📚 Year 10 CAIE Geography: Formula & Theorem Quick Reference Handbook | Year 10 CAIE 地理：公式定理速查手册
This quick reference handbook covers essential formulas, models and theorems required for CAIE IGCSE / Year 10 Geography. It provides a concise summary of population equations, river hydrology calculations, map skills, urban land use theories, climate models and fieldwork statistics. Each formula is presented with a clear explanation to help you revise effectively for your examinations.
本速查手册涵盖了 CAIE IGCSE / Year 10 地理课程所需的公式、模型和定理。它简要总结了人口方程、河流水文计算、地图技能、城市土地利用理论、气候模型和田野调查统计。每个公式都附有清晰的解释，帮助您高效备考。
1. Population Change Formulas | 人口变化公式
Crude Birth Rate (CBR) = (Number of live births ÷ Mid-year total population) × 1000
粗出生率 (CBR) = (活产婴儿数 ÷ 年中总人口) × 1000
Crude Death Rate (CDR) = (Number of deaths ÷ Mid-year total population) × 1000
粗死亡率 (CDR) = (死亡人数 ÷ 年中总人口) × 1000
Natural Increase Rate (NIR) = CBR – CDR (expressed per 1000 people per year). To convert to a percentage: NIR% = (CBR – CDR) ÷ 10
自然增长率 (NIR) = CBR – CDR（以每年每千人为单位）。转换为百分比：NIR% = (CBR – CDR) ÷ 10
Population Growth Rate = (Natural increase + Net migration) ÷ Total population × 100
人口增长率 = (自然增长 + 净移民) ÷ 总人口 × 100
Doubling Time (in years) ≈ 70 ÷ Population growth rate (%)
人口翻倍时间 (年) ≈ 70 ÷ 人口增长率 (%)
2. Migration and Net Migration Rate | 迁移与净迁移率
Net Migration = Number of immigrants – Number of emigrants
净迁移 = 入境移民数 – 出境移民数
Net Migration Rate = (Net migration ÷ Mid-year population) × 1000
净迁移率 = (净迁移 ÷ 年中人口) × 1000
Population change = (Births – Deaths) + (Immigration – Emigration)
人口变化 = (出生人数 – 死亡人数) + (迁入人数 – 迁出人数)
3. Dependency Ratio and Age Structure | 抚养比与年龄结构
Youth Dependency Ratio = (Population aged 0–14 ÷ Population aged 15–64) × 100
少儿抚养比 = (0–14 岁人口 ÷ 15–64 岁人口) × 100
Old-age Dependency Ratio = (Population aged 65+ ÷ Population aged 15–64) × 100
老年抚养比 = (65 岁及以上人口 ÷ 15–64 岁人口) × 100
Total Dependency Ratio = [(Population 0–14 + Population 65+) ÷ Population aged 15–64] × 100
总抚养比 = [(0–14 岁人口 + 65 岁及以上人口) ÷ 15–64 岁人口] × 100
Sex Ratio = (Number of males ÷ Number of females) × 100
性别比 = (男性人口 ÷ 女性人口) × 100
4. Demographic Transition Model (DTM) | 人口转变模型
The Demographic Transition Model describes how birth and death rates change as a country develops. Stage 1: high fluctuating birth and death rates, low growth. Stage 2: death rate falls rapidly while birth rate remains high, leading to rapid population increase. Stage 3: birth rate starts to decline, growth rate slows. Stage 4: both rates are low, population stabilises. Stage 5 (optional): birth rate falls below death rate, natural decrease.
人口转变模型描述了一个国家在发展过程中出生率和死亡率的变化。第一阶段：高出生率和高死亡率波动，增长缓慢；第二阶段：死亡率迅速下降，出生率保持高位，人口快速增长；第三阶段：出生率开始下降，增速放缓；第四阶段：出生率和死亡率都处于低位，人口稳定；第五阶段（可选的）：出生率低于死亡率，出现自然负增长。
5. Urban Land Use Models | 城市土地利用模型
Burgess Concentric Zone Model (1925): A city grows in a series of rings radiating from the CBD. Zone 1: CBD; Zone 2: transition zone (industry, low-income housing); Zone 3: working-class housing; Zone 4: middle-class housing; Zone 5: commuter zone.
