📚 Year 7 CAIE Geography: Formulas & Theorems Quick Reference Handbook | 公式定理速查手册
This quick reference handbook brings together all the essential formulas and key geographical principles you need for Year 7 CAIE Geography. From map scales and gradient calculations to population indicators and important Earth processes, having these tools at your fingertips will help you solve quantitative questions and understand the 'why' behind landforms and human patterns. Use this guide to memorize the formulas, understand the supporting theorems, and apply them accurately in your assessments.
本速查手册汇集了七年级剑桥国际地理课程中所有重要的计算公式和关键地理原理。从地图比例尺、坡度计算到人口指标，再到重要的地球作用过程，掌握这些工具能帮助你解决定量问题，理解地貌与人类活动的成因。利用本手册熟记公式、领会相关定理，并在测试中准确应用。
1. Map Scale Formula | 地图比例尺公式
Scale is the relationship between a distance measured on a map and the same distance on the Earth's surface. Every map user must be able to convert map lengths into real-world distances. The formula is straightforward and underpins all map-reading tasks.
比例尺表示地图上的距离与地球表面对应距离之间的关系。每一位地图使用者都必须能够将图上的长度转换为实际距离。这个公式简单明了，是所有读图任务的基础。
Scale = Map Distance ÷ Actual Distance
比例尺 = 图上距离 ÷ 实际距离
1 cm represents 500 m). For example, if a map shows 1 cm to 1 km, the scale is 1:100,000. To find an actual distance, rearrange the formula: Actual Distance = Map Distance × Scale Denominator.
当两个距离使用相同单位时，结果可以写成数字比例尺（如 1:50,000）或文字比例尺（如 1 厘米代表 500 米）。例如，若地图上 1 厘米代表 1 千米，比例尺为 1:100,000。要求实际距离时，将公式变形：实际距离 = 图上距离 × 比例尺分母。
In Year 7, you will often work with linear scales (bar scales) too. Simply measure the length on the map and compare it against the bar to read off the real distance without calculation.
在七年级课程中，你也会经常使用线段比例尺。只需量出图上长度并与线段对比，便可直接读出实地距离，无需计算。
2. Gradient (Slope) Calculation | 坡度（梯度）计算
Gradient describes the steepness of a slope. It is commonly expressed as a ratio of vertical rise to horizontal distance. This theorem is used when analysing contour lines on topographic maps or studying river valleys.
梯度描述斜坡的陡峭程度，通常用垂直升高与水平距离的比值表示。当你分析地形图上的等高线或研究河谷时，需要用到这一定理。
Gradient = Vertical Rise (Height Difference) ÷ Horizontal Distance
梯度 = 垂直高差 ÷ 水平距离
as 1:20. Gradient can also be given as a percentage: multiply the fraction by 100 (here 5%). The smaller the second number in the ratio, the steeper the slope.
例如，若一座小山在 2000 米水平距离内上升了 100 米，则梯度为 100 ÷ 2000 = 1/20，写作 1:20。梯度也可用百分比表示：将分数乘以 100（此处为 5%）。比值中第二个数字越小，坡度越陡。
On a map, closely spaced contour lines indicate a steep gradient, while widely spaced contours show a gentle gradient. This visual rule helps you interpret landforms without performing calculations every time.
在地图上，等高线密集表示坡度陡峭，等高线稀疏则表示坡度平缓。这条视觉规律让你无需每次计算就能解读地形。
3. Population Density Formula | 人口密度公式
Population density measures how crowded a place is. It is one of the most fundamental formulas in human geography and appears regularly in data response questions.
人口密度衡量一个地方的拥挤程度，是人文地理中最基本的公式之一，经常出现在数据分析考题中。
Population Density = Total Population ÷ Total Land Area (km²)
人口密度 = 总人口 ÷ 总面积（平方千米）
The result is expressed as 'people per square kilometre' (people/km²). For instance, a country with 10 million people and an area of 500,000 km² has a population density of 20 people/km². Always check that the area unit is square kilometres; if it is given in m² or hectares, convert to km² first.
