📚 Year 9 WJEC Geography: Formula & Theorem Quick Reference Handbook | 九年级 WJEC 地理：公式定理速查手册
Welcome to the Year 9 WJEC Geography Formula & Theorem Quick Reference Handbook. This guide brings together all the essential mathematical relationships, population equations, map work calculations, and physical geography formulas you will encounter across the WJEC Geography specification. From measuring gradients on an Ordnance Survey map to calculating the natural increase of a population, each entry is explained simply and paired with a worked example to help you build confidence for classwork, homework, and assessments. Keep this handbook handy as a one‑stop revision tool for your geographical skills and knowledge.
欢迎使用九年级 WJEC 地理公式定理速查手册。本指南汇集了 WJEC 地理教学大纲中你将遇到的所有基本数学关系、人口方程、地图作业计算和自然地理公式。从测量地形图上坡度到计算人口自然增长，每一条都用简单的语言加以解释，并配以实际例子，帮助你建立对课堂作业、家庭作业和评估的信心。请将本手册作为地理技能和知识的一站式复习工具常备手边。
1. Map Scale Conversions | 地图比例尺换算
The relationship between a distance on a map and the corresponding distance on the ground is given by the map scale. To convert a measured map distance to real‑world distance, multiply the map length by the scale factor. For a scale expressed as 1:50 000, this means 1 cm on the map represents 50 000 cm (or 0.5 km) on the ground. The formula is: real distance = map distance × scale denominator. Always remember to convert units appropriately after multiplication to express the answer in kilometres or metres.
地图上的距离与地面实际距离之间的关系由地图比例尺给出。要将测量出的地图距离转换为实际距离，用地图长度乘以比例尺分母即可。若比例尺表示为 1:50 000，意味着地图上的 1 厘米代表实地 50 000 厘米（即 0.5 公里）。公式为：实地距离 = 地图距离 × 比例尺分母。务必在乘法运算后适当换算单位，将答案表示为千米或米。
For example, if a road measures 8.4 cm on a 1:25 000 map, the real distance is 8.4 × 25 000 = 210 000 cm = 2.1 km. When working from a linear scale bar, simply read off the ground distance directly without calculation. Converting between scales is also useful: a larger scale (e.g. 1:10 000) shows more detail but covers a smaller area than a smaller scale (e.g. 1:250 000).
例如，一条道路在 1:25 000 地图上长 8.4 厘米，实际距离为 8.4 × 25 000 = 210 000 厘米 = 2.1 公里。若使用直线比例尺，直接读出地面距离即可，无需计算。比例尺之间也有换算：大比例尺（如 1:10 000）展示的细节更多，但覆盖面积比小比例尺（如 1:250 000）更小。
2. Measuring Straight‑Line Distance | 测量直线距离
On an Ordnance Survey (OS) map, straight‑line distances are measured between two points using the edge of a piece of paper or a ruler. Once the map distance in centimetres is obtained, apply the scale formula: real distance (km) = (map distance in cm × scale denominator) ÷ 100 000. This formula immediately gives the result in kilometres because there are 100 000 cm in a kilometre. For curved routes, a piece of string or the edge of the paper can be rotated along the path; then straighten the string and measure against the linear scale.
在地形测量图上，用纸边或直尺测量两点之间的直线距离。得到以厘米为单位的地图距离后，应用比例尺公式：实地距离（千米） = (地图距离厘米 × 比例尺分母) ÷ 100 000。因一公里含有 100 000 厘米，该式直接给出公里结果。对于弯曲路线，可用细绳或纸边沿路径转动，然后将细绳拉直，对照直线比例尺测量。
For example, two villages are 12.5 cm apart on a 1:50 000 OS map. The real straight‑line distance is (12.5 × 50 000) ÷ 100 000 = 6.25 km. Always show your working in examinations and use the correct units. Remember that straight‑line distance is less than the actual distance you would travel along winding roads or footpaths.
例如，两个村落在 1:50 000 地形图上相距 12.5 厘米。直线实地距离为 (12.5 × 50 000) ÷ 100 000 = 6.25 公里。考试中务必展示计算步骤并使用正确单位。请记住，直线距离小于沿弯曲道路或小径的实际旅行距离。
3. Calculating Area on a Map | 地图上面积的计算
When calculating the area of a feature shown on a map, first measure the dimensions in centimetres, convert to real‑world distances using the scale, and then compute the area. For a rectangular field represented by length L cm and width W cm on a 1:50 000 map, the real area in square kilometres is found by real L (km) = L cm × 0.5, real W (km) = W cm × 0.5, and area = real L × real W. If the shape is irregular, you can overlay a grid of squares and count the number of full and partial squares within the boundary.
