📚 Year 9 CAIE Geography: Formula & Theorem Quick Reference Handbook | Year 9 CAIE 地理：公式定理速查手册
Welcome to your essential quick reference guide for Year 9 CAIE Geography. This handbook brings together all the key formulas, equations, and calculation techniques you need to master for your assessments. From population dynamics and urbanisation to map skills and river measurements, each section provides a clear explanation in both English and Chinese to support bilingual learners and ensure no formula is left behind. Keep this guide handy for revision, homework, and exam preparation.
欢迎使用这本为你量身打造的 Year 9 CAIE 地理公式定理速查手册。这里汇集了所有你需要在测评中掌握的核心公式、方程和计算技巧。从人口动态、城市化到地图技能和河流测量，每个小节都提供了清晰的中英文双语解释，帮助双语学习者轻松掌握每一个公式。复习、作业和备考时，请随身携带这份指南。
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 total land area. This gives an average number of people per square kilometre, helping geographers compare levels of crowding between different places.
人口密度衡量一个地区的拥挤程度。它用该地区的总人口除以其土地总面积来计算，得出每平方公里平均人口数，帮助地理学家比较不同地方的拥挤水平。
Formula | 公式：
Population Density = Total Population ÷ Land Area (km²)
n density is 2,000,000 ÷ 500 = 4,000 people per km². Remember to use the correct land area – not include bodies of water unless the question specifically asks for a different measure.
例如，如果一座城市有2,000,000人口，面积为500平方公里，那么人口密度为2,000,000 ÷ 500 = 4,000人/平方公里。请记得使用正确的土地面积——除非题目另有要求，否则不包括水域面积。
2. Birth Rate, Death Rate, and Natural Increase | 出生率、死亡率与自然增长率
These three vital rates help explain how a population changes over time without considering migration. The birth rate tells us how many live births occur per 1,000 people in one year, while the death rate shows how many deaths occur per 1,000 people. Their difference gives the natural increase rate.
这三项关键指标帮助我们解释人口在没有迁移因素下的自然变化。出生率表明每年每千人中活产婴儿的数量，死亡率表明每年每千人中的死亡数量。两者之差即为自然增长率。
Formulas | 公式：
Birth Rate (BR) = (Number of live births ÷ Total population) × 1,000
Death Rate (DR) = (Number of deaths ÷ Total population) × 1,000
Natural Increase Rate (NIR) = BR – DR (per 1,000 people)
rcentage. For instance, a birth rate of 20 and a death rate of 8 give NIR = 12 per 1,000, which is 1.2% per year.
许多考题会要求你将自然增长率转换为百分比。由于这些比率是每千人的，除以10即可得到百分数。例如，出生率20，死亡率8，NIR = 12‰，相当于每年1.2%。
3. Urbanisation Rate | 城市化率
Urbanisation is the increasing proportion of a country’s population living in towns and cities. The urbanisation rate is a simple percentage that shows how urbanised a place is. It is a key indicator when comparing regions at different levels of development.
城市化是指城镇人口在一个国家总人口中所占比例的上升。城市化率是一个简单的百分比，表明一个地方的城市化程度，是比较不同发展水平地区的关键指标。
Formula | 公式：
Urbanisation Rate (%) = (Urban population ÷ Total population) × 100
urban dwellers out of a total of 90 million, the urbanisation rate is (54 ÷ 90) × 100 = 60% ou must compare these percentages at two different years or calculate the absolute change.
如果一个国家总人口9000万，城市人口5400万，城市化率为(54 ÷ 90) × 100 = 60%。要衡量城市化进程的速度，需要比较两个不同年份的百分比或计算绝对变化量。
4. Dependency Ratio | 抚养比 / 受扶养人口比率
The dependency ratio shows the number of people who are likely to be economically dependent on the working-age population. Dependents are usually defined as those aged 0–14 (young dependents) and 65 and over (elderly dependents). The working-age group is 15–64. The dependency ratio is a powerful tool to analyse the social and economic pressures on a population.
抚养比反映了需要依靠劳动年龄人口经济支持的人群数量。受扶养者通常指 0–14 岁（幼年抚养）和 65 岁及以上（老年抚养）人群。劳动年龄组为 15–64 岁。抚养比是分析人口社会和经济压力的有力工具。
Formula | 公式：
Total Dependency Ratio = [(Population 0–14 + Population 65+) ÷ Population 15–64] × 100
50. This means there are 50 dependents for every 100 people of working age. A high dependency ratio suggests a greater financial burden on the workforce.
