📚 GCSE CIE Geography: Formula & Theorem Quick Reference | GCSE CIE 地理：公式定理速查手册
This quick reference handbook gathers all the essential formulas, key calculations, and fundamental models you need to master for GCSE CIE Geography (0460). Whether you are working with population data, reading Ordnance Survey maps, measuring river characteristics, or analysing development patterns, having these tools at your fingertips will save time in the exam and help you write precise, well‑supported answers. Each entry is presented with a clear definition, the formula or theorem itself, a worked example, and a brief explanation of when to use it. Keep this guide close while revising and practising past papers.
本速查手册汇集了 GCSE CIE 地理（0460）必须掌握的所有核心公式、关键计算方法和基础理论模型。无论你面对的是人口数据、地形图判读、河流特征测量还是发展模式分析，熟练运用这些工具都能帮你在考试中节省时间，写出精准、有据可依的答案。每个条目都配有清晰的定义、公式或定理本身、示例演算以及使用场景的简要说明。建议在复习和刷真题时随身查阅。
1. Population Density | 人口密度
Population density measures how crowded an area is. It is calculated by dividing the total population of a region by its land area. The result is usually expressed in people per square kilometre (persons/km²). This simple statistic allows geographers to compare settlement pressure, service demand, and land use intensity between different locations.
人口密度衡量一个地区的拥挤程度。它用区域总人口除以土地面积求得，结果通常以每平方千米人数（人/km²）表示。这一简单统计量可以帮助地理学者比较不同地点的居住压力、服务需求和土地利用强度。
Population Density = Total Population ÷ Land Area
人口密度 = 总人口 ÷ 土地面积
For example, if a city has 250 000 inhabitants and covers 50 km², its population density is 250 000 ÷ 50 = 5000 persons/km². Always remember to convert both population and area into consistent units before dividing.
例如，某城市拥有 250 000 居民，面积 50 km²，则人口密度为 250 000 ÷ 50 = 5000 人/km²。计算前务必确保人口和面积的单位统一。
2. Population Change Rates | 人口变化率
Understanding how a population grows or declines requires several related rates. The natural increase rate focuses only on births and deaths, while the net migration rate accounts for people moving in and out. Combining these gives the overall population growth rate, frequently expressed per 1000 people per year (‰).
要了解人口增长或减少的规律，需要若干相互关联的比率。自然增长率仅考虑出生和死亡，而净迁移率则统计迁入和迁出的人口。两者结合得到整体人口增长率，通常以每年每千人（‰）表示。
Birth Rate (CBR) = (Number of live births ÷ Total population) × 1000
出生率 = (活产婴儿数 ÷ 总人口) × 1000
Death Rate (CDR) = (Number of deaths ÷ Total population) × 1000
死亡率 = (死亡人数 ÷ 总人口) × 1000
Natural Increase Rate = Birth Rate – Death Rate
自然增长率 = 出生率 – 死亡率
Net Migration Rate = ((Immigrants – Emigrants) ÷ Total population) × 1000
净迁移率 = ((迁入人数 – 迁出人数) ÷ 总人口) × 1000
Population Growth Rate = Natural Increase Rate + Net Migration Rate
人口增长率 = 自然增长率 + 净迁移率
For instance, if a country has a birth rate of 20‰, a death rate of 8‰, and a net migration rate of 3‰, its population growth rate is (20 – 8) + 3 = 15‰, meaning the population grows by 15 people per 1000 each year. On a population pyramid, these rates shape the width of the base and the taper of the top.
例如，某国出生率为 20‰，死亡率为 8‰，净迁移率为 3‰，则人口增长率为 (20 – 8) + 3 = 15‰，即人口每年每千人增长 15 人。在人口金字塔上，这些比率决定着底部的宽度和顶部的收窄形态。
3. Dependency Ratio | 抚养比
The dependency ratio shows the economic burden on the working‑age population. It is calculated by adding the young (0–14 years) and the old (65+ years) dependent populations, then dividing by the working‑age population (15–64 years) and multiplying by 100. A high ratio suggests greater pressure on health services, education, and pensions.