计算地图上所示特征的面积时，首先以厘米为单位测量其尺寸，利用比例尺转换为实地距离，然后再计算面积。对于 1:50 000 地图上长 L 厘米、宽 W 厘米的长方形地块，实地面积（平方公里）求法为：实地长（公里）= L 厘米 × 0.5，实地宽（公里）= W 厘米 × 0.5，面积 = 实地长 × 实地宽。若形状不规则，可以叠加方格网，统计边界内的完整方格和部分方格数量。
The basic area formula is area = length × width. For a lake measuring 3.2 cm by 2.1 cm on a 1:25 000 map, the real dimensions are 3.2 × 0.25 = 0.8 km and 2.1 × 0.25 = 0.525 km, so the surface area is 0.8 × 0.525 = 0.42 km². You can also express area in hectares (1 ha = 0.01 km², or 100 m × 100 m).
基本面积公式为：面积 = 长度 × 宽度。对于 1:25 000 地图上一个长 3.2 厘米、宽 2.1 厘米的湖泊，实地尺寸为 3.2 × 0.25 = 0.8 公里，2.1 × 0.25 = 0.525 公里，因此水面面积为 0.8 × 0.525 = 0.42 平方公里。也可以用公顷表示面积（1 公顷 = 0.01 平方公里，或 100 米 × 100 米）。
4. Gradient and Slope Calculations | 坡度和坡度角计算
Gradient expresses the steepness of a slope as a ratio or percentage. On a map, gradient is determined by the vertical rise (change in height) divided by the horizontal run (distance). The formula is gradient = vertical interval (VI) / horizontal equivalent (HE). Both VI and HE must be in the same units. To express as a ratio (e.g. 1:10), write gradient = 1 : (HE / VI). The slope angle in degrees can be found using trigonometry: tan θ = VI / HE, so θ = tan⁻¹(VI/HE). In Year 9, you will most often be asked to state the gradient as a simplified ratio.
坡度以比值或百分比表示斜坡的陡峭程度。在地图上，坡度由垂直升高（高度变化）除以水平距离确定。公式为：坡度 = 垂直间距 (VI) / 水平距离 (HE)。VI 和 HE 必须采用相同单位。若以比值表示（如 1:10），则写为 坡度 = 1 : (HE / VI)。可用三角学求出以度数为单位的坡度角：tan θ = VI / HE，因此 θ = tan⁻¹(VI/HE)。九年级阶段最常见的要求是以简化比值的形式表达坡度。
For a cliff where the land rises from 20 m to 140 m over a horizontal distance of 600 m, VI = 140 – 20 = 120 m. HE = 600 m. Gradient = 120/600 = 1/5, so the gradient is 1:5. This means for every 5 metres you walk horizontally, you go up 1 metre. A steep gradient (e.g. 1:3) suggests a much more rapid gain in height than a gentle gradient (e.g. 1:20).
某悬崖从 20 米升高至 140 米，水平距离为 600 米，则 VI = 140 – 20 = 120 米，HE = 600 米。坡度 = 120/600 = 1/5，因此坡度为 1:5。这意味着每水平移动 5 米，高度上升 1 米。陡峭的坡度（如 1:3）表示高度增加远快于平缓的坡度（如 1:20）。
5. Measuring Bearing and Direction | 测量方位角和方向
On a map, a bearing is a compass direction measured in degrees clockwise from north (0°). To find the bearing from point A to point B, place the protractor with its centre over A, align 0° with grid north, and read the angle where the line AB intersects the protractor. The formula is simply bearing = clockwise angle from north. Bearings are always given as three figures: 045°, 120°, 270°. Cardinal and intercardinal directions (N, NE, E, SE, S, SW, W, NW) can be described qualitatively, but you must be able to measure precise bearings for map skills questions.
地图上的方位角是从指北方向（0°）开始按顺时针方向以度数衡量的罗盘方向。求从 A 点到 B 点的方位角时，将量角器中心对准 A 点，0° 线对准网格北方，读取直线 AB 与量角器相交的角度。公式就是 方位角 = 从北方起算的顺时针角度。方位角始终用三位数字给出：045°、120°、270°。基本方位和间方位（北、东北、东、东南、南、西南、西、西北）可以进行定性描述，但在读图技能题中必须能测量精确的方位角。
For example, the bearing from a church to a windmill might be 115°. This means from the church you would face approximately ESE (east‑south‑east) to look directly at the windmill. When describing direction in a written answer, link it to a compass point: ‘the river flows in a south‑westerly direction’ or ‘the prevailing wind blows from the north‑west’.
