📚 Year 9 CIE Geography: Formula & Theorem Quick Reference Handbook | CIE 9年级地理：公式定理速查手册
This quick reference handbook brings together the essential formulae, key numerical relationships and fundamental concepts Year 9 students need for the CIE geography course. Whether you are working with maps, analysing population data, describing river characteristics or interpreting climate statistics, accurate calculation and confident use of these tools will strengthen your answers in both structured questions and fieldwork investigations.
本速查手册汇集了CIE 9年级地理课程中学生必须掌握的核心公式、关键数量关系和基本概念。无论是进行地图判读、分析人口数据、描述河流特征还是解读气候统计，准确计算并熟练运用这些工具，都能让你的结构化问题作答和实地考察报告更有说服力。
1. Map Scale | 地图比例尺
Map scale expresses the relationship between a distance measured on a map and the corresponding distance on the ground. It can be given as a ratio, a written statement or a linear scale. Using the correct scale is the foundation of all map-based measurements.
地图比例尺表示图上距离与对应的实地距离之间的比值关系，可以用数字比例尺、文字比例尺或直线比例尺三种形式呈现。正确使用比例尺是所有地图量算的基础。
Scale = Map distance / Ground distance
For a 1:50 000 map, 1 cm on the map represents 50 000 cm (0.5 km) on the ground. Always convert units to the same system before calculating. A common mistake is mixing centimetres and kilometres — multiply or divide by 100 000 as needed.
以1:50 000地图为例，图上1 cm代表实地50 000 cm（0.5 km）。计算前必须统一单位，最常见错误是将厘米和千米混用——按需乘或除以100 000即可完成换算。
2. Measuring Straight‑line and Curved Distances | 直线距离与曲线距离测量
Distances on a map are measured as straight lines or along winding features such as roads and rivers. A ruler gives the direct length; a piece of string or the edge of a paper strip is used to trace curved routes before measuring against the scale.
地图上的距离测量包括直线距离和沿道路、河流等弯曲地物的实际长度。直尺用于量取直线长度；棉线或纸条边缘可贴合曲线轨迹，之后比对比例尺读出距离。
Actual distance = Measured length on map × Scale factor
m with string on the same map, the real distance is 18.7 × 25 000 = 467 500 cm = 4.675 km.
例如，在1:25 000地图上，12.4 cm的直线代表12.4 × 25 000 = 310 000 cm = 3.1 km。若用棉线量得同图上一段弯曲道路长18.7 cm，则实地距离为18.7 × 25 000 = 467 500 cm = 4.675 km。
3. Calculating Area of Regular and Irregular Shapes | 规则与不规则图形面积计算
Area on a map can be found by applying simple geometric formulae to regular shapes or by counting grid squares for irregular patches such as a forest or a lake. Always convert the map area back to real ground area using the square of the scale.
地图上面积可通过几何公式计算规则图形，或用方格法估算森林、湖泊等不规则斑块的面积。无论哪种方法，最后都需将图上面积按比例尺的平方换算为实地面积。
Ground area = Map area × (Scale factor)²
00 cm². Convert to m² by dividing by 10 000: 60 000 m², or to hectares by dividing by 10 000 again: 6 ha.
在1:10 000地图上，长3 cm、宽2 cm的矩形图上面积为6 cm²。比例尺系数为10 000，故实地面积 = 6 × (10 000)² = 600 000 000 cm²。除以10 000得60 000 m²，再除以10 000即6 ha。
4. Gradient | 坡度（梯度）
Gradient describes the steepness of a slope, a road or a stream section. It is the ratio of vertical rise to horizontal distance, both expressed in the same units, and is commonly simplified to a ratio such as 1:10.
坡度用于描述斜坡、道路或河段的陡缓程度，是垂直高差与水平距离的比值，两者须用相同单位，并常化简为1:10这样的比值形式。
Gradient = Vertical interval (VI) / Horizontal equivalent (HE)
0 m length, gradient = 15/600 = 1/40, i.e. 1:40. A smaller second number means a steeper slope.