抚养比反映了劳动年龄人口承担的经济负担。其计算方法是将少儿抚养人口（0–14 岁）与老年抚养人口（65 岁以上）相加，除以劳动年龄人口（15–64 岁），再乘以 100。抚养比高意味着对医疗、教育和养老金等服务的压力更大。
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
Example: A village has 200 children, 500 working‑age adults, and 100 elderly people. The dependency ratio is ((200 + 100) ÷ 500) × 100 = 60. This means that for every 100 people of working age, there are 60 dependants. Geographers use this figure when discussing ageing populations or youthful populations in different stages of the Demographic Transition Model.
示例：某村庄有 200 名儿童、500 名劳动年龄成人和 100 名老人。抚养比为 ((200 + 100) ÷ 500) × 100 = 60。这意味着每 100 名劳动年龄人口要抚养 60 人。地理学者在讨论人口转变模型中不同阶段的老年型或少儿型人口时，常会引用这一数值。
4. Map Scale and Distance | 地图比例尺与距离
Ordnance Survey (OS) maps and other large‑scale maps use a representative fraction to relate map distance to real‑world distance. Understanding scale lets you convert a measurement on the map into actual ground distance, which is essential for fieldwork, route planning, and describing site and situation.
英国地形测量局（OS）地图及其他大比例尺地图使用比例分数来表示图上距离与实际距离的关系。理解比例尺可以让你将图上的测量结果转换为真实的地面距离，这对野外实地调查、路线规划和描述地点与区位至关重要。
Real Distance = Map Distance × Scale Denominator
实际距离 = 图上距离 × 比例尺分母
On a 1 : 25 000 map, 1 cm represents 25 000 cm = 250 m. Therefore, a line measuring 6 cm on the map represents 6 × 250 = 1500 m, or 1.5 km. Always start by reading the scale statement, then use the unit conversion: 100 cm = 1 m, 1000 m = 1 km. For curved distances, use a piece of string or the edge of a paper, then measure against the map’s linear scale.
在 1∶25 000 的地图上，1 cm 代表实地 25 000 cm = 250 m。因此，图上一条 6 cm 的线段代表实际 6 × 250 = 1500 m，即 1.5 km。请始终先读比例尺说明，然后进行单位转换：100 cm = 1 m，1000 m = 1 km。测量曲线距离时，可用细绳或纸边沿路线摆放，再与地图的直线比例尺对照。
5. Area Calculation on Maps | 地图面积计算
Estimating area from a map is a common examination skill. Regular shapes such as rectangles and triangles can be calculated with simple geometry. For irregular shapes — fields, lakes, urban zones — the grid‑square method gives a reliable approximate area. Accurate area figures help with land‑use surveys and environmental impact assessments.
从地图上估算面积是一项常见的考试技能。矩形、三角形等规则形状可以用简单的几何公式计算。而对于农田、湖泊、城区等不规则形状，方格法可以提供可靠的面積近似值。精确的面积数据有助于开展土地利用调查和环境影响评估。
Area (regular shape) = length × width (rectangle), ½ × base × height (triangle)
面积（规则形状）= 长 × 宽（矩形），½ × 底 × 高（三角形）
Using grid squares: on a 1 : 25 000 map, each 1 cm² square represents 250 m × 250 m = 62 500 m² (0.0625 km²). Count the number of complete squares inside the area, then add the parts of partially covered squares. Multiply the total square count by the area represented by one square. For example, if a forest covers 18 whole squares and approximately 6 half‑squares (equivalent to 3 more whole squares), the estimated area is 21 × 0.0625 km² = 1.3125 km².
