📚 KS3 WJEC Geography: Formula & Theorem Quick Reference Handbook | KS3 WJEC 地理：公式定理速查手册
This quick reference handbook gathers all the essential formulas, mathematical relationships and key geographical rules you need for KS3 WJEC Geography. From map skills and gradient calculations to population statistics, river discharge and climate data, these revision cards will sharpen your fieldwork and data interpretation skills. Each formula is paired with practical examples, clear definitions and tips for avoiding common mistakes.
本速查手册汇集了 KS3 WJEC 地理课程中所有关键的公式、数学关系和重要地理规律。从地图技能与坡度计算，到人口统计、河流流量与气候数据，这些复习卡片将助你提高野外考察与数据解读能力。每条公式均配有实用示例、清晰定义和避免常见错误的提示。
1. Map Scale and Distance Conversion | 比例尺与距离转换
Map scale describes the ratio between a distance measured on a map and the same distance on the ground. It is usually given as a representative fraction, for example 1:25 000, which means 1 unit on the map equals 25 000 of the same units in reality. A linear scale bar is also common. To convert a map distance to the actual ground distance, multiply the measured length by the scale denominator.
比例尺描述了图上距离与对应实地距离的比值。它通常以分数形式给出，如 1:25 000，表示图上 1 个单位长度相当于实地 25 000 个相同单位的长度。直线比例尺也十分常见。要将图上距离转换为实地距离，只需将测量长度乘以比例尺分母即可。
Ground Distance = Map Distance × Scale Denominator
实地距离 = 图上距离 × 比例尺分母
Always ensure you convert units correctly. For instance, on a 1:50 000 Ordnance Survey map, 1 cm represents 50 000 cm on the ground. Divide by 100 to get metres (500 m) and by 100 000 to get kilometres (0.5 km). So 4 cm on the map equals 4 × 50 000 = 200 000 cm, or 2 km on the ground.
务必注意单位换算的正确性。例如，在一幅 1:50 000 的 Ordnance Survey 地形图上，1 厘米代表实地 50 000 厘米。除以 100 得到 500 米，再除以 1000 即 0.5 千米。因此图上 4 厘米相当于 4 × 50 000 = 200 000 厘米，即 2 千米。
2. Measuring Curved Distance on a Map | 地图曲线距离的测量
Not all distances on a map are straight lines – rivers, roads and footpaths often curve. To measure a winding route, place a piece of string along the curve, mark the end point, then straighten the string and measure its length against the scale bar or a ruler. The actual ground distance can be found by multiplying the straightened string length by the scale denominator, just as for a straight line.
地图上的距离并非全是直线——河道、道路和步道常常蜿蜒曲折。要测量一段弯曲线路，可将一根细绳沿曲线放置，在终点做标记，然后将绳子拉直，用比例尺或直尺测量其长度。实地距离的计算方法与直线相同，只需将拉直后的绳长乘以比例尺分母即可。
Real Distance = Straightened String Length × Scale Denominator
实地距离 = 拉直绳长 × 比例尺分母
A common alternative is using the straight edge of a piece of paper: pivot the paper along each straight section of the curve, mark and measure the total length. Remember to record all measurements in the same unit before converting.
一种常用的替代方法是用纸片的直边：沿曲线的每一段转折旋转纸片，标起点与终点，测出总长度，再进行换算。换算前务必将所有测量值统一为相同单位。
3. Gradient of a Slope | 坡度梯度
Gradient describes the steepness of a slope. It is the ratio of vertical rise (change in height) to horizontal run (horizontal distance), with both quantities measured in the same unit. Gradient is often simplified to a ratio in the form 1:n, which tells you how many units you must travel horizontally to rise by 1 unit.
坡度描述斜坡的陡峭程度。它是垂直高差（高度变化）与水平距离的比值，两者必须以相同单位计量。坡度常简化为 1:n 的比值形式，表示每上升 1 个单位，需要水平移动 n 个单位。
Gradient = Rise ÷ Run
坡度 = 垂直高差 ÷ 水平距离
Ratio form: 1 : (Run ÷ Rise)
比值形式：1 : (水平距离 ÷ 垂直高差)
Example: if a hill rises 30 m over a horizontal distance of 400 m, the gradient is 30 ÷ 400 = 0.075. The simplified ratio is 1 : (400 ÷ 30) = 1:13.3. This means for every 13.3 m travelled horizontally, you climb 1 m. The smaller the second number, the steeper the slope.