例如，从教堂到风车的方位角可能是 115°。这意味着你站教堂处面朝大约东东南方向时正好看向风车。在书面回答中描述方向时，应将其与罗盘点联系起来：“河流向西南方向流动”或“盛行风从西北方向吹来”。
6. Six‑Figure Grid References | 六位数字网格坐标
A six‑figure grid reference pinpoints a specific location within a 100 m × 100 m grid square on an OS map. The rule is ‘go along the corridor and then go up the stairs’: first record the eastings (horizontal) then the northings (vertical). For a four‑figure reference such as 17 34, the easting is 17 and the northing is 34. To make it six‑figure, divide the grid square into tenths along both axes. For a point estimated to be four‑tenths across and six‑tenths up, the reference becomes 174 346. The formula is: six‑figure ref = easting(1/10) northing(1/10). No mathematical calculation beyond estimation and counting is required, but you must be confident placing the extra digits.
六位数字网格坐标可将位置精确定位到地形测量图上 100 米 × 100 米的网格方格内。规则是“先走走廊再上楼梯”：首先记录东向值（水平方向），再记录北向值（垂直方向）。对于四位数字坐标如 17 34，东向值为 17，北向值为 34。要将其变为六位数字坐标，沿两轴将网格方格分别十等分。若某点估计位于水平方向十分之四、垂直方向十分之六的位置，则坐标变为 174 346。公式为：六位坐标 = 东向值(十分位) 北向值(十分位)。除估算和计数外，不需任何数学计算，但你务必有信心添加额外数字。
Practice finding a church at 215 487: the eastings are 21 and 5/10, northings are 48 and 7/10. This houses the feature in the lower‑right portion of grid square 21 48. Grid references always have even digits for eastings and northings in full coordinates; a full reference like 215487 has no spaces when written, but we often see 215 487 for clarity.
练习寻找位于 215 487 的教堂：东向值为 21 又十分之五，北向值为 48 又十分之七。这使该地物落在网格方格 21 48 的右下区域。完整的网格坐标中，东向和北向的数字位数总是相同；写全坐标时如 215487 不留空格，但为清晰起见，我们常写作 215 487。
7. Population Density | 人口密度
Population density describes how crowded a place is. It is calculated by dividing the total population of an area by the total land area. The formula is population density = total population ÷ land area (km²). The unit is people per km². This measure allows for comparisons between countries, regions, and cities of different sizes. A high population density indicates many people living in each square kilometre, while a low density suggests more space per person. Knowing the density helps geographers understand pressure on resources, housing demand, and service provision.
人口密度用来描述一个地方的拥挤程度。它的计算方法是用该地区的总人口除以总土地面积。公式为：人口密度 = 总人口 ÷ 土地面积（平方公里）。单位为人/平方公里。这一度量方式可以对面积不同的国家、地区和城市进行比较。高人口密度意味着每平方公里居住大量人口，而低密度则表示人均拥有更多空间。了解密度有助于地理学家理解资源压力、住房需求以及服务供应情况。
For instance, if a region has a population of 2.4 million and covers an area of 8 000 km², its population density is 2 400 000 ÷ 8 000 = 300 people per km². The United Kingdom has around 280 people per km², whereas Canada has about 4 people per km². Remember that population density is an average; within any area there are places of much higher and much lower density, such as city centres versus rural farmland.
例如，某地区人口为 240 万，面积为 8 000 平方公里，则其人口密度为 2 400 000 ÷ 8 000 = 300 人/平方公里。英国人口密度约为 280 人/平方公里，而加拿大仅为约 4 人/平方公里。请记住人口密度是一个平均值；在任何区域内，都会存在密度极高和极低的地方，比如市中心与农村农田之间的对比。
8. Birth Rate, Death Rate, and Natural Increase | 出生率、死亡率和自然增长
Demographic change is measured using birth rates and death rates. The crude birth rate is the number of live births per 1 000 people in a population per year. The crude death rate is the number of deaths per 1 000 people per year. The formula for birth rate is: (number of births ÷ total population) × 1 000. For death rate: (number of deaths ÷ total population) × 1 000. Natural increase (or natural change) is the difference between the birth rate and the death rate: natural increase = birth rate – death rate. If the result is positive, the population grows naturally; if negative, it is in natural decrease. These rates are expressed per 1 000, not as percentages.