方格法：在 1∶25 000 地图上，每个 1 cm² 方格代表 250 m × 250 m = 62 500 m²（0.0625 km²）。数出区域内完整的方格数，再加上部分覆盖的方格估算值。将方格总数乘以每个方格所代表的实际面积。例如，某片林地覆盖了 18 个整格和大约 6 个半格（相当于 3 个整格），则估算面积为 21 × 0.0625 km² = 1.3125 km²。
6. Gradient (Slope) Calculation | 坡度计算
Gradient describes the steepness of a slope, an important factor in settlement location, farming, transport route planning, and river valley profiles. It is expressed as a ratio of vertical rise to horizontal distance, or as a percentage. On a contour map, the vertical rise is the difference in height between two points, and the horizontal distance is measured along the line connecting them.
坡度用以描述斜坡的陡峭程度，是聚落选址、农业生产、交通线路规划和河谷剖面分析中的重要因素。坡度用垂直高差与水平距离的比值表示，也可用百分比表达。在等高线地图上，垂直高差即两点间的高程差，水平距离则为连接两点的直线在实地中的长度。
Gradient = Vertical Rise ÷ Horizontal Distance
坡度 = 垂直高差 ÷ 水平距离
If a footpath climbs 80 m over a horizontal distance of 400 m, the gradient is 80 ÷ 400 = 0.2, often written as 1 : 5 (dividing both sides by 0.2). As a percentage, 0.2 × 100 = 20%. Always ensure the rise and distance are in the same unit; convert km to m if necessary. Maps with close contour lines indicate steep slopes, while widely spaced contours suggest gentle slopes.
若某条步道在 400 m 的水平距离内上升了 80 m，则坡度为 80 ÷ 400 = 0.2，常写作 1∶5（分子分母同除以 0.2）。换算为百分比，0.2 × 100 = 20%。务必确保高差和水平距离使用相同的单位；必要时将 km 转换为 m。等高线密集的地方坡陡，等高线稀疏的地方坡缓。
7. Vertical Exaggeration | 垂直夸大
When drawing a cross‑section from a map, the vertical scale is often enlarged to make relief features visible. Vertical exaggeration (VE) is the ratio of the vertical scale to the horizontal scale. Understanding VE helps you interpret the true slope of the land; a high VE makes gentle undulations look like dramatic mountains.
在根据地图绘制剖面图时，垂直比例尺常常被放大，以便看清地形起伏。垂直夸大（VE）就是垂直比例尺与水平比例尺的比值。理解 VE 有助于你正确判读坡度；过高的 VE 会让平缓的起伏看起来像陡峭的山脉。
Vertical Exaggeration = Vertical Scale ÷ Horizontal Scale
垂直夸大 = 垂直比例尺 ÷ 水平比例尺
Suppose a cross‑section uses a vertical scale of 1 cm to 100 m (1 : 10 000) on a map with a horizontal scale of 1 : 50 000. Convert both to representative fractions: VS = 1/10 000, HS = 1/50 000. VE = (1/10 000) ÷ (1/50 000) = (1/10 000) × (50 000/1) = 5. Thus, the vertical dimension is exaggerated 5 times. When you describe the shape of a valley from such a profile, remember to say that the real slopes are five times gentler than they appear.
假设某剖面图使用 1 cm 代表 100 m 的垂直比例尺（1∶10 000），而地图的水平比例尺为 1∶50 000。将两者转换为代表分数：VS = 1/10 000，HS = 1/50 000。VE = (1/10 000) ÷ (1/50 000) = 5。因此，垂直方向被夸大了 5 倍。当你在根据这样的剖面图描述河谷形态时，要记得说明实际的坡度是图中所见的五分之一。
8. River Discharge (Flow Rate) | 河流流量
River discharge is the volume of water flowing past a given point in a given time, usually measured in cubic metres per second (cumecs, m³/s). It is a fundamental measure in hydrology, used to compare river regimes, assess flood risk, and plan water resources. Discharge is calculated by multiplying the cross‑sectional area of the channel by the average flow velocity.