结果的单位是“人/平方千米”。例如，一个人口 1000 万、面积 50 万平方千米的国家，其人口密度为 20 人/平方千米。务必确认面积单位是平方千米；如果题目给出的是平方米或公顷，要先换算成平方千米。
Areas with high physical constraints, like deserts or mountain ranges, tend to have low population density, while fertile plains and coastal lowlands often show high density.
受沙漠、山脉等自然条件限制的地区人口密度往往较低，而肥沃平原和沿海低地的人口密度通常较高。
4. Birth Rate and Death Rate Formulas | 出生率与死亡率公式
Crude Birth Rate (CBR) and Crude Death Rate (CDR) are vital statistics that help geographers understand population change. Both are normally expressed per 1,000 people per year.
粗出生率与粗死亡率是帮助地理学家理解人口变化的重要统计指标，通常用每年每千人中的数量来表示。
Birth Rate = (Number of Live Births in a Year ÷ Total Population) × 1,000
出生率 =（一年内活产婴儿数 ÷ 总人口）× 1,000
Death Rate = (Number of Deaths in a Year ÷ Total Population) × 1,000
死亡率 =（一年内死亡人数 ÷ 总人口）× 1,000
1,000. By comparing CBR and CDR, you can quickly see whether a population is growing rapidly or slowly.
因此，一个总人口 50 万、一年内有 2000 名活产婴儿的国家，其出生率为 (2000 ÷ 500000) × 1000 = 4‰。通过对比出生率和死亡率，你能迅速判断人口是在快速增长还是缓慢增长。
5. Natural Increase Rate | 自然增长率
Natural increase (or decrease) is the difference between the number of live births and deaths. It excludes migration, showing the purely biological component of population change.
自然增长（或减少）是指活产婴儿数与死亡人数之间的差额。它不包含迁移因素，只反映人口变化的生物学部分。
Natural Increase Rate = Birth Rate – Death Rate
自然增长率 = 出生率 – 死亡率
The result is also expressed per 1,000 people. If the birth rate is 32 per 1,000 and the death rate is 8 per 1,000, the natural increase rate is 24 per 1,000 (or 2.4% when converted to a percentage). A negative value means the population is naturally declining.
结果同样以每千人表示。若出生率为 32‰，死亡率为 8‰，则自然增长率为 24‰（换算成百分比为 2.4%）。若结果为负数，则说明人口在自然减少。
When combined with net migration, natural increase helps calculate total population change, a key skill for population geography in Year 7.
将自然增长与净迁移结合起来，就能计算人口总量的变化，这是七年级人口地理中的一项关键技能。
6. Measuring Distances on Maps | 地图量距
mple tools. The underlying formula remains: Actual Distance = Map Distance × Scale Factor.
除了使用数字比例尺，你还可以借助简单工具测量直线距离和曲线距离。其根本公式仍是：实际距离 = 图上距离 × 比例尺倍数。
For straight lines, use a ruler to measure the length in centimetres, then multiply by the scale factor (e.g., for 1:25,000, multiply by 0.25 to get kilometres directly). For curved routes, lay a piece of string along the path, straighten it, and measure its length against the linear scale or ruler.
对于直线，用直尺量出厘米长度，再乘以比例尺倍数（例如 1:25,000 的地图，乘以 0.25 即可直接得到千米）。对于曲线，可沿路径放一根细绳，拉直后与线段比例尺或直尺对照得出长度。
A reliable theorem: if the map includes a grid and grid spacing is known (e.g., 1 km per square on a 1:50,000 map), you can estimate distances by counting grid squares diagonally or along a road.
一个可靠的方法：如果地图带有网格且已知网格间距（如 1:50,000 地图上每个方格代表 1 千米），你可以通过沿对角线或公路数格子的方式估算距离。
7. Contour Lines and Landform Interpretation | 等高线与地形判读
This section presents key theorems rather than a single formula. Contour lines join points of equal height. Their patterns reveal the shape and steepness of the land.