若山坡水平距离400 m内升高80 m，则坡度 = 80/400 = 1/5，记作1:5。一段铁路在600 m水平距离内爬升15 m，坡度 = 15/600 = 1/40。比值中后项越小，坡度越陡。
5. Population Density | 人口密度
Population density measures how crowded an area is by comparing the total number of people to the land area they occupy. It is a fundamental concept in population geography and helps explain settlement patterns, resource pressure and regional contrasts.
人口密度通过对比总人口与土地面积来衡量一个地区的拥挤程度，是人口地理学的基础概念，有助于解释聚落格局、资源压力和区域差异。
Population density = Total population / Total land area (km²)
a density of 18 000 000 / 90 000 = 200 people per km². Always use the same land area unit — convert square miles to square kilometres if necessary (1 mi² ≈ 2.59 km²).
某国人口1800万，国土面积90 000 km²，人口密度 = 18 000 000 / 90 000 = 200人/km²。务必统一面积单位，若给出平方英里，需换算（1 mi² ≈ 2.59 km²）。
6. Crude Birth Rate, Death Rate and Natural Increase | 粗出生率、粗死亡率和自然增长
These basic demographic indicators are expressed per 1000 people per year. Natural increase is simply the difference between birth rate and death rate, ignoring migration. The rate of natural increase is often converted into a percentage to describe population growth.
这些基础人口指标以每年每千人为单位表示。自然增长即出生率与死亡率之差，暂不考虑迁移。自然增长率常转换为百分比以描述人口增长快慢。
Crude Birth Rate (CBR) = (Number of births / Total population) × 1000
Crude Death Rate (CDR) = (Number of deaths / Total population) × 1000
Natural increase (per 1000) = CBR − CDR
Natural increase (%) = (Natural increase per 1000 / 10)
If CBR = 28‰ and CDR = 9‰, natural increase = 19 per 1000, i.e. 1.9% s produce rapid population growth, typical of Stage 2 of the Demographic Transition Model.
若CBR = 28‰，CDR = 9‰，自然增长为每千人19，即年增长率1.9%。高出生率伴随下降的死亡率导致人口快速增长，这是人口转变模型第二阶段典型特征。
7. Dependency Ratio | 抚养比
The dependency ratio compares the economically dependent part of the population (aged 0–14 and 65+) with the working‑age population (15–64). It gives an idea of the pressure on the productive group to support the young and the elderly.
抚养比将经济依赖人口（0–14岁和65岁以上）与劳动适龄人口（15–64岁）进行对比，反映生产性人群抚养少儿和老年人口所承受的压力。
Dependency ratio = [(Population 0–14 + Population 65+) / Population 15–64] × 100
A population with 400 000 children, 300 000 elderly and 1 400 000 working‑ his means there are 50 dependents for every 100 people of working age. High dependency ratios (above 70–80) are common in many low‑income countries with youthful populations and in ageing societies.
若少儿40万，老年30万，劳动适龄人口140万，抚养比 = (400 000 + 300 000) / 1 400 000 × 100 = 50，即每100名劳动年龄人口负担50名依赖人口。高抚养比（70–80以上）常见于年轻型低收入国家和老龄化社会。
8. Temperature Range and Mean Temperature | 气温较差与平均气温
Temperature range shows the spread between the highest and lowest values in a day, month or year. Mean temperature is usually calculated by averaging daily maximum and minimum readings, or the values for all months if calculating the annual mean.
气温较差表示一天、一月或一年内最高值与最低值的差幅。平均气温通常取每日最高温与最低温的均值，若计算年均温则对各月均温再进行平均。
Daily range = Maximum temperature − Minimum temperature
Mean daily temperature = (Max + Min) / 2
Annual range = Highest monthly mean − Lowest monthly mean
rge annual ranges indicate continentality; small ranges are typical of equatorial or maritime locations.
