比例尺计算

Scale expresses the relationship between a distance on a map and the corresponding distance on the ground. It is written as a numeric scale, a line scale, or a statement scale. To find real distance, multiply the measured map distance by the scale denominator.

比例尺表示地图上距离与实地距离的比值，可写成数字比例尺、直线比例尺或文字比例尺。计算实地距离时，用图上距离乘以比例尺分母即可。

Real distance = map distance × scale denominator

实地距离 = 图上距离 × 比例尺分母

For area, because length is squared, the conversion factor must also be squared.

计算面积时，由于长度具有平方关系，换算系数也必须进行平方。

Real area = map area × (scale denominator)²

实地面积 = 图上面积 × （比例尺分母）²

Gradient measures the steepness of a slope and is a common map skill question on Paper 2.

坡度用于衡量地面倾斜程度，是 Paper 2 常见的读图计算题考点。

Gradient = vertical height / horizontal distance

坡度 = 垂直高度 / 水平距离

Type \| 类型	Example \| 示例	Use \| 用途
Numeric scale 数字比例尺	1 : 50,000	Distance & area conversion 距离与面积换算
Linear scale 直线比例尺	0 – 1 – 2 km shown as a line	Quick measurement on the map 图上快速量测
Gradient 坡度	V.I. / H.E. = 100 m / 400 m = 1 : 4	Slope steepness assessment 坡度分析

2. Water Balance & Runoff | 水分平衡与径流

The water balance equation links precipitation, evaporation and changes in storage. It is the core framework for drainage basin studies.

水分平衡方程将降水、蒸发和储水变化联系起来，是流域研究的基础框架。

P = Q + E ± ΔS

P = Q + E ± ΔS

Where P is precipitation, Q is runoff or streamflow, E is evapotranspiration, and ΔS is the change in water stored in soil, groundwater, lakes and vegetation.

其中 P 为降水量，Q 为径流量或河流流量，E 为蒸散量，ΔS 为土壤水、地下水、湖泊和植被中储存量的变化。

Runoff coefficient is the proportion of rainfall that becomes direct river discharge. It is used to compare catchment responses and is especially relevant in urbanisation and flood-management questions.

径流系数表示降水中转化为直接径流的比例，用于比较不同流域的水文响应，在城市化和洪水管理题目中尤为重要。

Runoff coefficient = total runoff / total precipitation

径流系数 = 总径流量 / 总降水量

Discharge is measured in cumecs and can be derived from the stream cross-section.

流量以 cumecs（立方米每秒）为单位，可由河流横断面计算得出。

Q = A × v

流量 Q = 断面积 A × 平均流速 v

To convert discharge collected over a period into a depth of water over the catchment, use Runoff depth = Q × time / catchment area.

若需将一段时间内径流量换算为流域水深，可采用：径流深 = 流量 × 时间 / 流域面积。

3. River Hydraulics & Morphometry | 河流水力学与形态计量

Understanding river flow requires linking velocity, channel shape and friction. The hydraulic radius is a primary measure of channel efficiency, and it increases when the channel is wider and deeper relative to its wetted perimeter.

理解河流流动需要联系流速、河道形态与摩擦力。水力半径是衡量河道效率的重要指标，当河道相对湿周更宽更深时，水力半径增大。

R = A / P, where P is the wetted perimeter

水力半径 R = 过水断面积 A / 湿周 P

The Manning equation estimates mean velocity in an open channel under uniform flow. It combines hydraulic radius, channel slope and a roughness coefficient.

曼宁公式用于估计明渠均匀流条件下的平均流速，综合了水力半径、河道坡度和粗糙系数。

v = (1 / n) × R^(2/3) × S^(1/2)

Here v is velocity in m/s, n is Manning’s roughness coefficient, R is hydraulic radius in metres, and S is the gradient of the channel bed as a dimensionless ratio.

其中 v 为流速（米/秒），n 为曼宁粗糙系数，R 为水力半径（米），S 为河床坡度（无量纲比值）。

Drainage density measures how finely a drainage network dissects a basin. Compare or compute it using the total stream length divided by basin area.

河网密度用于衡量水系对流域的切割程度，计算时用河流总长度除以流域面积。

Drainage density = total length of all streams / basin area

河网密度 = 河流总长度 / 流域面积

Bifurcation ratio measures how a stream network branches. It equals the number of streams of one order divided by the number of streams of the next highest order.

分叉比表示水系分支的程度，等于某一级河流数量除以上一级更高阶河流的数量。

Bifurcation ratio = Nᵤ / Nᵤ₊₁

分叉比 = 第 u 级河流数量 / 第 u+1 级河流数量

4. Population Change & Momentum | 人口变化与动力

Population natural increase is calculated from crude birth and death rates. It is often expressed per 1000 people per year, then converted to a percentage.