示例：一座山丘在 400 米的水平距离内上升了 30 米，则坡度为 30 ÷ 400 = 0.075。简化比值为 1 : (400 ÷ 30) ≈ 1:13.3。这意味着每水平移动约 13.3 米，高度上升 1 米。比值中第二个数字越小，表示坡度越陡。
4. Vertical Exaggeration on Cross Sections | 地形剖面图的垂直放大
When drawing a cross-section of the land, the vertical scale is often made larger than the horizontal scale to make relief features stand out. Vertical exaggeration (VE) tells you how many times the vertical scale has been stretched compared with the horizontal scale. To calculate VE, you must first express both scales as ratios (or as scale denominators) and then divide the horizontal scale denominator by the vertical scale denominator.
绘制地形剖面图时，垂直比例尺通常比水平比例尺大，以便突显地势起伏。垂直放大率 (VE) 表示垂直比例尺相较于水平比例尺被放大了多少倍。计算 VE 时，需先将两者表示为比值（或比例尺分母），再用水平比例尺分母除以垂直比例尺分母。
Vertical Exaggeration = Horizontal Scale Denominator ÷ Vertical Scale Denominator
垂直放大率 = 水平比例尺分母 ÷ 垂直比例尺分母
For a map with a horizontal scale of 1:50 000 and a cross-section drawn with a vertical scale of 1:10 000, the VE would be 50 000 ÷ 10 000 = 5. This means hills and valleys appear 5 times steeper than they really are. VE helps visualize gentle slopes but can exaggerate the landform; always state the VE alongside your cross-section.
若水平比例尺为 1:50 000，剖面图垂直比例尺为 1:10 000，则 VE 为 50 000 ÷ 10 000 = 5。这意味着山丘与谷地在图上看起来比实际陡峭 5 倍。VE 有助于观察缓坡，但也会夸大真实地貌；务必在剖面图旁注明 VE 值。
5. Population Density | 人口密度
Population density measures how crowded an area is. It is the average number of people living per square kilometre (km²). The formula requires the total population of the area and its total land area in square kilometres. This statistic allows meaningful comparison between regions of different sizes and helps explain human pressure on resources.
人口密度衡量一个地区的人口稠密程度。它是每平方千米 (km²) 平均居住的人口数量。计算时需要该地区的总人口与总面积（单位：平方千米）。该统计指标可用于比较不同大小区域之间的人口集中程度，并帮助分析人类对资源的压力。
Population Density = Total Population ÷ Total Area (km²)
人口密度 = 总人口 ÷ 总面积（km²）
If a city has 1.2 million people and covers 400 km², its population density is 1 200 000 ÷ 400 = 3 000 people per km². Rural areas often have densities below 50 people per km², while megacities can exceed 10 000 per km². Remember that density is an average; it does not show variations within the area.
若某城市拥有 120 万人，占地 400 km²，则人口密度为 1 200 000 ÷ 400 = 3 000 人/km²。农村地区的人口密度常低于 50 人/km²，而超大城市可能超过 10 000 人/km²。需注意密度仅为平均值，无法显示区域内部的分布差异。
6. Birth Rate and Death Rate | 出生率与死亡率
Birth rate, also called crude birth rate (CBR), is the number of live births in a year for every 1 000 people in the population. Death rate, or crude death rate (CDR), is the annual number of deaths per 1 000 people. These rates ignore the age structure and are expressed per 1 000 to make comparisons between countries of different sizes fair.
出生率又称粗出生率 (CBR)，是指一年内每千人口中的活产婴儿数。死亡率或称粗死亡率 (CDR)，指每年每千人口中的死亡人数。这两个比率均排除年龄结构的影响，并以千分比表示，以便在不同人口规模的国家之间进行公平比较。
CBR = (Number of live births ÷ Total population) × 1 000
粗出生率 = (活产数 ÷ 总人口) × 1 000
CDR = (Number of deaths ÷ Total population) × 1 000
粗死亡率 = (死亡人数 ÷ 总人口) × 1 000
For example, a country of 5 million people records 125 000 births in a year. The CBR is (125 000 ÷ 5 000 000) × 1 000 = 25 per 1 000. A high birth rate is typical in Stage 2 of the Demographic Transition Model, while a low death rate often reflects better healthcare. Always use the mid-year population for accuracy in official statistics.
