📚 Year 12 CAIE Geography: Formulas & Theorems Quick Reference | Year 12 CAIE 地理：公式定理速查手册
This quick-reference handbook brings together the essential quantitative tools, theorems and key models required for the Year 12 CAIE Geography (9696) syllabus. From hydrological budgets and population change to urban hierarchies and spatial interaction, mastering these formulas will sharpen your data-response skills and deepen your analytical essays. Use this guide alongside your case studies for quick revision before tests and the final examination.
本速查手册汇集了 Year 12 CAIE 地理（9696）教学大纲所要求的核心定量工具、定理与重要模型。从水文平衡与人口变动到城市等级体系与空间相互作用，掌握这些公式将助你在数据分析题和论述分析中事半功倍。请结合你的案例学习使用本指南，便于在单元测验和统考前快速复习。
1. River Discharge & Water Balance | 河流径流量与水平衡
Discharge (Q) is the volume of water passing a channel cross‑section per unit time. The fundamental formula is Q = A × V, where A is the cross‑sectional area (m²) and V is the mean flow velocity (m s⁻¹). Q is expressed in cubic metres per second (m³ s⁻¹) or cumecs. Accurate discharge measurements underpin flood‑warning systems and dam design.
Q = A × V
径流量（Q）是指单位时间内通过河道断面的水体体积。基本公式为 Q = A × V，A 为过水断面面积（m²），V 为平均流速（m s⁻¹）。Q 的单位是立方米每秒（m³ s⁻¹），又称 cumecs。准确的流量数据是洪水预警和水库设计的基础。
The hydrological (water) budget for a drainage basin is expressed as P = Q + E + ΔS, where P = total precipitation, Q = runoff (streamflow leaving the basin), E = evapotranspiration, and ΔS = net change in storage (soil moisture, groundwater, lakes, snowpack). This equation closes the water balance and reveals whether a basin is gaining or losing stored water over a given period.
P = Q + E + ΔS
流域水文（水量）平衡方程表述为 P = Q + E + ΔS，其中 P 为总降水量，Q 为径流量（流出流域的河道流量），E 为蒸散发量，ΔS 为蓄水量净变化（土壤水、地下水、湖泊、积雪等）。这一方程闭合了水量收支，能够揭示研究时段内流域储水是增加还是减少。
2. Channel Geometry: Gradient & Sinuosity | 河道几何：坡降与弯曲度
Stream gradient represents the slope of a river reach. It is calculated as the vertical drop (rise) divided by the horizontal distance (run): Gradient = Δh / L. The result can be given as a percentage (multiply by 100) or as a dimensionless ratio, e.g. 1 in 150. Steep gradients favour vertical erosion and coarse bedload transport, while gentle gradients encourage lateral migration and deposition.
Gradient = Δh / L
河流坡降（比降）表示河段的倾斜程度，用垂直落差（Δh）除以水平距离（L）计算：比降 = Δh / L。结果可表示为百分比（乘以100）或无量纲比值，如 1:150。陡坡有利于下切侵蚀和粗颗粒推移质搬运，缓坡则促进侧向迁移和沉积。
Sinuosity measures how much a channel meanders. Sinuosity index (SI) = channel length (CL) ÷ straight‑line valley length (VL). A perfectly straight channel has SI = 1.0. Values between 1.0 and 1.5 describe sinuous channels, while SI > 1.5 identifies a meandering river. High sinuosity influences flow velocity, flood‑risk patterns and the formation of oxbow lakes.
SI = CL / VL
弯曲度衡量河道的蜿蜒程度。弯曲指数 SI = 河道长度（CL）÷ 河谷直线长度（VL）。完全顺直的河道 SI = 1.0；1.0–1.5 为微弯河道，大于 1.5 则为典型曲流河。高弯曲度影响流速、洪水泛滥格局以及牛轭湖的形成。
3. Crude Birth Rate, Death Rate & Natural Increase | 粗出生率、粗死亡率与自然增长
The Crude Birth Rate (CBR) is the number of live births in a year per 1000 people: CBR = (B ÷ P) × 1000. The Crude Death Rate (CDR) is similar: CDR = (D ÷ P) × 1000, where D is the number of deaths and P is the total mid‑year population. Both are expressed per mille (‰).
CBR = (B / P) × 1000
粗出生率（CBR）指某一年每千人的活产婴儿数：CBR = (B ÷ P) × 1000。粗死亡率（CDR）同理：CDR = (D ÷ P) × 1000，其中 D 为死亡人口，P 为年中总人口。两者均以千分比（‰）表示。
Natural increase (per 1000) is simply the difference: NI = CBR – CDR. To convert this into an annual growth percentage, the rate of natural increase (RNI %) is obtained by dividing the difference by 10: RNI (%) = (CBR – CDR) ÷ 10. A positive RNI indicates population growth excluding migration.