例如，如果一个地区有300名幼年被抚养者、200名老年被抚养者和1000名劳动年龄人口，抚养比为[(300+200) ÷ 1000] × 100 = 50。这意味着每100名劳动年龄人口需要抚养50名受扶养者。抚养比高意味着劳动力负担更重。
5. Gradient (Slope) | 坡度 / 斜率
Gradient is a map skill that measures how steep a slope is. It is usually expressed as a ratio, such as 1:20, which means for every 20 metres of horizontal distance, the land rises or falls by 1 metre. Understanding gradient is crucial for interpreting contour lines on topographic maps and assessing land use potential.
坡度是衡量斜坡陡峭程度的地图技能。它通常以比率表示，如 1:20，即水平距离每 20 米，地面升高或降低 1 米。理解坡度对解读地形图等高线、评估土地用途至关重要。
Formula | 公式：
Gradient = Vertical Difference (Height) ÷ Horizontal Distance
distance (after scale conversion) is 600 m, the gradient is 30 ÷ 600 = 1/20, so the ratio is 1:20. Always ensure both measurements are in the same units before calculating.
先将高差除以水平距离，再化简为分子为 1 的比率。例如，两点间高差为 30 米，根据比例尺换算后的水平距离为 600 米，则坡度为 30 ÷ 600 = 1/20，即 1:20。计算前务必确保两者的单位一致。
6. Map Scale | 地图比例尺
Map scale expresses the relationship between a distance on a map and the corresponding distance on the ground. In CAIE exams, you will work with linear scales (bar scales) and representative fractions (e.g., 1:50,000). You must be able to convert map distances into real-world distances and vice versa.
地图比例尺表示图上距离与相应实地距离的关系。在 CAIE 考试中，你将使用直线比例尺和数字比例尺（如 1:50,000）。你必须能够将图上距离换算为实际距离，反之亦然。
Key conversions | 关键换算：
Real Ground Distance (cm) = Map Distance (cm) × Scale Denominator
d, which is 1,000 m or 1 km. Also learn to measure curved distances using a string or the edge of a piece of paper and then applying the scale.
然后将会厘米转换为米（÷100）或公里（÷100,000）。对于 1:25,000 的地图，图上 4 厘米代表实地 4 × 25,000 = 100,000 厘米，即 1,000 米或 1 公里。还要学会用细线或纸边测量曲线距离，再套用比例尺。
7. Area Calculation on Maps | 地图上面积计算
shapes. For irregular shapes, the grid square method is most reliable: count all full squares and combine partial squares to estimate the total.
计算地形图上某一地物的面积是一项常见的 GIS 技能。规则图形可以用网格法或长乘宽法，不规则图形最可靠的是网格法：统计所有完整方格，并合并不完整方格来估算总面积。
Steps | 步骤：
- 0 map where each grid square is 2 cm × 2 cm, the real side is 1 km, so one square = 1 km².
- Count full squares within the boundary, then estimate partial squares (count every two half squares as one full square, or use ticks).
- Multiply total grid squares by the area per square.
1. 确定比例尺，计算一个网格方格的实际面积。例如，在 1:50,000 地图上，每个方格边长 2 厘米，实地边长 1 公里，因此一个方格 = 1 平方公里。
2. 统计边界内完整方格，然后估算不完整方格（两个半格计为一格，或用打点法）。
3. 将估算方格总数乘以每格的面积。
8. River Discharge (Stream Flow) | 河流流量
Discharge is the volume of water flowing past a given point in a river per unit of time, usually expressed in cubic metres per second (cumecs). It is a fundamental measurement in hydrology and links to flooding, erosion, and water resource management. Discharge increases downstream as tributaries join the main channel.
流量是指单位时间内通过河流某一断面的水量，通常以立方米/秒表示。它是水文学的基本测量指标，与洪水、侵蚀和水资源管理密切相关。随着支流汇入，流量通常向下游增加。
Formula | 公式：
Discharge (Q) = Cross-sectional Area (A) × Mean Velocity (V)
s-section area of 12 m² and the water is flowing at 0.8 m/s, then Q = 12 × 0.8 = 9.6 m³/s.