例如，一个 500 万人口的国家全年记录活产数 125 000，则 CBR 为 (125 000 ÷ 5 000 000) × 1 000 = 25‰。高出生率常见于人口转变模式的第二阶段，而低死亡率则反映医疗水平的提高。在官方统计中，为保证精确性，通常使用年中人口。
7. Natural Increase and Rate of Natural Increase | 自然增长与自然增长率
Natural increase is the difference between the number of live births and the number of deaths in a population over a given period, usually one year. It does not include migration. The rate of natural increase (RNI) can be expressed as a crude value per 1 000 people or as a percentage. Subtracting the death rate from the birth rate gives the rate per 1 000 directly.
自然增长指在一定时期（通常为一年）内，人口的出生数与死亡数之差，不包含迁移因素。自然增长率 (RNI) 可以千分比或百分比表示。将出生率减去死亡率，直接得到千分比形式的自然增长率。
Natural Increase (people) = Births − Deaths
自然增长人数 = 出生人数 − 死亡人数
RNI (per 1 000) = CBR − CDR
自然增长率（千分比） = 粗出生率 − 粗死亡率
RNI (%) = (CBR − CDR) ÷ 10
自然增长率（百分比） = (粗出生率 − 粗死亡率) ÷ 10
If the CBR is 20 per 1 000 and the CDR is 6 per 1 000, the RNI is 14 per 1 000, which equals 1.4% per year. A negative RNI occurs when deaths exceed births, leading to a natural decrease – this is seen in some countries in Stage 5 of the Demographic Transition Model.
若 CBR 为 20‰，CDR 为 6‰，则 RNI 为 14‰，相当于每年 1.4%。当死亡人数超过出生人数时，RNI 为负值，出现人口自然减少，这在人口转变模式第五阶段的部分国家中有所体现。
8. River Discharge | 河流流量
River discharge (Q) is the volume of water flowing through a river channel at a given point per second. It is measured in cubic metres per second (m³/s or cumecs). To calculate discharge, you need the cross-sectional area of the channel (A, in m²) and the average flow velocity (V, in m/s). Understanding discharge helps predict flood risk and river capacity.
河流流量 (Q) 指每秒流过河道某一断面的水体体积，单位为立方米每秒 (m³/s 或 cumecs)。计算流量需要两个数据：河道横截面积（A，单位为 m²）和平均流速（V，单位为 m/s）。掌握流量有助于预测洪水风险和河流输水能力。
Q = A × V
流量 = 横截面积 × 平均流速
To find A, measure the width and several depths across the channel, then calculate the wetted area. Velocity can be measured with a flowmeter or by timing a floating object over a set distance (and multiplying by a correction factor, typically 0.85 for surface float). Although KS3 fieldwork may simplify the method, the relationship Q = A × V remains fundamental.
计算 A 需测量河面宽度和多个点的水深，然后算出过水面积。流速可用流速仪或通过浮标漂移计时法（通常乘以 0.85 的校正系数）测定。虽然 KS3 阶段的野外考察可能简化方法，但 Q = A × V 的核心关系始终不变。
9. Mean Temperature and Temperature Range | 平均气温与气温较差
Climate graphs summarise temperature and precipitation data for a location. Two simple statistics help describe the temperature regime: the mean annual temperature and the annual temperature range. The mean annual temperature is the average of all twelve monthly mean temperatures. The annual range is the difference between the hottest and coldest month averages.
气候图汇总了一个地点的气温与降水量数据。有两个简单的统计量可描述气温状况：年平均气温与年气温较差。年平均气温是 12 个月平均气温的总和除以 12。年较差则为最热月与最冷月平均气温之差。
Mean Annual Temperature = Sum of 12 monthly mean temperatures ÷ 12
年平均气温 = 12 个月平均气温之和 ÷ 12
Annual Temperature Range = Hottest month mean − Coldest month mean
年气温较差 = 最热月平均气温 − 最冷月平均
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