河流流量是指单位时间内流过某一断面的水体体积，通常以立方米每秒（m³/s）表示。它是水文学中的基本指标，用于比较河流情势、评估洪水风险和规划水资源。流量的计算方法是用河道横截面积乘以平均流速。
Discharge (Q) = Cross‑sectional Area (A) × Average Velocity (v)
流量 (Q) = 横截面积 (A) × 平均流速 (v)
For example, if a river channel has a cross‑sectional area of 12 m² and the water is flowing at an average velocity of 0.8 m/s, the discharge is 12 × 0.8 = 9.6 m³/s. To obtain reliable results in a fieldwork investigation, you need several velocity readings across the channel using a flowmeter, then average them before multiplying by the wetted area.
例如，某河道横截面积为 12 m²，水流平均流速为 0.8 m/s，则流量为 12 × 0.8 = 9.6 m³/s。在野外调查中获得可靠数据时，需要用流速仪在河道不同位置多次测速，取平均值后再乘以过水断面面积。
9. Cross‑sectional Area of a River | 河道横截面积
The cross‑sectional area of a river channel is the size of the slice through the water, perpendicular to the flow. It is measured by recording the depth at regular intervals across the channel and treating the shape as a series of trapezoids or rectangles. Accurate area values are essential for calculating discharge and understanding channel efficiency.
河道横截面积是指垂直于水流方向的水体剖面的大小。测量时需要沿河道宽度以固定间隔记录水深，并把该剖面近似为一系列梯形或矩形。精确的面积值是计算流量和理解河道效率必不可少的基础。
Cross‑sectional Area = Σ (Width of segment × Average depth of segment)
横截面积 = Σ (分段宽度 × 分段平均水深)
In the simplest case, if the channel is roughly rectangular, area = width × average depth. For a more typical trapezoidal channel, measure depth at 0, ¼, ½, ¾, and full width, then use the trapezoidal rule. For instance, widths of segments are all 2 m and depths at five points are 0 m, 0.3 m, 0.5 m, 0.4 m, 0 m. The area of the two end triangles (zero depth) is negligible; the three middle trapezoids sum to a total of about 2.4 m². Always tabulate your readings neatly to show the examiner your method.
最简单的情形是河道近似矩形，面积 = 宽度 × 平均水深。对于更典型的梯形河道，可在 0、¼、½、¾ 以及全宽处测量水深，然后采用梯形法则计算。假设各分段宽度均为 2 m，五个测点水深依次为 0 m、0.3 m、0.5 m、0.4 m、0 m，则两端三角形（深度为 0）面积忽略不计，中间三个梯形面积之和约为 2.4 m²。请始终将测量数据以表格形式整齐列出，向阅卷官展示你的计算过程。
10. Velocity of River Flow | 河流流速
Velocity tells how fast the water is moving. It is a key factor in river erosion, transportation, and deposition. Average velocity is obtained by measuring the time a float or dye takes to travel a known distance, or by using a flowmeter that records revolutions per second. The standard formula links distance, time, and speed.
流速表明水体移动的快慢，是影响河流侵蚀、搬运和沉积的关键因素。平均流速可通过测量浮标或染色剂经过已知距离的时间获得，或利用流速仪记录每秒转数来求得。标准的公式将距离、时间和速度联系在一起。
Velocity (v) = Distance (d) ÷ Time (t)
流速 (v) = 距离 (d) ÷ 时间 (t)
If a float travels 10 m in 25 seconds, the surface velocity is 10 ÷ 25 = 0.4 m/s. Because friction with the bed and banks slows water, the true average velocity is often about 0.8–0.9 of the surface velocity in a small stream. When writing up a geographical investigation, always explain how you accounted for this reduction and why you took multiple readings.
