📚 CCEA Year 11 Geography: Formula and Theorem Quick Reference Handbook | CCEA 11年级地理：公式定理速查手册
This concise reference guide brings together the essential formulae, key metrics and conceptual models that every CCEA Year 11 Geography student is expected to master. From river discharge calculations to population change equations, these quantitative tools underpin fieldwork analysis, exam data-response questions and the deeper understanding of physical and human processes. Keep this handbook close when revising your river studies, population dynamics and map skills.
这份速查手册汇总了每一位 CCEA 11 年级地理学生必须掌握的关键公式、核心指标和概念模型。从河流流量计算到人口变化方程，这些定量工具是野外考察分析、考试数据分析题以及深入理解自然与人文过程的基石。复习河流地貌、人口动态和地图技能时，请随时查阅本手册。
1. River Discharge | 河流流量
River discharge is the volume of water passing a given point in the channel per unit of time. It is a fundamental measure of a river's energy and competence.
河流流量是指单位时间内通过河道某一断面的水体体积，是衡量河流能量和搬运能力的基本指标。
Q = A × v
where Q = discharge (cubic metres per second, m³/s), A = cross-sectional area of the channel (m²), and v = average flow velocity (m/s).
其中 Q = 流量（立方米/秒，m³/s），A = 河道横截面积（m²），v = 平均流速（m/s）。
To calculate A, multiply the average depth of the channel by its width. Velocity can be obtained with a flowmeter or by timing a float over a measured distance.
计算 A 时，用河道平均水深乘以河宽。流速可使用流速计测量，或通过浮标漂过已知距离的时间来计算。
- Ensure consistent units: width and depth in metres, velocity in metres per second.
- 单位务必统一：宽度和水深用米，流速用米/秒。
- Fieldwork tip: measure depth at several points across the channel, calculate the average depth, then multiply by width.
- 野外技巧：在河道多个位置测量水深，取平均水深再乘以河宽。
2. River Gradient | 河流比降
Gradient describes the steepness of a river channel along its course. It is closely linked to energy, erosion and deposition patterns.
比降描述河道沿程的陡峭程度，与河流能量、侵蚀和沉积模式密切相关。
Gradient = vertical fall ÷ horizontal distance
r a distance of 4,000 metres: Gradient = 20 m / 4,000 m = 0.005 or 5 metres per kilometre.
例如，一条溪流在4000米水平距离内下降了20米：比降 = 20 m / 4000 m = 0.005，或表示为5米/公里。
Gradient is usually expressed as a ratio, a fraction or in metres per kilometre. On Ordnance Survey maps it can be estimated from contour intervals.
比降通常用比率、分数或米/公里表示。在地形图上，可通过等高线间距估算比降。
- formula: Gradient = (contour difference in height) / (actual horizontal ground distance).
- 公式：比降 = （等高线高差）/（实际水平地面距离）。
- A steeper gradient means higher potential energy for erosion and transportation.
- 比降越陡，意味着更大的侵蚀和搬运势能。
3. Hydraulic Radius and Channel Efficiency | 水力半径与河道效率
The hydraulic radius helps assess how efficiently a river channel transmits water. It influences velocity, erosion and flood risk.
水力半径有助于评估河道输送水流的效率，同时影响流速、侵蚀和洪水风险。
R = A / P
where R = hydraulic radius (m), A = cross-sectional area (m²), and P = wetted perimeter (m), the length of the channel boundary in contact with water.
其中 R = 水力半径（m），A = 横截面积（m²），P = 湿周（m），即河道断面与水接触的边界长度。
A larger hydraulic radius indicates lower frictional resistance and more efficient flow, whereas a smaller R points to greater energy loss against the bed and banks.
水力半径越大，摩擦阻力越小，水流效率越高；R 较小时，意味着水流与河床及河岸的摩擦耗能更大。
| Channel shape | Typical A, P | Hydraulic radius | Efficiency |
|---|---|---|---|
| Broad, shallow | Large P relative to A | Small | Low |
| Semicircular, narrow | Small P relative to A | Large | High |
Translation: 宽浅河道 → 相对A较大的P → 水力半径小 → 效率低；半圆形窄深河道 → 相对A较小的P → 水力半径大 → 效率高
4. Population Density | 人口密度
Population density is a fundamental measure of how crowded a place is. It helps explain pressure on resources, services and infrastructure.
人口密度是衡量某地拥挤程度的基本指标，有助于解释资源、服务和基础设施所承受的压力。
Population density = total population ÷ land area (km² or mi²)
For instance, a region with 250,000 people living in 500 km² has a density of 500 people per km².
例如，一个拥有25万人口的地区，面积为500 km²，则人口密度为每平方公里500人。
Always be careful with units: total area must be in the same unit as the density expression. On CCEA exams, you may need to calculate density from data tables or maps.
务必注意单位：总面积单位须与密度表达式一致。在 CCEA 考试中，你可能需要根据表格或地图数据计算人口密度。
- High density often correlates with urbanisation and intensive land use.