人口变化通过出生率和死亡率来衡量。粗出生率是指某人口中每年每 1 000 人中的活产婴儿数。粗死亡率是每年每 1 000 人中的死亡人数。出生率的公式为：(出生人数 ÷ 总人口) × 1 000。死亡率的公式为：(死亡人数 ÷ 总人口) × 1 000。自然增长（或自然变化）是出生率与死亡率之间的差值：自然增长 = 出生率 – 死亡率。若结果为正，人口自然增长；若为负，则处于自然减少状态。这些比率用千分比表示，而非百分比。
For example, a country with 500 000 people records 12 500 births and 7 500 deaths in a year. Birth rate = (12 500 ÷ 500 000) × 1 000 = 25 per 1 000. Death rate = (7 500 ÷ 500 000) × 1 000 = 15 per 1 000. Natural increase = 25 – 15 = 10 per 1 000, meaning the population grows by 10 people for every 1 000 people already there. Natural increase does not include migration; it shows only the balance of births and deaths.
例如，一个人口为 50 万的国家一年内记录出生 12 500 人，死亡 7 500 人。出生率 = (12 500 ÷ 500 000) × 1 000 = 25 每千人。死亡率 = (7 500 ÷ 500 000) × 1 000 = 15 每千人。自然增长 = 25 – 15 = 10 每千人，这意味着每 1 000 人中增加 10 人。自然增长不包括迁移因素；它仅体现出生与死亡的平衡情况。
9. Population Growth Rate Including Migration | 含迁移的人口增长率
The total population growth rate combines natural change and net migration. Net migration is the number of immigrants (people moving in) minus the number of emigrants (people moving out). The annual population growth rate (as a percentage) can be approximated by the formula: growth rate (%) = ((births – deaths + net migration) ÷ total population) × 100. This measures the overall change of population size over one year. Some specifications require you to express the result per 1 000, in which case multiply by 1 000 instead of 100.
总人口增长率综合了自然变化和净迁移。净迁移为迁入人口（移入者）减去迁出人口（移出者）。年度人口增长率（以百分比表示）可由公式近似计算：增长率 (%) = ((出生人数 – 死亡人数 + 净迁移) ÷ 总人口) × 100。这衡量一年内人口总量的整体变化。有些教学大纲要求用千分比表示结果，此时乘以 1 000 而非 100。
Imagine a town of 80 000 people has 1 200 births, 800 deaths, 400 immigrants and 200 emigrants in a year. Net migration = 400 – 200 = 200. Total change = 1 200 – 800 + 200 = 600. Growth rate = (600 ÷ 80 000) × 100 = 0.75% per year. When the growth rate is very low or negative (e.g. in some rural areas of the UK), the population may be declining, which brings challenges such as school closures and an ageing society.
想象一个 8 万人的城镇，一年中有 1 200 名出生、800 名死亡、400 名迁入者和 200 名迁出者。净迁移 = 400 – 200 = 200。总变化 = 1 200 – 800 + 200 = 600。增长率 = (600 ÷ 80 000) × 100 = 每年 0.75%。当增长率很低或为负时（如英国某些农村地区），人口可能正在下降，这会带来学校关闭和社会老龄化等挑战。
10. Dependency Ratio | 抚养比
The dependency ratio is a measure showing the proportion of the population that is not economically active (dependents) compared to those who are economically active (the working population). Dependents are usually defined as people aged 0–14 (young dependents) and 65+ (old dependents). The formula is: dependency ratio = ((number of people aged 0–14 + number aged 65+) ÷ number of people aged 15–64) × 100. The result is a number, not a percentage, but it is often spoken of as ‘X dependents per 100 working‑age people’.
抚养比是衡量非经济活动人口（被抚养人口）与经济活动人口（劳动人口）对比关系的指标。被抚养人口通常指 0–14 岁（幼年抚养人口）和 65 岁以上（老年抚养人口）的人口。公式为：抚养比 = ((0–14 岁人口数 + 65 岁以上人口数) ÷ 15–64 岁人口数) × 100。计算结果是一个数字，并非百分比，但通常表述为“每 100 名劳动年龄人口负担 X 名被抚养人口”。
For a country with 180 000 young dependents, 120 000 old dependents, and 600 000 working‑age people, the dependency ratio = ((180 000 + 120 000) ÷ 600 000) × 100 = (300 000 ÷ 600 000) × 100 = 50. This means there are 50 dependents for every 100 people of working age. A rising dependency ratio can put pressure on healthcare, pensions, and education systems.