某沙漠测站七月均温35 °C、一月均温12 °C，年较差 = 35 − 12 = 23 °C。年均温 = 12个月均温之和 / 12。年较差大反映大陆性气候，年较差小则为赤道或海洋性气候特征。
9. Annual Precipitation and Precipitation Effectiveness | 年降水量与降水有效性
Total annual precipitation is simply the sum of 12 monthly rainfall totals. However, effective precipitation matters more for ecosystems and farming — it accounts for losses through evaporation and runoff. A simple index compares precipitation with temperature.
年降水总量即12个月降水量之和。但对生态系统和农业而言，有效降水更为重要，它扣除了蒸发和径流的损失。一个简单指数可将降水量与温度进行比较。
Annual precipitation = Σ monthly precipitation
P/T ratio (simplified) = Annual precipitation (mm) / Mean annual temperature (°C)
average temperature of 15 °C has a P/T ratio of 600/15 = 40. Values below 20 suggest semi‑ tions, while values above 100 indicate very humid environments. This is a rough guide only — seasonal distribution is equally important.
年降水量600 mm、年均温15 °C的地区，P/T比 = 600/15 = 40。该比值低于20提示半干旱环境，高于100表示非常湿润。这仅为粗略参考，降水的季节分配同样重要。
10. Environmental Lapse Rate | 气温垂直递减率
As altitude increases, air temperature usually drops at a fairly steady rate in the lower atmosphere. This environmental lapse rate is crucial for explaining temperature differences between lowland and upland stations and for mountain climate studies.
随着海拔升高，低层大气气温通常以较稳定的速率下降。这一气温垂直递减率对解释低地与高地测站间的温差以及山地气候研究至关重要。
Average environmental lapse rate ≈ 6.5 °C per 1000 m
Temperature at elevation = Sea‑level temperature − (Altitude in m / 1000) × 6.5 °C
If the sea‑ idity and air movement can alter the actual lapse rate, so treat this as an approximation.
若海平面气温为28 °C，海拔1500 m的城镇气温约 28 − (1.5 × 6.5) = 18.25 °C。局部条件、湿度和空气运动会影响实际递减率，因此将该值视作估算值。
11. River Discharge | 河流流量
Discharge is the volume of water flowing past a point in a river per unit time. It links the river’s cross‑sectional area with its velocity and is the key variable for understanding flood risk, erosion potential and water availability.
流量是单位时间内通过河流某断面的水体体积，将过水断面面积与流速联系起来，是理解洪水风险、侵蚀能力和水资源量的核心变量。
Discharge (Q) = Cross‑sectional area (A) × Mean velocity (V)
Q = A × V (m³/s)
If a stream’s cross‑ s (cumecs). Discharge increases downstream as more tributaries join; sudden increases during a storm can lead to flooding.
若河道过水断面面积12.5 m²，流速仪测得平均流速0.8 m/s，则Q = 12.5 × 0.8 = 10 m³/s（立方米/秒）。流量随汇入支流增多而沿程增加；暴雨期间流量骤然增大可能引发洪水。
12. Percentage Change in Urbanisation and Other Indicators | 城市化及其他指标的百分比变化
Percentage change is widely used to compare rates of growth — in urban population, GDP, tourist arrivals or any time‑series data. The formula is straightforward but must be applied carefully to the start and end values.
百分比变化广泛用于比较城市人口、GDP、旅游人次等时间序列数据的增长率。公式虽简单，但须谨慎区分起始值与结束值。
Percentage change = [(New value − Old value) / Old value] × 100
250 000 to 310 000 over ten years, the change is (310 000 − 250 000) / 250 000 × 100 = 24% ive result indicates decline. Always state the time period over which the change occurred.
若某城市人口十年间从250 000增至310 000，变化率 = (310 000 − 250 000) / 250 000 × 100 = 24%。若结果为负值则表示下降。务必标明变化发生的时间段。
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