伯吉斯同心圆模型 (1925)： 城市以中央商务区为中心呈同心圆环状扩展。1区：CBD；2区：过渡地带（工业、低收入住宅）；3区：工人阶级住宅；4区：中产阶级住宅；5区：通勤区。
Hoyt Sector Model (1939): Urban land uses develop in wedge-shaped sectors along transport routes. High-rent housing expands outward along desirable sectors, while industry follows railways or rivers.
霍伊特扇形模型 (1939)： 城市土地利用沿交通线路呈扇形发展。高租金住宅沿优质扇形区向外延伸，工业则沿铁路或河流布局。
Harris & Ullman Multiple Nuclei Model (1945): A city has several independent centres (nuclei) for different functions, such as a port, an industrial park, a university and a suburban business district. Growth occurs from multiple nodes.
哈里斯和厄尔曼多核心模型 (1945)： 城市存在多个独立的核心区域，如港口、工业园区、大学和郊区商业中心，增长从多个节点同时发生。
6. River Discharge and Channel Formulae | 河流流量与河道公式
Discharge (Q) = Cross-sectional area (A) × Mean velocity (V)
流量 (Q) = 过水断面面积 (A) × 平均流速 (V)
Cross-sectional area (A) ≈ Width (w) × Average depth (d̅). In fieldwork, depth is measured at regular intervals and averaged.
过水断面面积 (A) ≈ 河面宽度 (w) × 平均水深 (d̅)。实地测量时，以规则间隔测得水深后取平均值。
Velocity by float method = Distance travelled ÷ Time taken
浮标法测流速 = 移动距离 ÷ 所用时间
Hydraulic Radius (R) = A ÷ Wetted perimeter (P). Used for understanding channel efficiency, though not always required at IGCSE.
水力半径 (R) = 过水断面面积 (A) ÷ 湿周 (P)。用于理解河道效率，IGCSE 阶段不一定要求深入计算。
Sinuosity = Channel length ÷ Straight-line valley length. A sinuosity greater than 1.5 indicates a meandering river.
蜿蜒度 = 河道长度 ÷ 直线谷地长度。蜿蜒度大于 1.5 表明为曲流河。
Gradient = Vertical rise ÷ Horizontal distance (often expressed as a ratio, e.g. 1:50).
河道比降 = 垂直落差 ÷ 水平距离（通常以比例表示，如 1:50）。
7. Drainage Basin Water Balance | 流域水平衡
The water balance equation for a drainage basin: Precipitation (P) = Runoff (Q) + Evapotranspiration (E) + Change in storage (ΔS). In simple terms, the water entering a catchment leaves as streamflow or water vapour or is stored in soil, groundwater and vegetation.
流域水平衡方程：降水量 (P) = 径流量 (Q) + 蒸发蒸腾量 (E) + 储存变化量 (ΔS)。简单来说，进入流域的水以河川径流或水汽形式离开，或者储存在土壤、地下水和植被中。
Runoff ratio = (Runoff ÷ Precipitation) × 100, indicating how much precipitation becomes streamflow.
径流系数 = (径流 ÷ 降水) × 100，表示有多少降水转变为河川径流。
8. Map Skills: Gradient, Area and Scale | 地图技能：坡度、面积与比例尺
Gradient = Height difference (vertical interval) ÷ Horizontal distance (both in same units). Simplify to a ratio like 1:X.
坡度 = 高差 (垂直间隔) ÷ 水平距离（单位统一）。化简为 1:X 的比例。
Area from a grid square: 1 grid square (1 km × 1 km) = 1 km² on a 1:25 000 map. Count full squares and part squares to estimate irregular areas.
