📚 IGCSE CCEA Geography: Formula and Theorem Quick Reference Handbook | IGCSE CCEA 地理：公式定理速查手册
This quick reference handbook brings together the essential quantitative skills, formulas and calculation techniques required for the IGCSE CCEA Geography specification. From map work and graphical analysis to population dynamics and river fieldwork, mastering these formulas will help you interpret data accurately, answer exam questions with confidence and carry out geographical investigations effectively. Each section below presents a key skill or formula, explains when and how to use it, and provides a straightforward example.
本速查手册汇集了 IGCSE CCEA 地理大纲所要求的基本定量技能、公式和计算技巧。从地图作业和图表分析到人口动态和河流实地考察，掌握这些公式将帮助你准确解读数据、自信解答考题并有效开展地理探究。以下每一节都给出一个关键技能或公式，说明使用场景和方法，并提供简单示例。
1. Map Scales and Distance Conversion | 地图比例尺与距离换算
A map scale expresses the relationship between a distance on the map and the corresponding distance on the ground. The three main types are the statement scale (e.g. ‘1 cm represents 1 km’), the representative fraction or ratio scale (e.g. 1:50,000), and the linear scale (a drawn scale bar). To convert a map distance to the real‑world distance, multiply the measured length on the map by the scale factor.
地图比例尺表示图上距离与对应的实际地面距离之间的关系。三种主要类型是文字比例尺（例如“1 厘米代表 1 公里”）、数字比例尺或分数比例尺（例如 1∶50 000）以及直线比例尺（绘制的比例尺条）。要将图上距离转换为实际距离，可将图上量得的长度乘以比例尺分母。
The core formula is:
核心公式为：
Real distance = Map distance × Scale denominator
实际距离 = 图上距离 × 比例尺分母
When using a ratio scale, remember to keep units consistent. For example, on a 1:25,000 map, a measured distance of 4 cm becomes 4 × 25,000 = 100,000 cm, which is 1,000 m or 1 km.
使用数字比例尺时，要记得保持单位一致。例如，在一幅 1∶25 000 的地图上，量得 4 cm 的距离，计算结果为 4 × 25 000 = 100 000 cm，即 1 000 m 或 1 km。
2. Measuring Straight and Curved Distances | 直线距离与曲线距离的量测
Straight-line distances can be measured with a ruler and then converted using the scale. On an OS map, it is often easiest to work in centimetres and then change to metres or kilometres. For curved distances – such as a winding river or road – use a piece of string or the edge of a piece of paper to trace the curve, mark the end point, and then measure the straightened string or paper against the ruler. The same scale conversion formula applies.
直线距离可用直尺量测，然后利用比例尺换算。在使用英国地形测量局（OS）地图时，通常先以厘米为单位，再转换为米或千米。对于弯曲距离——例如蜿蜒的河流或道路——可用一根细线或纸张边缘沿着曲线描摹，标记终点，然后将拉直的细线或纸张用直尺测量。同样适用上述比例尺换算公式。
If roads or footpaths are measured in sections, always add the individual straight segments before applying the scale. This reduces rounding errors.
如果道路或人行道是分段测量，应在应用比例尺前先将各直线段相加。这样可以减少舍入误差。
3. Area Measurement | 面积测量
Estimating the area of regular or irregular shapes on a map is a common skill. For regular shapes such as rectangles, measure the length and width on the map, convert to real distances, and multiply. The formula is:
估算地图上规则或不规则图形的面积是一项常见技能。对于矩形等规则图形，量测图上长和宽，转换为实际距离，然后相乘。公式为：
Area = Length × Width (both as real distances)
面积 = 长度 × 宽度（均为实际距离）
For irregular shapes, the grid square method is often used on 1:25,000 or 1:50,000 maps. Count the number of complete grid squares covered by the area, then estimate and add the partial squares. Each grid square represents a known real area: on a 1:25,000 map, a 1 cm × 1 cm square (which is a 4 cm² grid square on many OS maps) corresponds to 1 km² in reality (since 1 cm = 0.25 km, area = 0.25 × 0.25 = 0.0625 km²? Wait, typical OS 1:25,000 maps use 4 cm grid squares representing 1 km on the ground, so each 4 cm × 4 cm square represents 1 km × 1 km = 1 km²). Check the map’s grid interval and calculate accordingly.
对于不规则图形，在 1∶25 000 或 1∶50 000 地图上常使用方格法。数出图形覆盖的完整方格数目，再估计并加上部分方格。每个方格代表已知的实际面积：在 1∶25 000 地图上，一个 4 cm × 4 cm 的格网方格对应实际 1 km × 1 km，即 1 km²。应核对地图的格网间距并相应计算。
Another method is to use a dot grid or transparent overlay, counting dots that fall inside the shape to estimate area. The final area in square kilometres is often required.
