📚 Year 9 CAIE Geography Formula & Theorem Quick Reference Handbook | CAIE 9 年级地理公式定理速查手册
Welcome to your go-to quick reference for the essential formulas, calculations and key theoretical models in the Year 9 CAIE Geography course. Whether you are revising map skills, population dynamics, river processes or climate factors, this handbook brings together all the must-know equations and concepts in one place. Each entry is paired with clear explanations in English and Chinese to support bilingual learners.
欢迎使用专为 Year 9 CAIE 地理课程准备的核心公式、计算方法和重要理论模型速查手册。无论你正在复习地图技能、人口动态、河流过程还是气候因子，这本手册都将所有必知必会的公式和概念汇集一处。每个条目都配有清晰的中英双语解释，帮助双语学习者掌握关键内容。
1. Map Scale and Distance | 地图比例尺与距离
Map scale describes the relationship between a length on the map and the corresponding length on the ground. It is usually given as a ratio or representative fraction, for example 1:50 000. This means 1 cm on the map stands for 50 000 cm (or 0.5 km) on the Earth's surface.
地图比例尺描述地图上的长度与地面上相应长度之间的关系。它通常以比率或比例尺分数的形式给出，例如 1:50 000，这意味着地图上 1 厘米代表地表上的 50 000 厘米（或 0.5 千米）。
The key formula to find real distances is: Actual Distance = Map Distance x Scale Denominator vert centimetres into metres or kilometres. Remember that 100 cm = 1 m, 100 000 cm = 1 km.
计算实际距离的关键公式是：实际距离 = 地图距离 x 比例尺分母。完成乘法运算后，需要将厘米转换为米或千米。请记住：100 厘米 = 1 米，100 000 厘米 = 1 千米。
distance = 4 x 25 000 = 100 000 cm = 1 km. If you are asked to find the map distance, use Map Distance = Actual Distance / Scale Denominator.
示例：在 1:25 000 的地图上，两点之间的地图距离为 4 厘米。实际距离 = 4 x 25 000 = 100 000 厘米 = 1 千米。如果要求你计算地图距离，请使用 地图距离 = 实际距离 / 比例尺分母。
2. Gradient Calculation | 坡度计算
Gradient measures the steepness of a slope. It compares the vertical difference (rise) between two points with the horizontal distance (run) that separates them. Both measurements must be in the same unit.
坡度衡量斜坡的陡峭程度。它比较两点之间的垂直高度差（升高）与两点之间的水平距离（水平跨度）。两个距离必须使用相同的单位。
The formula for gradient is: Gradient = Vertical Difference / Horizontal Distance. The result is often simplified to a ratio such as 1 in X, which is written as 1:X.
坡度计算公式为：坡度 = 垂直高差 / 水平距离。计算结果通常简化为比率，例如 1 in X，并写作 1:X。
/25 = 1:25. This means for every 25 metres travelled horizontally, the land rises 1 metre.
示例：一座小山在 500 米的水平距离内上升了 20 米。坡度 = 20 / 500 = 1/25 = 1:25。这意味着每水平前进 25 米，地面上升 1 米。
3. Population Density | 人口密度
Population density shows how crowded an area is. It is found by dividing the total population by the land area. The formula is: Population Density = Total Population / Land Area (km²).
人口密度显示一个地区的拥挤程度。它由总人口除以土地面积得出。公式为：人口密度 = 总人口 / 土地面积（平方公里）。
ces with high densities tend to be urbanised, while low densities are typical of rural regions.
例如，一个有 600 000 人的城市，面积为 300 平方公里，其人口密度为 600 000 / 300 = 每平方公里 2000 人。人口密度高的地区通常是城市化地区，而低密度则是农村地区的典型特征。
4. Crude Birth Rate and Death Rate | 粗出生率与粗死亡率
The Crude Birth Rate (CBR) is the number of live births per year per 1000 people in the population. Its formula is: CBR = (Number of live births / Total population) x 1000. Similarly, the Crude Death Rate (CDR) is calculated as: CDR = (Number of deaths / Total population) x 1000.
粗出生率（CBR）是每年每 1000 人中活产婴儿的数量。其公式为：CBR =（活产婴儿数 / 总人口）x 1000。同样，粗死亡率（CDR）的计算公式为：CDR =（死亡人数 / 总人口）x 1000。
These rates are per 1000, not percentages, so a CBR of 25 means 25 babies per 1000 people each year. High birth rates are often linked to limited access to family planning, while low death rates suggest good healthcare.
