📚 KS3 CAIE Geography: Formula & Theorem Quick Reference Handbook | KS3 CAIE 地理：公式定理速查手册
This handbook provides a rapid revision of essential formulas, calculations and key theorems required for the KS3 CAIE Geography syllabus. Use it to master map skills, population dynamics, climate statistics and river measurements with confidence.
本手册为 KS3 CAIE 地理课程提供必背公式、计算与重要定理的快速复习。掌握地图技能、人口统计、气候数据与河流测量，助你自信应考。
1. Map Scale and Distance | 地图比例尺与距离
Map scale expresses the ratio between a distance on the map and the actual distance on the ground. It is shown as a representative fraction (e.g. 1:50,000) or a linear scale.
地图比例尺表示图上距离与实际地面距离之比，通常以数字比例尺（如 1:50,000）或直线比例尺表示。
The formula to convert map distance to real distance is: Actual Distance = Map Distance × Scale Denominator. For a map with a scale of 1:50,000, every 1 cm on the map equals 50,000 cm (0.5 km) on the ground.
将图上距离换算为实际距离的公式：实际距离 = 图上距离 × 比例尺分母。在 1:50,000 地图上，图上 1 厘米等于实地 50,000 厘米（0.5 千米）。
To express actual distance in kilometres, divide by 100,000 after multiplying map distance in centimetres.
若要以千米为单位表示实际距离，用图上厘米数乘以比例尺分母后再除以 100,000。
Actual Distance (km) = (Map Distance in cm × Scale) ÷ 100,000
实际距离（千米）=（图上距离cm × 比例尺分母）÷ 100,000
Example: On a 1:25,000 map, two villages are 4 cm apart. Real distance = 4 × 25,000 ÷ 100,000 = 1 km.
示例：在 1:25,000 地图上，两村图上距离 4 cm。实际距离 = 4 × 25,000 ÷ 100,000 = 1 km。
2. Area Calculation from Grid Squares | 利用方格网计算面积
On maps with grid lines, area can be estimated by counting the number of whole and partial squares within a boundary.
在有方格网的地图上，可通过计算边界内完整和部分方格的数量来估算面积。
Each grid square represents a known ground area. For example, on a 1:50,000 map with a 2 cm grid, each square represents 1 km².
每个方格代表已知的地面面积。例如，1:50,000 地图上 2 cm 网格，每个方格代表 1 km²。
Area = (Number of full squares + ½ × number of partial squares) × area represented by one square.
面积 =（完整方格数 + ½ × 不完整方格数）× 单个方格所代表的面积。
Area (km²) = Total square count × Ground area per square
面积（平方千米）= 方格总数 × 每格代表的地面面积
3. Gradient (Slope) Calculation | 坡度计算
Gradient measures how steep a slope is. It is the ratio of the vertical rise to the horizontal distance travelled.
坡度用于衡量斜坡的陡峭程度，是垂直升高与水平距离之比。
On a map, gradient is calculated using the contour interval (vertical) and the map distance between contours (horizontal).
在地图上，利用等高线间距（垂直）和等高线间的图上水平距离来计算坡度。
Gradient = Vertical Interval (m) ÷ Horizontal Distance (m)
坡度 = 垂直高差（m）÷ 水平距离（m）
The result is often simplified to a ratio such as 1:10, meaning for every 1 metre of rise, there is 10 metres of horizontal travel.
结果常简化为如 1:10 的比率，表示每垂直上升 1 米，水平前进 10 米。
4. Population Density | 人口密度
Population density describes how crowded an area is. It is the number of people per unit area, usually per square kilometre.
人口密度描述一个地区的拥挤程度，指单位面积内的人口数量，通常为每平方千米人数。
Population Density = Total Population ÷ Total Land Area (km²)
人口密度 = 总人口 ÷ 总面积（平方千米）
High density indicates many people in a small space; low density means few people spread over a large area.
高密度表示小空间内人口众多；低密度表示人口稀少且分布在大面积区域。
5. Birth Rate, Death Rate and Natural Increase | 出生率、死亡率与自然增长
These demographic indicators help analyse population change. Rates are usually expressed per 1,000 people per year.
这些人口统计指标用于分析人口变化，比率通常以每年每千人表示。
Crude Birth Rate (CBR) = (Number of live births ÷ Total population) × 1,000
粗出生率（CBR）=（活产婴儿数 ÷ 总人口）× 1,000
Crude Death Rate (CDR) = (Number of deaths ÷ Total population) × 1,000
粗死亡率（CDR）=（死亡人数 ÷ 总人口）× 1,000
Natural increase is simply the difference between birth rate and death rate, ignoring migration.