另一种方法是使用点格网或透明覆盖膜，统计落入图形内部的点数来估算面积。通常要求以平方千米为单位的最终面积。
4. Gradient Calculation | 坡度计算
Gradient describes the steepness of a slope, often expressed as a ratio or fraction. On a map, it is calculated between two points with known heights. The formula is:
坡度描述斜坡的陡峭程度，通常用比率或分数表示。在地图上，它根据两个已知高程点之间计算得出。公式为：
Gradient = Vertical difference (rise) ÷ Horizontal distance (run)
坡度 = 垂直高差（上升） ÷ 水平距离（水平移动）
Both measurements must be in the same unit. For example, if the rise is 100 m and the run is 2,000 m, the gradient is 100 / 2,000 = 1/20, which is often written as 1:20 (meaning one unit up for every 20 units along).
两项测量值必须采用相同单位。例如，若高差为 100 m，水平距离为 2 000 m，则坡度为 100/2 000 = 1/20，常写作 1∶20（表示每水平移动 20 个单位，垂直上升 1 个单位）。
To find the horizontal distance from a map, measure the straight-line distance between the two points, or follow the path if measuring a slope along a road, and convert using the scale. Contour lines provide the heights. Always subtract the lower height from the higher one to get the vertical difference.
要在地图上求取水平距离，可测量两点间的直线距离，或沿道路测量坡度，再利用比例尺换算。等高线提供高程信息。始终用较高点的高程减去较低点的高程得到垂直高差。
5. Vertical Exaggeration | 垂直夸大
When drawing a cross‑section, the vertical scale is often larger than the horizontal scale to make relief features visible. Vertical exaggeration (VE) expresses how much the vertical scale has been stretched compared with the horizontal scale. The formula is:
绘制地形剖面时，垂直比例尺通常比水平比例尺大，以便显现地势起伏。垂直夸大（VE）表示垂直比例尺相对于水平比例尺放大了多少倍。公式为：
VE = Vertical scale ÷ Horizontal scale
垂直夸大 = 垂直比例尺 ÷ 水平比例尺
Both scales must be expressed as representative fractions or ratios. For instance, if the horizontal scale is 1:50,000 and the vertical scale is 1:10,000 (meaning 1 cm represents 10,000 cm, i.e. 100 m), then VE = (1/10,000) ÷ (1/50,000) = 5. This tells you that the heights appear five times larger than they would if drawn at the same scale.
两个比例尺必须都用数字比例尺或比率表示。例如，若水平比例尺为 1∶50 000，垂直比例尺为 1∶10 000（即 1 cm 代表 10 000 cm，也就是 100 m），则 VE = (1/10 000) ÷ (1/50 000) = 5。这表明地形高度比按同一比例尺绘制时放大了 5 倍。
Always state VE as a simple number followed by ‘times’ (e.g. 5×). An accurate cross‑section must note the VE so that the reader knows the slopes are steepened.
VE 通常以一个简单数字加“倍”来表示（例如 5×）。准确的剖面图必须注明垂直夸大，以便读者了解坡度已被夸张。
6. Population Density | 人口密度
Population density measures how crowded an area is. It is the average number of people per unit area, usually expressed per square kilometre. The basic formula is:
人口密度衡量一个区域的拥挤程度。它是单位面积上的平均人数，通常以每平方千米表示。基本公式为：
Population density = Total population ÷ Total land area
人口密度 = 总人口 ÷ 土地总面积
For example, if a country has 65 million people and a land area of 500,000 km², its population density is 65,000,000 / 500,000 = 130 people per km². This is a crude measure and does not reflect variations within the area.
例如，如果一个国家有 6 500 万人口，土地面积为 50 万平方千米，则其人口密度为 65 000 000 / 500 000 = 130 人/平方千米。这是一个粗略的度量，无法反映区域内部的差异。
7. Crude Birth Rate and Crude Death Rate | 粗出生率与粗死亡率
Crude birth rate (CBR) is the number of live births per 1,000 people in a population per year. Crude death rate (CDR) is the number of deaths per 1,000 people per year. They are calculated as follows:
粗出生率（CBR）是指某地一年中每 1 000 人中的活产婴儿数。粗死亡率（CDR）是指每 1 000 人中的死亡人数。计算公式如下：
CBR = (Total live births ÷ Total population) × 1,000
粗出生率 =（活产婴儿总数 ÷ 总人口）× 1 000
CDR = (Total deaths ÷ Total population) × 1,000
粗死亡率 =（死亡总人数 ÷ 总人口）× 1 000
The multiplication by 1,000 is simply to give a rate per thousand, which is easier to interpret than a tiny decimal. These ‘crude’ rates do not take account of the age structure, so comparisons should be made carefully.
乘以 1 000 只是为了得到一个千分比率，比一个微小的十进制小数更易于理解。这些“粗”率没有考虑年龄结构，因此进行比较时需谨慎。
8. Natural Increase and Rate of Natural Increase | 自然增长与自然增长率
Natural increase is the difference between the number of live births and deaths in a given year. The rate of natural increase (RNI) expresses this as a percentage of the total population, typically using the crude rates.
