📚 GCSE CAIE Geography: Formula & Theorem Quick-Reference Handbook | GCSE CAIE 地理：公式定理速查手册
Mastering key geographical formulas and statistical methods is essential for success in both examination papers and fieldwork investigations. This quick-reference handbook brings together all the essential equations, conversions and indicators required by the CAIE IGCSE Geography syllabus, presented in a clear bilingual format for easy revision.
掌握关键的地理公式和统计方法对于在笔试和实地调查中取得成功至关重要。本速查手册汇集了 CAIE IGCSE 地理课程所要求的所有基本方程、换算和指标，以清晰的双语格式呈现，便于复习查阅。
1. Scale and Distance Conversion | 比例尺与距离换算
Map scale is the ratio between a distance measured on the map and the corresponding distance on the ground. Using the representative fraction (RF), such as 1 : 25,000, you can convert any map measurement into real-world kilometres.
地图比例尺是地图上测量的距离与相应实地距离的比率。利用代表分数 (RF)，例如 1 : 25,000，你可以将任何地图测量值转换为现实世界的千米数。
Actual Distance (km) = Map Distance (cm) × Scale Denominator ÷ 100,000
实际距离 (km) = 地图距离 (cm) × 比例尺分母 ÷ 100,000
Always remember that 1 km equals 100,000 cm. This formula works for both straight-line distances measured with a ruler and curved distances traced with a string or paper edge.
请牢记 1 km 等于 100,000 cm。该公式既适用于用直尺测量的直线距离，也适用于用绳子或纸边描绘的曲线距离。
2. Area Calculation | 面积计算
When you need to estimate the ground area of a feature shown on a map, such as a forest or a lake, you must account for the scale factor in both length and width. The area conversion uses the square of the scale denominator.
当你需要估算地图上所示要素（如森林或湖泊）的实地面积时，必须考虑长度和宽度两个方向的比例尺因子。面积换算要使用比例尺分母的平方。
Area (km²) = Map Area (cm²) × (Scale Denominator)² ÷ 10,000,000,000
实地面积 (km²) = 地图面积 (cm²) × (比例尺分母)² ÷ 10,000,000,000
One square kilometre contains 10¹⁰ square centimetres, so dividing by 10,000,000,000 converts the product into the correct unit. For irregular shapes, count full and partial grid squares carefully.
一平方公里含有 10¹⁰ 平方厘米，因此除以 10,000,000,000 可将乘积转换为正确的单位。对于不规则形状，请仔细统计完整的和部分的网格方格。
3. Slope and Gradient | 坡度与梯度
Slope describes the steepness of a land surface. It is commonly expressed either as a percentage or as a gradient ratio (e.g. 1 in 5). You need the vertical interval (rise) and the horizontal equivalent (run) taken from a contour map or measurement.
坡度描述地表的陡峭程度。它通常表示为百分比或梯度比（例如 1 比 5）。你需要从等高线图或测量中获取垂直间距（上升高度）和水平距离（水平移动距离）。
Slope (%) = (Rise ÷ Run) × 100
坡度 (%) = (上升高度 ÷ 水平距离) × 100
Gradient Ratio = 1 : (Run ÷ Rise)
梯度比 = 1 : (水平距离 ÷ 上升高度)
Always use the same units for rise and run. A slope of 10% means a vertical increase of 10 m for every 100 m horizontally. A gradient of 1 : 5 is steeper than 1 : 20.
上升和水平距离务必使用相同的单位。10% 的坡度意味着每水平 100 米垂直上升 10 米。1 : 5 的梯度比 1 : 20 更陡。
4. Birth Rate, Death Rate and Natural Increase | 出生率、死亡率与自然增长率
These vital rates are the foundation of population studies. They measure the number of births or deaths relative to the total population, usually expressed per 1,000 people per year.
这些基本比率是人口研究的基础。它们衡量出生或死亡人数相对于总人口的比值，通常表示为每年每千人的数量。
Birth Rate (per 1,000) = (Number of Live Births ÷ Mid‑year Population) × 1,000
出生率 (‰) = (年内活产数 ÷ 年中总人口) × 1,000
Death Rate (per 1,000) = (Number of Deaths ÷ Mid‑year Population) × 1,000
死亡率 (‰) = (年内死亡数 ÷ 年中总人口) × 1,000
Natural Increase = Birth Rate − Death Rate
自然增长率 = 出生率 − 死亡率
A positive natural increase means population growth; a negative figure indicates natural decrease. These rates do not include migration.