自然增长就是出生率与死亡率之差，不考虑迁移因素。
Natural Increase Rate = CBR − CDR (per 1,000)
自然增长率 = 粗出生率 − 粗死亡率（每千人）
6. Population Growth Rate | 人口增长率
Population growth rate includes natural change and net migration, reflecting the total change in population over a year.
人口增长率包括自然变动和净迁移，反映一年内人口的总变化。
Growth Rate (%) = [(Births − Deaths + Net Migration) ÷ Total Population] × 100
增长率（%）= [（出生数 − 死亡数 + 净迁移人口）÷ 总人口] × 100
A positive rate means the population is rising; a negative rate indicates decline.
正增长率表示人口增加；负增长率则表示人口减少。
7. Urbanisation Rate | 城市化率
Urbanisation measures the share of a country’s population living in urban areas. It is a key indicator of development.
城市化衡量一国居住在城市地区的人口比例，是发展水平的重要指标。
Urbanisation Rate (%) = (Urban population ÷ Total population) × 100
城市化率（%）=（城镇人口 ÷ 总人口）× 100
An increasing urbanisation rate suggests rural-to-urban migration and the growth of towns and cities.
城市化率上升表明农村向城市的人口迁移以及城镇的发展壮大。
8. River Discharge | 河流流量
Discharge is the volume of water flowing through a river channel at a given point per second, measured in cubic metres per second (m³/s or cumecs).
流量是河流某断面每秒流过的水体体积，单位为立方米每秒（m³/s）。
It is calculated by multiplying the cross-sectional area of the channel by the average flow velocity.
其计算方法是将河道横截面积乘以平均流速。
Discharge (Q) = Cross-sectional area (A) × Velocity (V)
流量（Q）= 横截面积（A）× 流速（V）
Width × average depth gives the approximate area. Velocity is measured by timing a float over a known distance.
宽度 × 平均水深可得出近似面积。流速通过测量浮标漂过已知距离的时间来计算。
9. Temperature Range (Diurnal and Annual) | 气温较差（日较差和年较差）
Temperature range shows the variation in temperature over a day or a year, helping to classify climates.
气温较差表示一天或一年内的温度变化幅度，有助于气候分类。
Diurnal range is the difference between the maximum and minimum temperature in a single day.
日较差是一天中最高气温与最低气温的差值。
Diurnal Temperature Range = Tₘₐₓ − Tₘᵢₙ
日较差 = 最高气温 − 最低气温
Annual range compares the average temperatures of the hottest and coldest months.
年较差则比较最热月与最冷月的平均气温。
Annual Temperature Range = Average temp of hottest month − Average temp of coldest month
年较差 = 最热月平均气温 − 最冷月平均气温
10. Precipitation Variability and Average | 降水量变率与平均值
Average precipitation is the long-term mean rainfall or snowfall for a location, while variability shows how much individual years differ from this average.
平均降水量是一个地点长期的降水平均值，而降水变率显示个别年份与平均值的差异程度。
Annual Precipitation Variability (%) = [(Annual total − Mean annual total) ÷ Mean annual total] × 100
年降水变率（%）= [（年总量 − 多年平均总量）÷ 多年平均总量] × 100
A high positive variability means a much wetter year than normal; negative means drier.
正值偏大表示该年比常年湿润得多；负值则表示更为干燥。
11. Bearings and Direction | 方位角与方向
Bearings are used in map reading to give accurate directions. They are measured clockwise from true north, from 0° to 360°.
方位角用于地图判读以给出精确方向，按顺时针从真北起算，0° 至 360°。
To find the bearing from point A to point B, draw a line connecting them, then measure the angle from north using a protractor.
确定 A 点到 B 点的方位角时，连接两点画线，然后用量角器从北方向顺时针量取角度。
Always express bearings with three digits, e.g. 045° for north-east, 225° for south-west.
方位角始终用三位数字表示，如东北方向为 045°，西南方向为 225°。
12. Doubling Time (Rule of 70) | 人口翻倍时间（70法则）
Doubling time estimates how many years it will take for a population to double in size, assuming a constant growth rate.
翻倍时间用于估算在恒定增长率下，人口数量翻一番所需的年数。
The ‘Rule of 70’ provides a quick approximation: divide 70 by the annual percentage growth rate.
“70法则”提供了一个快速近似值：用 70 除以年增长率百分数。
Doubling Time (years) ≈ 70 ÷ Annual growth rate (%)
翻倍时间（年） ≈ 70 ÷ 年增长率（%）
For example, if a country’s population grows at 2% per year, doubling time = 70 ÷ 2 = 35 years.
例如，若某国人口年增长率为 2%，则翻倍时间 = 70 ÷ 2 = 35 年。
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