人口自然增长由粗出生率与粗死亡率计算而来，通常以每千人每年表示，再转换为百分比。

Natural increase rate = CBR − CDR

人口自然增长率 = 粗出生率 − 粗死亡率

Doubling time estimates how long a population takes to double under a constant growth rate. The Rule of 70 is an approximation widely accepted in A-Level geography.

倍增时间估计人口在恒定增长率下翻番所需年数。70 法则是一种被 A-Level 地理广泛接受的近似计算方法。

Doubling time ≈ 70 / annual growth rate (%)

人口倍增时间 ≈ 70 / 年增长率（%）

Population density is a straightforward but vital measure for comparing settlement, agriculture and resource pressure.

人口密度是描述人口压力的基础指标，在比较聚落、农业与资源承载时非常重要。

Population density = total population / land area (km²)

人口密度 = 总人口 / 土地面积（km²）

Dependency ratio indicates the weight of young and elderly population on the working-age population. It is useful in evaluating demographic dividend or ageing burden.

抚养比反映少年儿童与老年人口对劳动年龄人口的压力，可用于评估人口红利和老龄化负担。

Dependency ratio = (population aged 0–14 + 65+) / population aged 15–64 × 100

抚养比 = （0–14 岁人口 + 65 岁及以上人口）/ 15–64 岁人口 × 100

Rate of natural change can be converted into a percentage by dividing the per-thousand figure by 10.

将千分率除以 10 即可转换为百分比，便于后续倍增时间计算。

5. Settlement & Gravity Models | 聚落与引力模型

The gravity model is used to estimate interaction between two places, such as migration, trade, traffic flow or service threshold. Interaction is proportional to the product of their populations and inversely proportional to distance, often squared.

引力模型用于估算两地之间的相互作用，如迁移、贸易、交通流或服务阈值。相互作用与两地人口乘积成正比，与距离（通常为平方）成反比。

I = (G × M₁ × M₂) / D²

Here I is interaction, M₁ and M₂ are population masses of the two settlements, D is distance, and G is a gravitational constant or calibration factor.

其中 I 为相互作用量，M₁ 与 M₂ 为两地人口规模，D 为距离，G 为引力常数或校准因子。

Reilly’s law of retail gravitation identifies the breakpoint where customers are equally likely to shop in either of two towns. It is a common stimulus-response question in settlement studies.

雷利零售引力法则用于确定消费者选择两城镇购物的均衡分界点，是聚落区位论的常见考点。

D_a = d / (1 + √(P_b / P_a))

Where D_a is the breaking-point distance from town A, d is the total distance between towns, and P_a and P_b are their populations.

其中 D_a 为从城镇 A 到分界点的距离，d 为两城镇间总距离，P_a 与 P_b 为两城镇人口。

Central place theory relies on threshold and range. Threshold is the minimum population needed to support a service; range is the maximum distance consumers will travel.

中心地理论依赖于门槛与腹地范围。门槛指维持某项服务所需的最小人口数，范围指消费者愿意前往的最远距离。

6. Agricultural Systems & Indices | 农业系统与指数

Agricultural geography requires quantitative measures of land use intensity, yield and crop efficiency. The most common exam calculation is cropping intensity, which compares gross cropped area with net sown area.

农业地理需要定量衡量土地利用强度、产量与作物效率。最常见的考试计算是复种指数，即将总播种面积与净播种面积比较。

Cropping intensity = gross cropped area / net sown area × 100

复种指数 = 总播种面积 / 净播种面积 × 100

Yield measures output per unit area, and is often compared across regions, farming types or time periods.

单产衡量单位面积的产出，通常用于比较不同地区、农业类型或时间阶段。

Yield = total output / cultivated area

单产 = 总产量 / 种植面积

Seed rate and plant density are also tested, especially in intensive land-use and green revolution case studies.

播种量和种植密度也是重要考点，尤其出现在集约农业和绿色革命案例分析中。

Plant density = number of plants / area (plants per m² or ha)

种植密度 = 植株数量 / 面积（株/m² 或株/ha）

Water use efficiency evaluates crop production per unit of water consumed, an increasingly important metric under climate change and irrigation demand.

水分利用效率评估每消耗单位水量的作物产出，在气候变化与灌溉需求增加的背景下越来越重要。

WUE = crop yield / water used

水分利用效率 WUE = 作物产量 / 用水量

7. Economic Multiplier & Location Quotient | 经济乘数与区位熵

The multiplier effect explains how an initial injection of spending creates additional rounds of income and employment. In geography, it is often used to assess the impact of tourism, infrastructure or manufacturing investment.

乘数效应解释初始支出如何通过多轮收入与就业创造放大经济影响。在地理学中，常用于评估旅游、基础设施或制造业投资的带动作用。

k = 1 / (1 − MPC)

乘数 k = 1 / (1 − MPC)

Here MPC is the marginal propensity to consume. A higher MPC produces a larger multiplier; a higher leakage, through saving or importing, reduces it.