本节介绍的是核心定理而非单一公式。等高线连接高度相等的点，其形态能揭示地形的形状与坡度。
Theorem 1: Closely packed contour lines indicate a steep slope; widely spaced contours indicate a gentle slope.
定理一：等高线密集表明陡坡；等高线稀疏表明缓坡。
Theorem 2: When contour lines form V-shapes pointing uphill, they show a valley; V-shapes pointing downhill indicate a ridge or spur.
定理二：等高线呈 V 形且尖端指向高处时，表示山谷；V 形尖端指向低处时，表示山脊或山嘴。
Theorem 3: Concentric closed contours represent hills or mountains. If the inner contour is the highest, it is a hill; if there is a depression, hachure marks or the inner contour being lower will indicate it.
定理三：闭合的同心等高线代表丘陵或山峰。若内部等高线数值最高，则为山丘；若有洼地，则会有晕线或内部等高线数值较低来区分。
These theorems allow you to sketch profiles and predict the appearance of the landscape from a typical 1:25,000 or 1:50,000 Ordnance Survey-style map.
掌握这些定理后，你就能根据 1:25,000 或 1:50,000 地形测量式地图绘制剖面图，并预判地貌外观。
8. The Water Cycle Principles | 水循环原理
The water cycle is a closed system, constantly moving water between the atmosphere, lithosphere, hydrosphere and biosphere. Its processes act as a theorem of continuity.
水循环是一个闭合系统，水在大气圈、岩石圈、水圈和生物圈之间持续运动。其中的过程构成了一个连续性定理。
Key processes:
关键过程：
Evaporation: liquid water to water vapour; Transpiration: water vapour released by plants; Condensation: vapour to liquid droplets forming clouds; Precipitation: water falling as rain, snow, or hail; Surface runoff and infiltration: water moving across or into the ground.
蒸发：液态水转变为水蒸气；蒸腾：植物释放水蒸气；凝结：水蒸气变成液态水滴形成云；降水：以雨、雪或冰雹形式降落的水；地表径流与下渗：水在地表流动或渗入地下。
e in Storage. While complete calculations are for higher years, Year 7 students should understand the links.
通过各路径转移的水量可概括为水量平衡方程：降水 = 径流 + 蒸发 + 蓄水量变化。尽管完整计算是高年级的内容，七年级学生应理解其中的联系。
9. River Erosion, Transportation and Deposition | 河流侵蚀、搬运与沉积
Rivers shape landscapes through erosion, transport and deposition. These processes follow predictable patterns – essentially physical theorems – that determine valley shape and landforms.
河流通过侵蚀、搬运和沉积作用塑造地貌。这些作用遵循可预测的模式，本质上就是物理定理，决定了河谷形态和地貌特征。
Erosion techniques: Hydraulic action (force of water), abrasion (rocks rubbing against river bed and banks), attrition (rocks hitting each other and breaking), and solution (dissolving of soluble minerals).
侵蚀方式：水力作用（水流的力量）、磨蚀（石块磨蚀河床和河岸）、磨耗（石块相互碰撞破碎）和溶蚀（可溶性矿物的溶解）。
Transportation: Traction (large boulders rolled), saltation (smaller stones bounced), suspension (fine particles carried within the water), and solution (dissolved materials).
搬运方式：推移（大石块滚动）、跃移（较小石块弹跳）、悬移（细颗粒在水中悬浮）和溶解搬运（溶解物质随水运动）。
Deposition theorem: Sediment is deposited when a river loses energy. This happens when the slope decreases, the channel widens, or the river enters a lake or sea. Heavier materials are dropped first, creating graded bedding.