过水断面面积单位为平方米，流速单位为米/秒。如果河道过水断面面积 12 平方米，水流速度 0.8 米/秒，则流量 Q = 12 × 0.8 = 9.6 立方米/秒。
9. Cross-sectional Area of a River Channel | 河道过水断面面积
To find the cross-sectional area, geographers measure the width and multiple depths across a channel. The area is then estimated by dividing the channel into sections (the 'dividing line' method) or by using a simplified formula for a roughly rectangular or trapezoidal channel.
为求出过水断面面积，地理学家需测量河道宽度和不同点的深度。然后通过将河道分成若干区域（等分法）或使用近似矩形/梯形的简化公式来估算面积。
Simplified formula for a rectangular approximation | 矩形近似公式：
Area (A) ≈ Channel Width (W) × Mean Depth (D)
In field studies, depth is measured at regular intervals across the channel using a ruler or meter stick. The mean depth is calculated by adding up all the depth readings and dividing by the number of readings. This mean depth multiplied by the width (wetted perimeter) gives the approximate cross-sectional area.
实地考察中，使用直尺或测杆每隔一定距离测量一次水深。平均水深 = 各测点水深之和 ÷ 测点数。平均水深乘以宽度（湿周）即得近似过水断面面积。
10. Mean Depth and Width of a River | 平均水深与河宽
These two simple but essential field measurements underpin many river calculations, such as cross-sectional area and discharge. Width is measured with a tape measure stretched across the channel perpendicular to the banks. Mean depth is computed from a series of vertical depth readings.
这两个简单却关键的实地测量指标是许多河流计算（如过水断面面积和流量）的基础。宽度用卷尺垂直河岸横跨河道测量。平均深度通过一系列垂直水深读数计算得出。
Formulas | 公式：
Mean Depth = Sum of all depth readings ÷ Number of readings
Width = Distance between left bank and right bank water’s edge (in metres)
When recording, always note the units and take multiple readings to improve reliability. An unusually deep pool or very shallow riffle should not be omitted – they are used in the average. For safe wading, never exceed the water depth that is safe for your height and always wear appropriate footwear.
记录时务必注明单位并进行多次测量以提高可靠性。不应忽略特别深的潭或特别浅的浅滩——它们都应计入平均值。涉水时注意安全，水深不得超过个人安全身高，并穿好合适的鞋子。
11. Rule of 70 (Population Doubling Time) | 70法则（人口倍增时间）
The Rule of 70 is a quick way to estimate how many years it will take for a population to double, given a constant annual percentage growth rate. It is widely used in demography and development geography, illustrating the powerful effect of compound growth.
70法则是一种在给定固定年百分比增长率的情况下，快速估算人口翻倍所需年数的方法。它广泛应用于人口学和发展地理学，体现了复利增长带来的巨大效应。
Formula | 公式：
Doubling Time (years) ≈ 70 ÷ Annual Growth Rate (%)
For example, a country with a 2% 5 years. Note that this rule assumes the growth rate remains constant, which is rarely the case over long periods, but it is a valuable estimation tool for exam questions.
例如，人口年增长率为 2% 的国家，大约在 70 ÷ 2 = 35 年后人口翻倍。注意，此法则假设增长率长期恒定，尽管现实中很少如此，但仍是一个极有价值的考试估测工具。
12. Crop Yield and Food Production Efficiency | 作物产量与粮食生产效率
In agricultural geography, crop yield is a measure of the amount of a crop that was harvested per unit of land area. It allows comparison of farming productivity between different regions or farming systems. Yield can be expressed in tonnes per hectare (t/ha) or kilograms per square metre.
在农业地理中，作物产量是衡量单位面积土地收获量的指标，可用来比较不同地区或农业系统的生产能力。产量可以用吨/公顷或公斤/平方米表示。
Formula | 公式：
Crop Yield = Total Harvested Quantity (kg or t) ÷ Cultivated Area (ha or m²)
For instance, if a farmer harvests 8,000 kg of rice from 2 hectares, the yield is 4,000 kg/ha. When studying food security, you may also need to calculate food production per capita: Total Food Production ÷ Total Population. This links back to population data and helps assess whether a country can feed its people.
例如，一个农民从 2 公顷地上收获 8,000 公斤水稻，产量为 4,000 公斤/公顷。在研究粮食安全时，还可能需计算人均粮食产量：总粮食产量 ÷ 总人口。这又回到了人口数据，有助于评估一个国家是否能养活其人民。
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