若一个浮标 25 秒移动了 10 m，则表面流速为 10 ÷ 25 = 0.4 m/s。由于河床和岸壁的摩擦会减缓水流，在小溪流中真实的平均流速通常约为表面流速的 0.8–0.9 倍。在撰写地理调查报告时，务必说明你是如何处理这一折减的，以及为何要多次读数。
11. Climate Data Calculations | 气候数据计算
Climate graphs display temperature and precipitation data, and exam questions often ask you to calculate mean annual temperature, annual temperature range, and total annual rainfall. These statistics support descriptions of climate zones, comparisons between locations, and discussions of seasonal patterns.
气候图展示气温和降水资料，考试题目常要求计算年均温、年温差和年总降水量。这些统计量为描述气候带、比较不同地点和讨论季节规律提供了数据支撑。
Mean Annual Temperature = Sum of 12 monthly mean temperatures ÷ 12
年均温 = 12 个月的月均温总和 ÷ 12
Annual Temperature Range = Warmest monthly mean – Coldest monthly mean
年温差 = 最暖月平均气温 – 最冷月平均气温
Total Annual Precipitation = Sum of 12 monthly precipitation totals
年降水量 = 12 个月的月降水量总和
For a station with monthly temperatures of 5, 7, 10, 14, 18, 22, 25, 24, 20, 15, 10, 6 °C, the mean annual temperature is (5+7+10+14+18+22+25+24+20+15+10+6) ÷ 12 = 176 ÷ 12 = 14.7 °C. The annual range is 25 – 5 = 20 °C. If total monthly precipitation figures add to 650 mm, the station receives 650 mm of rainfall annually. Always state the units clearly – °C for temperature, mm for precipitation.
假设某气象站各月平均气温分别为 5、7、10、14、18、22、25、24、20、15、10、6 °C，则年均温为 (5+7+10+14+18+22+25+24+20+15+10+6) ÷ 12 = 176 ÷ 12 = 14.7 °C。年温差为 25 – 5 = 20 °C。若月降水量合计为 650 mm，则该站年降水量为 650 mm。请务必明确标注单位——气温用 °C，降水量用 mm。
12. Demographic Transition Model | 人口转变模型
While not a formula in the algebraic sense, the Demographic Transition Model (DTM) is a foundational theorem in population geography. It describes how a country’s population growth changes through five stages as it develops economically. The model links birth rates, death rates, and total population size, and it helps explain the demographic history of countries as well as predict future trends.
虽然人口转变模型（DTM）不是一个代数意义上的公式，但它是人口地理学中一项基础性的理论定理。该模型描述了一个国家随着经济发展，其人口增长如何经历五个阶段的变化。它将出生率、死亡率和总人口规模联系起来，有助于解释各国的人口发展历史并预测未来趋势。
- Stage 1: High fluctuating birth and death rates, low and stable population. Rare today. | 第一阶段：高出生率和高死亡率上下波动，人口低而稳定。如今极为少见。
- Stage 2: Death rate falls sharply (improvements in medicine, sanitation), birth rate remains high, rapid population increase. | 第二阶段：死亡率急剧下降（医疗、卫生改善），出生率仍高，人口快速增长。
- Stage 3: Birth rate begins to decline (urbanisation, family planning), population growth slows. | 第三阶段：出生率开始下降（城市化、计划生育），人口增速放缓。
- Stage 4: Both birth and death rates are low, population stabilises at a high level. | 第四阶段：出生率和死亡率均低，人口在高位趋于稳定。
- Stage 5: Birth rate falls below death rate, natural decrease begins; observed in some highly developed countries. | 第五阶段：出生率低于死亡率，出现自然减少；可在部分高度发达国家看到。
The DTM is not a rigid law but a generalised pattern. When using it in an exam answer, support your arguments with specific named countries and statistics (e.g., UK in Stage 4, Niger in Stage 2). This shows you can apply the theorem to real‑world contexts.
人口转变模型并非一条僵硬的定律，而是一种概括性规律。在考试答题时使用该模型，要用具体国名和统计数据来支撑你的论述（如英国处于第四阶段，尼日尔处于第二阶段），以证明你能够将理论应用于真实世界的情境。
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