网格法计算面积： 在 1:25 000 地图上，一个网格正方形 (1 km × 1 km) = 1 km²。通过数完整和部分方格估算不规则面积。
Scale conversion: 1:50 000 means 1 cm on the map = 50 000 cm on the ground = 500 m. Use direct proportion to find real distances.
比例尺转换： 1:50 000 表示地图上 1 厘米 = 地面上 50 000 厘米 = 500 米。利用正比例求实际距离。
Bearing: Measured clockwise from north (0° to 360°) using a protractor on the map.
方位角： 在地图上用半圆仪从正北顺时针量度 (0° 到 360°)。
9. Fieldwork Statistical Measures | 田野调查统计测量
Mean (average) = Sum of all values ÷ Number of values
平均值 = 所有数值之和 ÷ 数值个数
Median = Middle value when data are arranged in order
中位数 = 数据按顺序排列后的中间值
Mode = Most frequently occurring value
众数 = 出现频率最高的值
Range = Maximum value – Minimum value
极差 = 最大值 – 最小值
Percentage (%) = (Part ÷ Whole) × 100
百分比 (%) = (部分 ÷ 整体) × 100
Interquartile Range (IQR) = Upper quartile (Q3) – Lower quartile (Q1). It measures the spread of the middle 50% of data.
四分位距 (IQR) = 上四分位数 (Q3) – 下四分位数 (Q1)，衡量中间 50% 数据的分散程度。
10. Economic Development Models | 经济发展模型
Clark-Fisher Sector Model: As an economy develops, employment shifts from primary (agriculture) to secondary (manufacturing) and then to tertiary (services). In advanced economies, quaternary (knowledge-based) jobs become significant.
克拉克-费希尔产业模型： 随着经济发展，就业从第一产业（农业）转向第二产业（制造业），再转向第三产业（服务业）。在发达经济体中，第四产业（知识型）工作岗位变得重要。
Rostow’s Stages of Growth: 1. Traditional society; 2. Pre-conditions for take-off; 3. Take-off; 4. Drive to maturity; 5. Age of high mass consumption. Development follows a linear sequence of investment and industrialisation.
罗斯托经济增长阶段： 1. 传统社会；2. 起飞准备阶段；3. 起飞；4. 走向成熟；5. 大众高消费时代。发展遵循投资和工业化的线性序列。
11. Global Climate and Circulation Theories | 全球气候与环流理论
Hadley Cell: Warm air rises at the equator (low pressure, ITCZ), moves polewards at high altitude, sinks around 30° N/S (high pressure, subtropical deserts), and returns to the equator as trade winds.
哈德莱环流： 暖空气在赤道上升（低压，热带辐合带），高空向两极移动，在南北纬 30° 左右下沉（高压，亚热带沙漠），以信风形式返回赤道。
Ferrel Cell and Polar Cell: The Ferrel cell operates between 30° and 60° driven by the movement of adjacent cells. The polar cell brings cold air from the poles to 60°, where it meets warmer air, forming the polar front.
费雷尔环流和极地环流： 费雷尔环流在 30° 至 60° 之间运行，由相邻环流带动。极地环流将冷空气从极地输送到 60°，与暖空气相遇形成极锋。
ITCZ (Intertropical Convergence Zone): A belt of low pressure near the equator where trade winds converge, causing rising air, cloud formation and heavy rainfall. It shifts seasonally with the overhead sun.
热带辐合带 (ITCZ)： 赤道附近的低压带，信风汇聚，空气上升，形成大量云系和强降雨。它随太阳直射点的移动而发生季节性变化。
Rain Shadow Effect: Moist air rises on the windward side of a mountain, cools and condenses causing precipitation. On the leeward side, dry air descends and warms, creating a rain shadow.
雨影效应： 湿空气在迎风坡抬升、冷却凝结形成降水。在背风坡，干燥空气下沉增温，形成雨影区。
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