自然增长是指某一年中活产婴儿数与死亡人数之差。自然增长率（RNI）将此差值表示为总人口的百分比，通常利用粗出生率和粗死亡率计算。
The formula for RNI as a percentage is:
自然增长率的百分比公式为：
RNI (%) = [(CBR − CDR) ÷ 1,000] × 100
自然增长率（%）= [（粗出生率 − 粗死亡率）÷ 1 000] × 100
Equivalently, you can take the birth rate minus death rate (per 1,000) and divide by 10 to get a percentage. For example, if CBR = 25 per 1,000 and CDR = 8 per 1,000, the RNI is (25 − 8)/10 = 1.7%. This excludes migration.
也可以直接将千分之几的出生率减去死亡率，再除以 10，得到百分比。例如，若粗出生率为 25‰，粗死亡率为 8‰，则自然增长率为 (25 − 8)/10 = 1.7%。该数值不包括迁移的影响。
9. Population Growth Rate Including Migration | 包括迁移的人口增长率
The overall population growth rate takes into account both natural change and net migration (immigrants minus emigrants). It is often expressed as a percentage per year.
总人口增长率同时考虑自然变动和净迁移（迁入人口减去迁出人口）。通常以年百分比表示。
Population growth rate (%) = [(Natural increase + Net migration) ÷ Total population] × 100
人口增长率（%）= [（自然增长 + 净迁移）÷ 总人口] × 100
Net migration is obtained by subtracting the number of emigrants from the number of immigrants. A positive net migration adds to population growth; a negative net migration reduces the overall growth rate.
净迁移通过迁入人数减去迁出人数得到。净迁移为正会促进人口增长，为负则会降低总增长率。
10. Dependency Ratio | 抚养比
The dependency ratio compares the economically dependent part of the population (typically those aged 0–14 and 65+) with the working-age population (15–64). It provides an indication of the pressure on the productive population. The formula is:
抚养比将人口中经济上需要扶养的部分（通常指 0–14 岁和 65 岁以上人口）与劳动适龄人口（15–64 岁）进行比较，用以反映劳动人口所承受的负担压力。计算公式为：
Dependency ratio = [(Population aged 0–14 + Population aged 65+) ÷ Population aged 15–64] × 100
抚养比 = [（0–14 岁人口 + 65 岁以上人口）÷ 15–64 岁人口] × 100
The result is a percentage: a ratio of 50 means there are 50 dependents for every 100 people of working age. A high dependency ratio may indicate greater pressure on social services and the economy.
结果是一个百分比：抚养比为 50 意味着每 100 名劳动适龄人口要抚养 50 名受扶养者。高抚养比可能预示着社会服务和经济面临较大压力。
11. River Discharge | 河流流量
River discharge is the volume of water passing a given point in the river per unit time, usually expressed in cubic metres per second (m³/s, also called cumecs). It is calculated as:
河流流量是指单位时间内通过河流某一断面的水体体积，通常以立方米/秒（m³/s，又称 cumecs）表示。计算公式为：
Discharge (Q) = Cross‑sectional area (A) × Mean velocity (v)
流量（Q）= 横截面积（A）× 平均流速（v）
To obtain the cross‑sectional area, the river channel is divided into a series of vertical sections. In each section, the depth is measured, and the width of the section is recorded. The area of each section is calculated as an approximate rectangle or trapezoid, and the sum gives the total cross‑sectional area. Velocity can be measured with a flowmeter or by timing a float over a known distance and multiplying by a correction factor (usually 0.8–0.9 for surface floats).
为求得横截面积，需将河道划分为若干个垂直区间。在每个区间测量水深，并记录区间宽度。每个区间的面积近似按矩形或梯形计算，各区间面积之和即为总横截面积。流速可用流速仪测量，也可通过浮标法计时，再乘以校正系数（表面浮标通常取 0.8–0.9）。
12. Cross‑Sectional Area of a River Channel (Trapezoid Approximation) | 河道横截面积的梯形近似计算
During fieldwork, you may need to estimate the cross‑sectional area of a stream using simple depth and width measurements. The area can be calculated by summing a series of trapezoids:
在实地考察中，你可能需要利用简单的水深和宽度测量值来估算溪流的横截面积。可通过累加一系列梯形面积来计算：
A = Σ [ (d₁ + d₂) / 2 × w ]
面积 = Σ [ (水深₁ + 水深₂) / 2 × 区间宽度 ]
Here, d₁ and d₂ are the depths at two adjacent measuring points, and w is the distance between them. The total channel width is the sum of all interval widths. This method gives a good approximation of the wetted cross‑section needed for discharge calculation.
式中，d₁ 和 d₂ 为相邻两测点的水深，w 为两点间的距离。河道总宽度为所有区间宽度之和。该方法能很好地近似计算流量所需的过水横截面积。
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