式中 MPC 为边际消费倾向。MPC 越高，乘数越大；储蓄、进口等漏出越多，乘数越小。

The location quotient measures whether a region specialises in a particular industry compared with the national average. Values above 1 indicate local specialisation and export orientation.

区位熵用于衡量某地区特定行业相对于全国平均水平的专门化程度。数值大于 1 表示该地区具有专门化和对外输出倾向。

LQ = (local industry employment / local total employment) ÷ (national industry employment / national total employment)

区位熵 LQ = （当地某行业就业 / 当地总就业） ÷ （全国某行业就业 / 全国总就业）

Basic employment multiplier is an alternative approach, derived from the ratio of total employment to basic employment.

基础就业乘数是另一种常见方法，可由总就业与基础产业就业之比求得。

Employment multiplier = total employment / basic employment

就业乘数 = 总就业 / 基础产业就业

8. Biodiversity & Sampling Indices | 生物多样性与取样指数

Simpson’s diversity index is a core statistical measure in ecological geography. It accounts for both species richness and evenness: higher values indicate greater diversity.

辛普森多样性指数是生态地理学中的核心统计指标，同时考虑物种丰富度与均匀度，数值越高代表多样性越大。

D = 1 − Σ(nᵢ / N)²

Here nᵢ is the number of individuals of species i, and N is the total number of individuals of all species. A floristic sample in a field study would produce values from 0 to 1, although CAIE sometimes uses the alternative reciprocal form.

其中 nᵢ 为第 i 种物种的个体数，N 为所有物种个体总数。野外样方调查得到的 D 值在 0 到 1 之间；CAIE 偶尔也使用另一种倒数形式公式。

Species frequency measures how often a species occurs across sample quadrats, while density measures individuals per area.

频度衡量某物种在样方中出现概率的普遍程度，而密度衡量单位面积内的个体数。

Frequency = (number of quadrats containing species / total quadrats) × 100

频度 = （出现该物种的样方数 / 样方总数） × 100

Density = number of individuals / total area sampled

密度 = 个体总数 / 采样总面积

Percentage cover is a quick visual estimate in vegetation sampling, and is used to compare plant communities across a transect.

盖度百分比是植被取样中常用的快速目测估算指标，用于比较样线沿线不同植物群落。

9. Statistical Testing in Geography | 地理统计检验

Students are expected to choose and calculate a statistical test appropriate to the data. The most common are mean, standard deviation, Spearman’s rank correlation, chi-square and the Mann-Whitney U test.

地理大纲要求学生能够选择并计算合适的统计检验。最常见的是均值、标准差、斯皮尔曼等级相关、卡方检验和曼-惠特尼 U 检验。

Standard deviation measures the dispersion of data around the mean. In geography it is used to compare the reliability of rainfall, river discharge, population density and regenerated economic data.

标准差衡量数据围绕均值的离散程度，常用于比较降水、河流流量、人口密度和经济再生数据的可靠性。

σ = √(Σ(x − x̄)² / n)

Spearman’s rank correlation tests for a monotonic relationship between two variables. Use when data are ordinal or not normally distributed.

斯皮尔曼等级相关用于检验两个变量之间的单调相关关系，适用于有序数据或不满足正态分布的数据。

R = 1 − 6Σd² / (n(n² − 1))

Here d is the difference between the two ranks for each item, and n is the number of paired observations. Critical values must be compared with the calculated R.

其中 d 为每组数据在两个变量中的等级差，n 为配对数。必须将计算所得 R 值与临界值表比较。

Chi-square tests whether observed frequencies differ significantly from expected frequencies. It is useful for settlement distributions, land-use change counts and questionnaire responses.

卡方检验用于判断观察频数与期望频数是否存在显著差异，适用于聚落分布、土地利用变化计数和问卷答复类数据。

χ² = Σ((O − E)² / E)

Remember to check the degrees of freedom, df = (rows − 1) × (columns − 1), and choose the correct critical value at the 0.05 significance level.

注意自由度 df = (行数 − 1) × (列数 − 1)，并在 0.05 显著性水平下选择相应临界值。

10. Coastal, Climate & Hazard Formulas | 海岸、气候与灾害公式

Physical geography papers often assess tsunamis, wave characteristics and carbon cycle fluxes. The tsunami wave celerity is directly related to water depth.

自然地理试卷常涉及海啸、波浪特征与碳循环通量。海啸波速与水深直接相关。

c = √(g × d)

Here c is wave celerity in m/s, g is gravitational acceleration (9.8 m/s²), and d is water depth in metres. The formula explains why tsunamis travel slowly in shallow water but accelerate in deep ocean.

其中 c 为波速（米/秒），g 为重力加速度（9.8 m/s²），d 为水深（米）。该公式解释了海啸在浅水区波速慢，在深海中波速快的原因。

For ordinary waves, speed can be found from

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A-Level CAIE Geography: Formula & Theorem Quick Reference | A-Level CAIE 地理：公式定理速查手册