某国有幼年抚养人口 18 万，老年抚养人口 12 万，劳动年龄人口 60 万，抚养比 = ((180 000 + 120 000) ÷ 600 000) × 100 = (300 000 ÷ 600 000) × 100 = 50。这意味着每 100 名劳动年龄人口需抚养 50 人。抚养比上升可能给医疗保健、养老金和教育体系带来压力。
11. River Discharge and Cross‑Sectional Area | 河流流量与横截面积
River discharge (Q) is the volume of water flowing past a point in the river channel per unit time, usually measured in cubic metres per second (cumecs). It is calculated by multiplying the cross‑sectional area of the channel by the mean velocity (speed) of the water. The formula is discharge (m³/s) = cross‑sectional area (m²) × velocity (m/s). The cross‑sectional area of a simple trapezoidal river channel can be found by taking depth measurements at regular intervals across the width and averaging them, then multiplying by the width: area = average depth × width. For a more precise method, add the area of vertical sections across the channel.
河流流量 (Q) 是指单位时间内流过河道某一点的体积，通常以立方米/秒（cumecs）计量。它由河道的横截面积乘以水流平均流速（速度）计算得出。公式为：流量 (m³/s) = 横截面积 (m²) × 流速 (m/s)。简单梯形河道的横截面积可以通过在宽度方向上等距测量水深并取平均值，再乘以宽度来求得：面积 = 平均水深 × 宽度。更精确的方法是累加河道各垂直分块的面积。
Worked example: a river with an average depth of 1.2 m and width 8 m has a cross‑sectional area of 1.2 × 8 = 9.6 m². If the water is flowing at 0.5 m/s, discharge = 9.6 × 0.5 = 4.8 cumecs. A high discharge after heavy rain can lead to flooding, because the channel cannot contain the extra volume of water. Geographers also use velocity measured with a flow meter or a float to study how rivers change downstream.
计算实例：一条河平均水深 1.2 米，宽度 8 米，则横截面积为 1.2 × 8 = 9.6 平方米。若水流速度为 0.5 米/秒，则流量 = 9.6 × 0.5 = 4.8 立方米/秒。大雨之后的高流量可能导致洪水，因为河道无法容纳额外水体。地理学家还使用流速计或浮标测量流速，以研究河流如何沿下游变化。
12. The Hydraulic Radius and Efficiency | 水力半径与效率
The hydraulic radius is a concept used to compare how efficiently different river channel shapes convey water. It is defined as the cross‑sectional area divided by the wetted perimeter (the length of the channel bed and banks in contact with the water). The formula is hydraulic radius (R) = cross‑sectional area (A) ÷ wetted perimeter (P). The unit is metres. A larger hydraulic radius suggests a more efficient channel, because a smaller proportion of the water is slowed down by friction with the bed and banks. This idea is particularly useful when explaining why a river’s velocity often increases downstream even when the slope decreases.
水力半径是一个用于比较不同河道形状输水效率的概念。它定义为横截面积除以湿周长（河道底面和两岸与水接触的长度）。公式为：水力半径 (R) = 横截面积 (A) ÷ 湿周长 (P)。单位为米。水力半径越大，表明河道效率越高，因为与水底和岸壁摩擦而减速的水流比例更小。在解释为何河流流速通常在下游增加（即使坡度降低时）这一点时，这一概念尤为有用。
For a rectangular channel 10 m wide with water depth 2 m, the cross‑sectional area is 10 × 2 = 20 m². The wetted perimeter is the width plus twice the depth: 10 + (2 × 2) = 14 m. So the hydraulic radius is 20 ÷ 14 ≈ 1.43 m. In a broader, shallower channel with the same area but greater wetted perimeter, the hydraulic radius would be smaller, indicating a less efficient channel. While Year 9 students are not required to perform complex hydraulic radius calculations, understanding the principle helps explain flood risk and river management.
对于一条宽 10 米、水深 2 米的矩形河道，横截面积为 10 × 2 = 20 平方米。湿周长为宽度加两倍深度：10 + (2 × 2) = 14 米。因此水力半径为 20 ÷ 14 ≈ 1.43 米。在一条更宽、更浅但面积相同、湿周长更大的河道中，水力半径会更小，表明河道效率较低。虽说九年级学生无需进行复杂的水力半径计算，但理解原理有助于解释洪水风险和河流管理。
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