这些指标是千分率，不是百分比，因此 CBR 为 25 表示每年每 1000 人有 25 个婴儿出生。高出生率通常与计划生育普及程度有限相关，而低死亡率则意味着良好的医疗保健。
5. Natural Increase Rate | 自然增长率
The rate of natural increase (RNI) shows how fast a population is growing without counting migration. It is the difference between the birth rate and death rate: Natural Increase Rate (per 1000) = CBR – CDR.
自然增长率（RNI）显示在未计入迁移的情况下，人口增长的速度。它是出生率与死亡率之差：自然增长率（千分率）= CBR – CDR。
To convert the natural increase rate into a percentage, divide by 10. For instance, a CBR of 30 and CDR of 10 gives a RNI of 20 per 1000, which is 2.0% per year. A negative RNI means the population is shrinking naturally.
将自然增长率转换为百分比时，需除以 10。例如，CBR 为 30，CDR 为 10，自然增长率为千分之二十，即每年 2.0%。自然增长率为负意味着人口自然缩减。
6. Doubling Time (Rule of 70) | 人口倍增时间（70 法则）
Doubling time estimates the number of years it will take for a population to double in size, assuming a constant growth rate. The simple rule is: Doubling Time (years) = 70 / Growth Rate (%).
人口倍增时间用于估算在保持恒定增长率的情况下，人口规模翻一番所需的年数。简单的计算法则为：倍增时间（年）= 70 / 增长率（%）。
For example, if a country's natural increase rate is 2% oubling time is about 70 / 2 = 35 years. This formula works best for growth rates below 10% and is a quick way to grasp the power of compound growth.
例如，如果一个国家的自然增长率为 2%，那么人口倍增时间大约是 70 / 2 = 35 年。该公式最适合低于 10% 的增长率，是快速理解复利增长威力的好方法。
7. River Discharge | 河流流量
River discharge is the volume of water flowing past a point in a river per unit of time, usually measured in cubic metres per second (cumecs). The equation is: Discharge (Q) = Cross-sectional Area (A) x Velocity (V).
河流流量是指单位时间内流过河流某一点的水量，通常以立方米每秒（cumecs）为单位。计算公式为：流量 (Q) = 横截面积 (A) x 流速 (V)。
Cross-sectional area is measured in m² and velocity in m/s. To find A, multiply the average width of the channel by its average depth. During floods, both the area and velocity increase, causing a rapid rise in discharge.
横截面积以平方米计算，流速以米/秒计算。要得出 A，将河道的平均宽度乘以其平均深度。在洪水期间，面积和流速都会增加，导致流量迅速增大。
8. Environmental Lapse Rate | 环境温度直减率
In the troposphere, temperature normally decreases as altitude increases. The average environmental lapse rate is about 6.5 °C per 1000 m, but for simple calculations we often use 0.6 °C per 100 m.
在对流层中，气温通常随着海拔升高而降低。平均环境温度直减率约为每升高 1000 米下降 6.5 °C，但在简单计算中我们常用 每升高 100 米下降 0.6 °C。
The formula to estimate temperature at a higher elevation is: T₂ = T₁ – (Lapse Rate x Δh), where T₁ is the temperature at the lower altitude, Δh is the change in height in hundreds of metres, and the lapse rate is 0.6 °C/100 m.
估算更高海拔处温度的公式为：T₂ = T₁ –（温度直减率 x Δh），其中 T₁ 是较低海拔处的温度，Δh 是以百米为单位的高度变化，温度直减率为 0.6 °C/100 m。
s: Height difference = 1000 m = 10 x 100 m. Temperature drop = 10 x 0.6 = 6 °C. So T₂ ≈ 24 – 6 = 18 °C.
示例：一个位于海拔 200 米的站点记录温度为 24 °C。海拔 1200 米处的温度估算如下：高度差 = 1000 米 = 10 个 100 米。降温幅度 = 10 x 0.6 = 6 °C。因此 T₂ ≈ 24 – 6 = 18 °C。
9. Demographic Transition Model | 人口转变模型
The Demographic Transition Model (DTM) describes how birth rates, death rates and total population change as a country develops. It is typically divided into four or five stages. Stage 1 (high fluctuating) has high birth and death rates, keeping the population low and stable. Stage 2 (early expanding) sees death rates fall sharply while birth rates stay high, causing rapid population growth. In Stage 3 (late expanding) birth rates begin to decline, and population growth slows. Stage 4 (low fluctuating) brings low, steady birth and death rates, with a stable population.