- 高密度通常与城市化及集约用地相关联。
- Low density may indicate rural, mountainous or desert areas.
- 低密度可能指向农村、山区或沙漠地带。
5. Birth Rate and Death Rate | 出生率与死亡率
The crude birth rate (CBR) and crude death rate (CDR) are expressed per 1,000 people per year, allowing comparisons between regions of differing population size.
粗出生率 (CBR) 和粗死亡率 (CDR) 按每千人每年表示，可在不同人口规模的区域间进行比较。
CBR = (number of live births in a year ÷ total population) × 1,000
CDR = (number of deaths in a year ÷ total population) × 1,000
on records 42,000 live births in a year: CBR = (42,000 / 2,000,000) × 1000 = 21 per 1,000.
例如，一个人口为200万的国家一年内记录活产婴儿42,000名：CBR = (42,000 / 2,000,000) × 1000 = 21‰。
Birth and death rates are essential components of the demographic transition model and influence natural population change.
出生率和死亡率是人口转变模型的重要组成部分，影响人口自然变化。
- High birth rates are typical of Stage 2, while low birth and death rates mark Stage 4 of the demographic transition.
- 高出生率典型出现在人口转变第2阶段，而低出生率和低死亡率则是第4阶段的特征。
- In CCEA, you may be asked to interpret data or comment on trends shown by these rates.
- 在CCEA考试中，你或许需要对数据做出解读，或评论这些比率所呈现的趋势。
6. Natural Increase and Population Change | 自然增长率与人口变化
Natural increase is the difference between birth rate and death rate, typically expressed per 1,000 population or as a percentage.
自然增长率是出生率与死亡率的差值，通常以每千人或百分比表示。
Natural increase = CBR − CDR
If CBR = 28 per 1,000 and CDR = 8 per 1,000, then natural increase is 20 per 1,000, or 2% per annum.
若 CBR = 28‰，CDR = 8‰，则自然增长率为 20‰，即每年 2%。
Overall population change also accounts for net migration:
Population change = (births − deaths) + (immigrants − emigrants)
Therefore, a country can grow even with low natural increase if net migration is strongly positive.
因此，即便自然增长率较低，若净迁移率呈大幅正值，总人口仍会增长。
- Use this formula to forecast future population sizes.
- 运用该公式可预测未来人口规模。
- In CCEA case studies, remember to distinguish between natural and total population growth.
- 在 CCEA 案例分析中，注意区分自然增长与总人口增长。
7. Net Migration Rate | 净迁移率
The net migration rate expresses the balance of people moving into and out of a region per 1,000 population.
净迁移率表示每千人中迁入与迁出人口的净差额。
Net migration rate = ((immigrants − emigrants) ÷ total population) × 1,000
nts in a year, the net migration rate = ((12,000 − 8,000) / 500,000) × 1000 = 8 per 1,000.
若某城市人口50万，一年内有12,000人迁入、8,000人迁出，净迁移率 = ((12,000 − 8,000) / 500,000) × 1000 = 8‰。
Net migration can reflect push-pull factors such as employment, safety or environmental conditions. In CCEA, it is frequently examined alongside natural change to explain urban growth or rural decline.
净迁移能够反映推拉因素，如就业机会、安全状况或环境条件。在 CCEA 考试中，净迁移常与自然变化一起用来解释城市增长或乡村衰败。
8. Dependency Ratio | 抚养比
The dependency ratio indicates the economic pressure on the productive population by the young and elderly.
抚养比反映少儿和老年人口给劳动适龄人口带来的经济负担。
Dependency ratio = ((population aged 0–14 + population aged 65+) ÷ population aged 15–64) × 100
lderly dependants and 6.2 million people of working age: ((2.6 + 1.2) / 6.2) × 100 ≈ 61.3.
若某国少儿抚养人口260万，老年抚养人口120万，劳动年龄人口620万，则抚养比 = ((2.6 + 1.2) / 6.2) × 100 ≈ 61.3。
This means there are roughly 61 dependants for every 100 working-age people. High ratios strain health, education and pension systems.
这大致意味着每100名劳动年龄人口需抚养61名非劳动年龄人口。高抚养比会给医疗、教育及养老金体系带来压力。
- A youthful dependency ratio indicates many children, common in Stage 2 countries.
- 少儿抚养比高表示儿童数量多，常见于第2阶段国家。
- An ageing dependency ratio reflects an older population, typical of Stage 4/5.
- 老年抚养比高反映人口老龄化，常见于第4/5阶段。
9. Level of Urbanisation | 城市化水平
Urbanisation is the process by which an increasing percentage of a population lives in towns and cities. This simple formula allows comparison across countries and over time.
城市化是指人口中居住于城镇的百分比不断上升的过程。这一简单公式便于跨国别和跨时间比较。
Urban population % = (urban population ÷ total population) × 100
wellers out of 7 million total inhabitants, its urbanisation level is (4.1/7) × 100 ≈ 58.6%.