NI = CBR – CDR | RNI (%) = (CBR – CDR) / 10
自然增长量（每千人）为粗出生率与粗死亡率之差：NI = CBR – CDR。要换算为年增长百分比，用两者的差值除以 10：RNI (%) = (CBR – CDR) ÷ 10。正值的自然增长率表明在不考虑迁移因素时人口处于增长状态。
4. Doubling Time (Rule of 70) | 人口倍增时间（70法则）
Demographers often estimate how long it will take for a population to double using the ‘Rule of 70’. Doubling time (in years) = 70 ÷ annual growth rate (in %). The rule assumes constant exponential growth and provides a quick approximation. For example, a country growing at 2 % per year would double its population in approximately 35 years (70 ÷ 2).
Doubling time (years) = 70 / growth rate (%)
人口学家常用“70 法则”估算人口翻番所需的时间。倍增时间（年）= 70 ÷ 年增长率（%）。该法则假定呈指数增长且增长率恒定，能给出快速近似值。例如，某国年增长率为 2%，则其人口大约在 35 年后翻番（70 ÷ 2）。
5. Dependency Ratio | 抚养比
The dependency ratio measures the demographic burden on the productive population. Total dependency ratio DR = [(P₀–₁₄ + P₆₅₊) ÷ P₁₅–₆₄] × 100, where P₀–₁₄ is the young (0–14), P₆₅₊ is the elderly (65 and over), and P₁₅–₆₄ is the working‑age population. A high ratio signals greater pressure on health, education and pension systems.
DR = [(P₀–₁₄ + P₆₅₊) / P₁₅–₆₄] × 100
抚养比衡量非劳动年龄人口对劳动年龄人口带来的负担。总抚养比 DR = [(P₀–₁₄ + P₆₅₊) ÷ P₁₅–₆₄] × 100，其中 P₀–₁₄ 为少儿人口（0–14 岁），P₆₅₊ 为老年人口（65 岁及以上），P₁₅–₆₄ 为劳动适龄人口。高抚养比意味着医疗、教育和养老体系承受更大的压力。
Child dependency ratio = (P₀–₁₄ ÷ P₁₅–₆₄) × 100, and old‑age dependency ratio = (P₆₅₊ ÷ P₁₅–₆₄) × 100. Analysing these separately helps governments plan age‑specific services.
Child DR = (P₀–₁₄ / P₁₅–₆₄) × 100 | Old‑age DR = (P₆₅₊ / P₁₅–₆₄) × 100
少儿抚养比 = (P₀–₁₄ ÷ P₁₅–₆₄) × 100，老年抚养比 = (P₆₅₊ ÷ P₁₅–₆₄) × 100。分别分析这两个比值有助于政府规划针对不同年龄段的公共服务。
6. Urbanisation Level & Rate | 城市化水平与速率
Level of urbanisation measures the share of a country’s population living in areas classified as urban. Urbanisation level (%) = (Urban population ÷ Total population) × 100. This indicator reflects the degree of demographic concentration in towns and cities.
Urbanisation level (%) = (U / P) × 100
城市化水平衡量一个国家居住在被划定为城镇地区的人口比例。城市化水平 (%) =（城镇人口 ÷ 总人口）× 100。这一指标反映了人口向城镇集聚的程度。
The rate of urbanisation refers to the speed at which the urban proportion is changing over time. Annual rate (%) = [(U₂ – U₁) / U₁] ÷ t × 100, or simply the difference in percentage points per year when comparing two censuses. Rapid urbanisation is typical of emerging economies experiencing rural–urban migration.
Urbanisation rate (%) ≈ (ΔU / U) / t × 100
城市化速率指城镇人口比例随时间变化的速度。年增长率 (%) = [(U₂ – U₁) / U₁] ÷ t × 100，或直接比较两次普查间每年变动的百分点。快速城市化是经历乡村向城市迁移的新兴经济体的典型特征。
7. Rank‑Size Rule & Primacy Index | 位序‑规模法则与首位度
Zipf’s rank‑size rule describes the distribution of city sizes in many countries. If the largest city has population P₁, then the population of the n‑th largest city (Pₙ) approximately equals P₁ / n. In an ideal rank‑size country, the second city is half the size of the largest, the third city is one‑third, and so on. Deviations from this pattern indicate primacy or a binary structure.
Pₙ = P₁ / n
齐普夫位序‑规模法则描述了许多国家的城市规模分布。若最大城市人口为 P₁，则第 n 大城市的人口 Pₙ 约等于 P₁ / n。在理想的位序‑规模体系中，第二大城市规模是首位城市的一半，第三大城市是首位城市的三分之一，依此类推。偏离此模式表明城市首位度过高或二元分布。
The primacy index is a simple ratio comparing the largest city with the second largest: Primacy index = P₁ / P₂. A
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