人口转变模型（DTM）描述了随着国家发展，出生率、死亡率和总人口如何变化。它通常分为四或五个阶段。第一阶段（高位波动）出生率和死亡率均高，人口维持在低而稳定的水平。第二阶段（早期扩张）死亡率急剧下降而出生率居高不下，导致人口快速增长。第三阶段（晚期扩张）出生率开始下降，人口增长放缓。第四阶段（低位波动）出生率和死亡率均稳定在低水平，人口趋于稳定。
This model helps geographers explain why some countries grow quickly and others grow slowly. It also links population change to economic development, urbanisation and the status of women. Keep in mind that original DTM was based on Western Europe, so other regions may follow different paths.
该模型帮助地理学者解释为何一些国家人口增长快，另一些增长慢。它将人口变化与经济发展、城市化和妇女地位联系起来。请记住，原始的 DTM 是基于西欧的经验，其他地区可能遵循不同的路径。
10. Urban Land Use Models (Burgess and Hoyt) | 城市土地利用模型（伯吉斯和霍伊特）
Burgess's Concentric Zone Model (1924) suggests cities grow outward in a series of rings. The Central Business District (CBD) sits at the core, surrounded by a zone of transition with old industry and low-cost housing. Outer rings contain working-class housing, then middle-class housing, and finally the commuter zone on the edge.
伯吉斯的同心圆模型（1924 年）认为城市以一系列同心圆环的方式向外扩展。中央商务区（CBD）位于核心，周围是过渡地带，分布着老旧工业和低成本住房。外围的同心圆环依次包含工人阶级住房、中产阶级住房，最终在边缘地带形成通勤者居住区。
Hoyt's Sector Model (1939) argues that cities develop in wedges or sectors radiating from the CBD. Transport routes influence the pattern: high-rent residential areas expand along major roads, while industry develops along railways or rivers. Low-income housing often lies close to industrial zones.
霍伊特的扇形模型（1939 年）认为城市沿着从 CBD 辐射出的楔形或扇形区域发展。交通线路影响着这一模式：高租金居住区沿主干道扩展，而工业区则沿铁路或河流发展。低收入住房通常靠近工业区分布。
Both models help explain land-use patterns, but real cities are more complex. They provide a starting point for analysing housing, industry and social segregation in urban areas.
这两个模型都有助于解释土地利用模式，但现实中的城市更为复杂。它们为分析城市地区的住房、工业和社会隔离提供了起点。
11. Factors Influencing Climate | 影响气候的因素
Several key factors shape the climate of a place. Latitude affects temperatures: regions near the equator receive more direct sunlight and are warmer, while polar areas are colder. Altitude is also crucial – temperature falls with height, as described by the lapse rate.
影响一地气候的关键因素有几个。纬度影响气温：赤道附近地区接收更多太阳直射光，温度较高，而极地地区较冷。海拔同样至关重要——如温度直减率所述，气温随高度升高而下降。
Distance from the sea (continentality) changes the temperature range; coastal areas have milder winters and cooler summers, whereas inland areas experience more extreme temperatures. Ocean currents can warm or cool adjacent land, and prevailing winds bring moisture or dryness depending on their source. Mountain ranges create rain shadows – windward slopes receive heavy rainfall, while leeward slopes remain dry.
距海远近（大陆度）会改变气温年较差；沿海地区冬季温和、夏季凉爽，而内陆地区则经历更为极端的气温。洋流能使其附近的陆地增温或降温，而盛行风带来的水汽或干燥程度取决于其源地。山脉形成雨影效应——迎风坡降雨充沛，背风坡则保持干燥。
These factors interact to produce the wide variety of climate types studied in geography, such as equatorial, desert, Mediterranean and tundra climates.
这些因素相互作用，形成了地理学中研究的多种气候类型，如赤道气候、沙漠气候、地中海气候和苔原气候。
12. Water Balance Equation | 水量平衡方程
The water balance, or hydrologic budget, is a simple accounting of
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