若某地区总人口700万，其中城市人口410万，则城市化水平 = (410万/700万) × 100 ≈ 58.6%。
CCEA often asks students to interpret urbanisation trends using graphs or to link the figure with models such as the Burgess or Hoyt urban land-use models.
CCEA 经常要求学生利用图表解读城市化趋势，或将该数据与 Burgess、Hoyt 等城市土地利用模型联系起来。
- Rapid urbanisation is characteristic of emerging economies.
- 快速城市化是新兴经济体的特征。
- Counter-urbanisation may see the percentage fall in some high-income countries.
- 逆城市化可能导致一些高收入国家的城市人口占比下降。
10. Map Scale and Real Distance | 地图比例尺与实际距离
All CCEA geography papers require competence with Ordnance Survey maps and scale calculations. Understanding scale is essential for measuring straight-line and winding distances.
所有 CCEA 地理试卷均考查对 Ordnance Survey 地图和比例尺计算的掌握。理解比例尺对测算直线距离和弯曲距离至关重要。
Real distance (km) = map distance (cm) × scale factor ÷ 100,000
For a map with scale 1:50,000, 1 cm on the map represents 50,000 cm on the ground, i.e. 0.5 km.
对于比例尺为1:50,000的地图，图上1厘米代表实地50,000厘米，即0.5公里。
p, actual straight-line distance = 7.4 × 0.5 = 3.7 km. For a 1:25,000 map, 1 cm = 0.25 km.
因此，若在1:50,000地图上两点相距7.4厘米，实际直线距离 = 7.4 × 0.5 = 3.7公里。若为1:25,000地图，1厘米 = 0.25公里。
For route distances along a road or river, use a piece of string or the edge of a paper to trace the curves, then measure against the scale bar.
若要测量沿道路或河流的曲线距离，可用细绳或纸条沿线描摹，再与直线比例尺比对测量。
- Always check whether the question asks for straight-line or actual travel distance.
- 务必要看清题目要求的是直线距离还是实际行进距离。
- Converting areas: area on ground = (map area) × (scale factor)² .
- 面积换算：实地面积 = （图上面积）×（比例尺分母）² 。
11. Hjulström Curve: Erosion, Transport and Deposition | Hjulström 曲线：侵蚀、搬运与沉积
The Hjulström curve is a graph that relates sediment particle size to the water velocity needed for erosion, transportation and deposition. It is a key physical geography concept on the CCEA specification.
Hjulström 曲线是一条关联沉积物粒径与侵蚀、搬运、沉积所需水流速度的图线，是 CCEA 课程规范中自然地理的核心概念。
The critical erosion velocity curve shows that fine sands and silts are eroded at lower velocities than clay because clay particles are cohesive. Larger particles like pebbles require higher velocities to be picked up, but once in motion, they may be transported at lower velocities than those needed for initiation of motion.
临界侵蚀速度曲线表明，细沙和粉砂被侵蚀所需流速低于黏土，因为黏土颗粒具有粘结性。像卵石那样的大颗粒需要更高流速才能起动，但一旦开始运动，它们可在比起动速度更低的流速下继续搬运。
Although not a formula, the Hjulström curve is a vital interpretive tool. Students may be asked to annotate or explain the relationship between velocity, grain size and process.
尽管不是公式，Hjulström 曲线仍是一种极其重要的解读工具。考生或许需要标注或阐释流速、粒径与地貌过程之间的关系。
- Deposition occurs when velocity drops below the settling velocity for a given particle size.
- 当流速降到给定粒径的沉降速度以下时，沉积发生。
- Fine clay may never settle out in turbulent flows; thus it is transported in suspension over long distances.
- 细颗粒的黏土在紊流中可能永远不会沉降，因此以悬移方式长距离搬运。
12. Bradshaw Model and Downstream Changes | Bradshaw 模型与下游变化
The Bradshaw model is a conceptual framework that describes how river characteristics change from source to mouth. While not a mathematical theorem, it helps formulate predictions and interpret fieldwork data.
Bradshaw 模型是描述河流从源头到河口特征如何变化的概念框架。虽然它并非数学定理，但有助于制定预测并解读野外数据。
The model predicts downstream increases in discharge, average velocity, hydraulic radius and channel depth, together with decreases in bedload size and channel gradient. Students often test these relationships during their CCEA fieldwork investigation.
该模型预测，向下游方向，流量、平均流速、水力半径和河道深度增加，而河床荷载粒径和河道比降减小。学生在 CCEA 野外调查中经常检验这些关系。
Understanding the Bradshaw model enables you to formulate hypotheses such as 'Velocity increases with distance downstream' and to present data in scatter graphs or cross-sections.
理解 Bradshaw 模型能让你提出类似“流速随下游距离增加而增大”的假设，并用散点图或断面图展示数据。
- Be critical: real-world variations may occur due to tributaries, geology or human intervention.
- 需持批判态度：由于支流、地质或人类干预，现实情况可能存在差异。
- Link observations to the model in CCEA evaluative questions.
- 在 CCEA 的评价类题目中，把观察结果与模型相联系。
Published by TutorHao | Geography Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图