📚 Year 9 CIE Geography: Formula & Theorem Quick Reference Handbook | Year 9 CIE 地理：公式定理速查手册
This quick reference handbook brings together all the essential formulas, key principles and important models covered in the Year 9 CIE Geography syllabus. Use it to reinforce your understanding, complete calculations accurately and interpret geographical data with confidence.
这份速查手册汇总了Year 9 CIE地理课程中所有的核心公式、关键原理和重要模型。用它来巩固理解、准确完成计算并自信地解读地理数据。
1. Population Change Formulas | 人口变化公式
Population change can be measured using crude birth rate, crude death rate, natural increase and net migration rate. Each indicator has a specific formula expressed per 1000 population or as a percentage.
人口变化可以通过粗出生率、粗死亡率、自然增长率和净迁移率来衡量。每个指标都有特定的公式，以每千人或百分比表示。
CBR = (B ÷ P) × 1000
Crude Birth Rate (CBR) is the number of live births (B) divided by total population (P), multiplied by 1000. It tells us how many births occur per 1000 people in one year.
粗出生率（CBR）等于活产婴儿数（B）除以总人口（P）再乘以1000。它表示一年中每千人中出生的婴儿数。
CDR = (D ÷ P) × 1000
Crude Death Rate (CDR) is the number of deaths (D) divided by total population, multiplied by 1000. Like CBR, it helps compare death rates across countries with different population sizes.
粗死亡率（CDR）等于死亡人数（D）除以总人口再乘以1000。与CBR相似，它可以比较不同人口规模的国家之间的死亡率。
NI = CBR – CDR
Natural Increase (NI) is the difference between birth rate and death rate, usually expressed per 1000 population. A positive NI means natural growth, while a negative NI indicates natural decrease.
自然增长率（NI）是出生率和死亡率之差，通常以每千人为单位。正的NI表示自然增长，负的NI表示自然减少。
The Natural Increase Rate (NIR) as a percentage is calculated as (CBR – CDR) / 10. This quick conversion gives the percentage annual natural change.
自然增长率百分比（NIR）的计算公式为 (CBR – CDR) / 10。这个快速换算得出年自然变化的百分比。
NMR = (I – E) ÷ P × 1000
Net Migration Rate (NMR) equals immigrants (I) minus emigrants (E), divided by total population, multiplied by 1000. A positive NMR means more people are moving in than out.
净迁移率（NMR）等于迁入人口（I）减去迁出人口（E），再除以总人口乘以1000。正的NMR表示迁入多于迁出。
2. Population Density & Distribution | 人口密度与分布
Population density is a fundamental measure of how crowded a place is. It helps geographers compare settlement patterns and pressure on resources.
人口密度是衡量一个地方拥挤程度的基本指标。它帮助地理学家比较定居模式和对资源的压力。
D = P / A
Population density (D) equals total population (P) divided by land area (A). The result is expressed as people per square kilometre (persons/km²).
人口密度（D）等于总人口（P）除以土地面积（A），结果以每平方千米人数（人/km²）表示。
Arithmetic density treats all land equally, while physiological density relates population to arable land only. In CIE Year 9 studies, arithmetic density is most commonly used.
算术密度将所有土地同等对待，而生理密度仅将人口与可耕地关联。在CIE Year 9学习中，算术密度最为常用。
Population distribution describes how people are spread across a given area. Combined with density, you can explain uneven patterns using factors such as relief, climate and economic opportunities.
人口分布描述人们在一个给定区域内的分布方式。结合密度，你可以用地形、气候和经济机会等因素解释不均匀的分布格局。
3. River Discharge & Velocity | 河流流量与流速
Discharge is the volume of water flowing through a river channel per unit time. It is a product of cross-sectional area and mean velocity.
流量是单位时间内流经河道的水体积，它等于横截面积与平均流速的乘积。
Q = A × v
Discharge (Q) is measured in cubic metres per second (cumecs). Cross-sectional area (A) is the width multiplied by mean depth (in m²), and velocity (v) is in m/s.
流量（Q）的单位是立方米每秒（cumecs）。横截面积（A）是河流宽度乘以平均深度（单位m²），流速（v）单位是m/s。
A = w × d
You can estimate A by measuring width (w) and average depth (d) at several points across the channel. Velocity is often measured with a flowmeter or float method.