正的自然增长率意味着人口增长；负值则表示人口自然减少。这些比率不包括迁移。
5. Population Density | 人口密度
Population density shows how crowded an area is. It is a simple but crucial measure for comparing different regions or countries.
人口密度显示一个区域的拥挤程度。它是比较不同地区或国家时一个简单却至关重要的指标。
Population Density (people per km²) = Total Population ÷ Land Area (km²)
人口密度 (人/平方公里) = 总人口 ÷ 陆地面积 (km²)
Remember to use the land area, excluding large bodies of water. Density figures can be misleading for countries with vast uninhabited deserts or highlands, so always link the number to physical and human factors.
请记住使用陆地面积，排除大片水域。对于拥有广阔无人沙漠或高地的国家，密度数字可能会产生误导，因此始终要将数字与自然和人文因素联系起来。
6. Dependency Ratio | 抚养比
The dependency ratio compares the economically dependent population (young and elderly) with the working-age population. It gives insight into the economic pressure on the productive sector of a society.
抚养比将经济上依赖他人的人口（少儿和老年人）与劳动适龄人口进行比较。它可以反映社会生产部门所面临的经济压力。
Dependency Ratio (%) = [(Pop. Aged 0‑14 + Pop. Aged 65+) ÷ Pop. 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
A high dependency ratio, often found in developing countries with youthful populations or in ageing developed nations, may strain public services such as education and healthcare.
高抚养比通常出现在年轻人口较多的发展中国家或老龄化的发达国家，可能会对教育和医疗等公共服务造成压力。
7. River Discharge and Velocity | 河流流量与流速
River discharge is the volume of water flowing through a cross-section of a channel per unit of time. It is a fundamental concept in hydrology and fluvial fieldwork.
河流流量是单位时间内通过河道某一横截面的水体体积。它是水文学和河流实地调查中的一个基本概念。
Discharge (Q) = Cross‑sectional Area (A) × Mean Velocity (V)
流量 (Q) = 横截面积 (A) × 平均流速 (V)
Discharge is measured in cubic metres per second (m³ s⁻¹). The cross-sectional area (in m²) is calculated by multiplying channel width by average depth. Velocity (m s⁻¹) is best obtained using a flowmeter or by timing a floating object over a known distance.
流量以立方米每秒 (m³ s⁻¹) 计量。横截面积（单位为 m²）通过河道宽度乘以平均水深计算得出。流速 (m s⁻¹) 最好使用流速仪或通过测量漂浮物经过已知距离的时间来获得。
8. Channel Gradient | 河道梯度
Channel gradient describes how steep a river channel is along its course. It influences flow velocity and erosive power and is a common measurement in river fieldwork.
河道梯度描述河流沿其河道走向的陡峭程度。它影响流速和侵蚀能力，是河流实地调查中常见的测量内容。
Channel Gradient = Vertical Drop (m) ÷ Channel Length (m)
河道梯度 = 垂直落差 (m) ÷ 河道长度 (m)
Alternatively: Gradient Ratio = 1 : (Channel Length ÷ Vertical Drop)
或表示为梯度比 = 1 : (河道长度 ÷ 垂直落差)
Use the same linear unit for both measurements. A channel that falls 5 m over a length of 500 m has a gradient of 0.01 or 1 in 100. This indicates a gentle slope typical of a lower course.
两项测量使用相同的线性单位。一条河道在 500 m 长度内落差 5 m，则其梯为 0.01 或 1 in 100。这表现为下游河段典型的平缓坡度。
9. Percentage Change | 百分比变化
Percentage change is widely used to describe population growth, land‑use shifts, economic data and results from fieldwork. It standardises comparisons between different starting values.
百分比变化广泛用于描述人口增长、土地利用变化、经济数据以及实地调查结果。它可以对不同的起始数值进行标准化比较。
Percentage Change (%) = [(New Value − Original Value) ÷ Original Value] × 100
变化百分比 (%) = [(新值 − 原值) ÷ 原值] × 100
A positive value indicates an increase, while a negative value signifies a decrease. For example, if a town’s population rises from 10,000 to 12,000, the percentage change is +20%.
正数表示增加，负数表示减少。例如，如果一个城镇
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