沉积定理：当河流失去能量时，泥沙便发生沉积。河道坡度减小、河道变宽或河流汇入湖泊、海洋时都会发生沉积。较重的物质先沉降，形成粒序层理。
10. Plate Tectonics Theory | 板块构造理论
The Earth's lithosphere is broken into tectonic plates that float on the semi-molten asthenosphere. This theory is a core theorem explaining the global distribution of volcanoes, earthquakes, and mountain belts.
地球的岩石圈分裂成多个构造板块，漂浮在半熔融的软流圈之上。这一理论是解释全球火山、地震和山脉带分布的核心定理。
Plate boundary types:
板块边界类型：
Divergent (constructive): plates move apart, magma rises, forming new crust (e.g., Mid-Atlantic Ridge).
离散型（建设性）：板块相互远离，岩浆上升形成新地壳（如大西洋中脊）。
Convergent (destructive): plates move towards each other; oceanic plate subducts, causing volcanoes and deep ocean trenches.
汇聚型（破坏性）：板块相向移动；海洋板块俯冲，引起火山活动和深海沟。
Conservative: plates slide past each other, generating earthquakes along faults (e.g., San Andreas Fault).
守恒型：板块相互滑动，沿断层产生地震（如圣安德烈亚斯断层）。
The theory explains why the Pacific 'Ring of Fire' is a zone of intense tectonic activity and helps predict where hazards may occur.
该理论解释了太平洋“火环”为何是地壳活动剧烈的地带，也有助于预测灾害可能发生的区域。
11. Climate Graphs Interpretation (Theorems) | 气候图表解读（定理）
A climate graph displays average monthly temperature and precipitation. While no formula is used to create it, reading and interpreting the graph involves several practical theorems.
气候图表展示月平均气温和降水量。虽然制作时不涉及公式，但阅读和解读图表涉及几条实用定理。
Temperature curve theorem: In the Northern Hemisphere, the temperature curve peaks in July/August and dips in January/February; the Southern Hemisphere is opposite. The annual temperature range (highest minus lowest) indicates continentality – larger ranges suggest inland locations.
气温曲线定理：北半球气温曲线在 7/8 月达到高峰，1/2 月降至低谷；南半球相反。气温年较差（最高减最低）反映大陆度——年较差越大，位置越靠近内陆。
Precipitation bar theorem: High bars in the summer months with a winter dip suggest a temperate or monsoon climate. Even precipitation throughout the year indicates an equatorial or maritime climate. Rain shadow areas show low totals on the leeward side of mountains.
降水柱状定理：夏季月份柱状高而冬季低，表明温带或季风气候；全年降水均匀则表明赤道或海洋性气候。山地背风坡的雨影区降水总量较少。
Use these theorems to match a climate graph to its location and to describe the climate without needing extra data.
运用这些定理，你便可将气候图表与其地理位置匹配，并在无需额外数据的情况下描述气候特征。
12. Settlement Site Factors (Location Theorems) | 聚落选址因素（区位定理）
Settlement site selection follows a set of geographical theorems. Although not expressed as an equation, the factors can be evaluated using simple site analysis.
聚落的选址遵循一系列地理定理。尽管不以方程式表达，但可利用简单的区位分析来评估这些因素。
Key site factors theorem:
关键区位因素定理：
Water supply: early settlements grew near rivers, springs, or wells. Flat land: gently sloping ground was easier to build on and farm. Defence: hilltops or meander bends offered natural protection. Building materials: nearby forests and stone quarries provided wood and stone. Fuel supply: wood for heating and cooking was crucial. Bridging points and route centres: river crossings and road junctions encouraged trade and growth.
水源：早期聚落多靠近河流、泉水或水井。平坦土地：缓坡地面易于建造房屋和耕作。防御：山顶或曲流弯道提供天然保护。建筑材料：附近的森林和采石场提供木材和石材。燃料供给：取暖和烹饪用的木材至关重要。架桥点和交通中心：河流渡口和道路交汇处促进贸易与聚落成长。
Applying these theorems, you can explain why
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