你可以通过在断面上测量多个点的宽度（w）和平均深度（d）来估算A。流速通常用流速仪或浮标法测量。
4. River Gradient & the Bradshaw Model | 河流坡降与布拉德肖模型
River gradient is the steepness of the river slope, typically decreasing downstream. The gradient formula is simple but crucial for understanding energy and erosion.
河流坡降是河床的陡峭程度，通常向下游递减。这一简单的梯级公式对于理解河流能量和侵蚀至关重要。
Gradient = Δh / d
Gradient equals the vertical drop (Δh) divided by the horizontal distance (d). It can be expressed as a ratio, e.g. 1:100, meaning a 1 m fall for every 100 m horizontal.
坡降等于垂直高差（Δh）除以水平距离（d），可以表示为比例，如1:100，即每100米水平距离下降1米。
The Bradshaw Model is a conceptual theorem showing how different river characteristics change from source to mouth. It predicts increases in discharge, width, depth, mean velocity and decreases in gradient, bedload size and channel roughness.
布拉德肖模型是一个概念性定理，展示不同河流特征从源头到河口的变化。它预测流量、宽度、深度、平均流速增加，而坡降、推移质粒径和河道糙率减小。
| Characteristic | Downstream change (Bradshaw) |
|---|---|
| Discharge | Increases |
| Width and depth | Increase |
| Gradient | Decreases |
| Bedload particle size | Decreases |
The model helps you predict that upper courses are steep, narrow and rough, while lower courses are wider, deeper and smoother, with higher discharge and finer sediments.
该模型帮助你预测上游河段陡峭、狭窄且粗糙，而下游河段更宽、更深且平滑，流量更高，沉积物更细。
5. Coastal Processes: Wave Energy Key Principles | 海岸过程：波浪能量关键原理
Although Year 9 does not require complex wave equations, understanding the principle of wave energy distribution is essential. Constructive and destructive waves behave according to energy thresholds.
尽管Year 9不要求复杂的波浪方程，但理解波浪能量分布的原理至关重要。建设性波浪和破坏性波浪根据能量阈值表现不同。
A key theorem states that wave energy is proportional to the square of wave height: E ∝ H². Thus a doubling of wave height quadruples the energy available for erosion or deposition.
一个关键原理是波浪能量与波高的平方成正比：E ∝ H²。因此波高加倍会使可用于侵蚀或沉积的能量增加四倍。
Constructive waves (low energy, long wavelength) have a wave steepness less than about 0.01 and favour deposition. Destructive waves (high energy, short wavelength) have steepness above 0.01 and promote erosion.
建设性波浪（低能量、长波长）波陡约小于0.01，有利于沉积。破坏性波浪（高能量、短波长）波陡大于0.01，促进侵蚀。
Wave steepness is the ratio of wave height to wavelength (H/L). This principle helps predict which coastal landforms develop under specific wave conditions.
波陡是波高与波长之比（H/L）。这一原理有助于预测特定波浪条件下会发育出哪些海岸地貌。
6. Climate & Weather Formulas: Temperature & Precipitation | 气候与天气公式：温度与降水
Weather data analysis relies on several straightforward formulas. Annual mean temperature, temperature range and total precipitation are the building blocks of climate graphs.
天气数据分析依赖于几个直观的公式。年均温、气温较差和总降水量是绘制气候图的基本要素。
Mean Tₐₙₙᵤₐₗ = Σ monthly means / 12
Annual mean temperature is the sum of all 12 monthly average temperatures divided by 12. Monthly means are typically calculated from daily maxima and minima.
年平均气温等于12个月的月平均气温总和除以12。月平均气温通常由日最高温和日最低温计算得出。
Temperature range = Tₘₐₓ – Tₘᵢₙ
Diurnal range is the difference between daily maximum and minimum. Annual range is the difference between the hottest and coldest monthly mean temperatures.
日较差是日最高温与日最低温之差。年较差是最热月平均气温与最冷月平均气温之差。
Total precipitation = Σ monthly rainfall
Total annual precipitation is simply the sum of all monthly rainfall totals. Rainfall totals are shown by the bar graph on a climate chart.
年总降水量就是所有月降水量之和。降水量用气候图表中的柱状图表示。
°F = (°C × 9 / 5) + 32
ired skill. For example, 25°C becomes (25 × 9/5) + 32 = 77°F. Reverse conversion: °C = (°F – 32) × 5/9.
摄氏与华氏温度换算是必备技能。例如，25°C换算为 (25 × 9/5) + 32 = 77°F。反向换算：°C = (°F – 32) × 5/9。
7. Map Skills: Scale, Gradient & Area | 地图技能：比例尺、坡度和面积
Mapwork formulas are essential for interpreting topographic maps. They allow you to convert map measurements into real-world distances and slopes.
地图技能公式是解读地形图的基础。它们让你能够将图上测量值转化为实际距离和坡度。
Scale = Map distance / Ground distance
Representative fraction (RF) uses the same units, e.g. 1:25 000 means 1 cm on the map equals 25 000 cm (250 m) on the ground. Always convert units correctly.
比例尺（RF）使用相同单位，如1:25000表示图上1厘米等于实地25000厘米（250米）。务必正确换算单位。
Gradient = Vertical interval / Horizontal equivalent
Gradient is calculated by dividing the difference in height between two points by the horizontal distance between them. Express it as a ratio 1:X for easy comparison.
坡度由两点间的高差除以水平距离得出。为便于比较，常表示为1:X 的比例。
readth. For irregular shapes, the number of grid squares (each 1 km²) and partial squares are counted to estimate area.
规则图形的面积用长乘宽。不规则图形则数网格平方数（每个1 km²）并估算不足整格的部分来求面积。
Bearing = angle measured clockwise from north
Bearings are given as three-figure numbers from 000° to 360°. Use a protractor aligned with the grid north line to measure the angle to the feature.
方位角以三位数字表示，范围从000°到360°。将量角器对准格网北方向线，量取到目标的夹角。
8. Demographic Transition Model (DTM) Theorem | 人口转变模型 (DTM) 定理
The Demographic Transition Model is a foundational theorem explaining how birth and death rates change as a country develops. It is divided into five stages.
人口转变模型是一个基本定理，解释了一个国家在发展过程中出生率和死亡率如何变化。它分为五个阶段。
| Stage | Birth rate | Death rate | Natural change |
|---|---|---|---|
| 1 | High and fluctuating | High and fluctuating | Stable or slow increase |
| 2 | High | Falling rapidly | Rapid increase |
| 3 | Falling | Slowly falling | Increase slows |
| 4 | Low | Low | Stable or slow increase |
| 5 | Very low | Low | Natural decrease possible |
The DTM theorem states that all countries pass through these stages, though the timing varies. It links population change to economic development, medical advances and social changes.
DTM 定理指出所有国家都会经历这些阶段，只是时间不同。它将人口变化与经济发展、医疗进步和社会变迁联系起来。
9. Settlement Hierarchy: Threshold & Range Principle | 定居点等级：门槛与范围原理
Settlement hierarchy theory is based on the principle that higher-order services need a larger threshold population and a wider range to survive.
定居点等级理论基于一个原理：高等级服务需要较大的人口门槛和较广的服务范围才能维持。
Threshold is the minimum number of people required to support a service. A small village shop has a low threshold, while a large hospital or university requires a much higher threshold.
门槛是支撑一项服务所需的最低人口数量。乡村小商店的门槛低，而大型医院或大学需要高得多的门槛。
Range is the maximum distance people are willing to travel for a service. Low-order goods have a short range; high-order goods have a long range. This principle explains the spacing and distribution of settlements.
范围是人们愿意为一项服务出行最远的距离。低级商品范围短，高级商品范围大。这一原理解释了定居点的间距与分布。
Christaller's central place theorem, in its simplified form, states that settlements form a hexagonal pattern based on threshold and range, with higher-order centres spaced farther apart.
克里斯泰勒的中心地定理简化形式表明，定居点根据门槛和范围形成六边形模式，高级中心相距更远。
10. Hydrological Cycle Balance Equation | 水循环平衡方程
The water balance equation expresses the fundamental theorem of conservation of water mass in a drainage basin over a given period.
水平衡方程表达了在一个给定时期内，流域水量守恒的基本定理。
P = Q + ET + ΔS
Precipitation (P) equals total runoff (Q) plus evapotranspiration (ET) plus the change in water storage (ΔS) in soil, groundwater, lakes and vegetation. All components are usually expressed in millimetres of water depth.
降水量（P）等于总径流量（Q）加上蒸散发（ET），再加上土壤、地下水、湖泊和植被中的贮水变化量（ΔS）。所有分量通常以毫米水深表示。
Runoff includes surface runoff, throughflow and groundwater flow. The balance equation helps predict floods and droughts. A positive ΔS means storage is increasing; a negative ΔS indicates depletion.
径流包括地表径流、壤中流和地下水流
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