📚 IGCSE CIE Biology: Formula Summary Handbook | IGCSE CIE 生物：公式汇总手册
This handbook brings together all the essential formulas you need to master for the IGCSE CIE Biology examination. Calculations appear frequently in Papers 2, 4 and 6, so knowing exactly when and how to apply each equation will boost your confidence and save you time. For each formula you will find a clear explanation, worked examples and common pitfalls to avoid.
本手册汇集了您在IGCSE CIE生物考试中必须掌握的所有重要公式。计算题频繁出现在试卷2、4和6中，因此准确掌握每个公式的适用场景和使用方法将增强您的信心并节省时间。每个公式都附有清晰的解释、例题和常见易错点。
1. Magnification Calculations | 放大倍率计算
Magnification tells you how many times larger an image appears compared to the real object. The core equation is:
放大倍率表示图像比实物大了多少倍。核心公式为：
Magnification = Image size ÷ Actual size
Image size is the length measured directly from a drawing or micrograph, while actual size is the true length of the specimen. It is vital to convert both measurements to the same unit before performing the division, otherwise the answer will be incorrect.
图像大小是从绘图或显微照片中直接测量的长度，而实际大小是标本的真实长度。进行除法计算前务必将两者单位统一，否则答案会出错。
If a student draws a cell with a diameter of 50 mm and the real cell diameter is 0.005 mm, the magnification would be:
如果学生画出一个直径为50 mm的细胞，而真实细胞直径为0.005 mm，则放大倍率为：
Magnification = 50 mm ÷ 0.005 mm = 10 000 ×
You can also rearrange the formula to find actual size: Actual size = Image size ÷ Magnification. Always include the multiplication sign (×) after the number or write ‘times’ to show that magnification has no units.
你也可以重新排列公式求出实际大小：实际大小 = 图像大小 ÷ 放大倍率。计算后记得写上“×”或“times”，表示放大倍率没有单位。
2. Unit Conversions for Microscopy | 显微镜单位换算
Measurements in microscopy often involve millimetres (mm), micrometres (µm) and nanometres (nm). Converting confidently between them is essential for magnification calculations.
显微镜测量常涉及毫米、微米和纳米。熟练进行单位换算对放大倍率计算至关重要。
| Conversion | Factor |
|---|---|
| 1 mm to µm | × 1000 |
| 1 µm to nm | × 1000 |
| 1 mm to nm | × 1 000 000 |
To convert a length from mm to µm, multiply by 1000. To go from µm to mm, divide by 1000. The same logic applies to µm and nm.
将毫米转换为微米，乘以1000。将微米转换为毫米，除以1000。同样的逻辑适用于微米和纳米。
Example: A chloroplast measures 0.006 mm in length. To express this in µm: 0.006 × 1000 = 6 µm. This value can then be used in the magnification formula.
例：一个叶绿体长0.006 mm。换算成微米：0.006 × 1000 = 6 µm。这个值可以带入放大倍率公式中。
3. Cardiac Output | 心输出量
Cardiac output is the volume of blood pumped by one ventricle per minute. It is calculated from the stroke volume and heart rate.
心输出量是每分钟一个心室泵出的血液体积，由每搏输出量和心率计算得出。
Cardiac output = Stroke volume × Heart rate
Stroke volume is the volume of blood ejected per beat (usually in cm³ or mL) and heart rate is beats per minute (bpm). Cardiac output is therefore expressed in cm³ min⁻¹ or L min⁻¹.
每搏输出量是每次心跳射出的血液体积（通常以cm³或mL计），心率为每分钟心跳次数（bpm）。心输出量单位通常为cm³ min⁻¹或L min⁻¹。
If a person has a stroke volume of 70 cm³ and a heart rate of 72 bpm, then:
如果某人的每搏输出量为70 cm³，心率为72 bpm，则：
Cardiac output = 70 cm³ × 72 bpm = 5040 cm³ min⁻¹
During exercise both stroke volume and heart rate increase, raising cardiac output significantly to deliver more oxygen to muscles.
运动时，每搏输出量和心率都会增加，心输出量显著升高，从而向肌肉输送更多氧气。
4. Respiratory Quotient (RQ) | 呼吸商
The respiratory quotient indicates which substrate is being respired. It is the ratio of carbon dioxide produced to oxygen used.
呼吸商可指示呼吸底物的种类，是产生二氧化碳量与消耗氧气量之比。
RQ = Volume of CO₂ produced ÷ Volume of O₂ consumed
An RQ of 1.0 suggests carbohydrate respiration, about 0.7 indicates lipids, and around 0.9 points to protein. These values arise from the different amounts of oxygen required to fully oxidise each substrate.
RQ为1.0表明呼吸底物为碳水化合物，约0.7为脂质，约0.9为蛋白质。这些数值源于不同底物完全氧化所需的氧气量不同。
| Substrate | Typical RQ |
|---|---|
| Carbohydrate | 1.0 |
| Lipid | 0.7 |
| Protein | 0.9 |
RQ is determined using a respirometer. The values also depend on whether aerobic or anaerobic respiration is occurring – anaerobic respiration produces CO₂ without using O₂, so RQ can be very high.
RQ用呼吸计测量。其值还取决于进行的是有氧呼吸还是无氧呼吸——无氧呼吸产生CO₂而不消耗O₂，因此RQ可能极高。
5. Population Size Estimation (Lincoln Index) | 种群大小估计（林肯指数）
When counting every individual is impossible, ecologists use the capture–mark–recapture method and the Lincoln Index to estimate population size.
当不可能计数所有个体时，生态学家使用标记重捕法和林肯指数来估算种群大小。
N = (M × C) ÷ R
N = estimated total population size, M = number of individuals captured and marked in the first sample, C = total number captured in the second sample, R = number of marked individuals recaptured in the second sample.
N代表估算的种群总数量，M为第一次捕获并标记的个体数，C为第二次捕获的总个体数，R为第二次捕获中带有标记的个体数。
This method assumes that marked individuals mix randomly, no births, deaths or migration occur between samples, and marking does not affect survival or recapture chance. Violating these assumptions makes the estimate inaccurate.
该方法假设标记个体随机混合，两次采样期间没有出生、死亡或迁移，且标记不影响生存或被重捕的机会。违背这些假设会导致估算不准。
Example: 40 woodlice are marked and released (M=40). Later, 50 woodlice are collected (C=50), of which 10 are marked (R=10). N = (40 × 50) ÷ 10 = 200.
例：标记并释放了40只潮虫（M=40）。之后采集到50只（C=50），其中10只带有标记（R=10）。则N = (40 × 50) ÷ 10 = 200。
6. Population Density | 种群密度
Population density describes how crowded a population is within a given area. It is often determined using quadrats for stationary organisms.
种群密度描述给定区域内种群的拥挤程度。对于固着生物，通常使用样方法测定。
Population density = Number of individuals ÷ Area
Units are typically individuals per square metre (ind. m⁻²). By placing quadrats randomly and counting individuals inside, you can calculate the mean number per quadrat and then scale up to the entire habitat area.
单位通常为每平方米个体数（ind. m⁻²）。随机放置样方并计数其中的个体，计算出每个样方的平均数量，再按比例推算整个栖息地的数量。
For example, if ten 1 m² quadrats have a total of 120 daisies, the mean is 12 per m². If the field area is 500 m², the estimated total population is 12 × 500 = 6000 daisies.
例如，10个1 m²样方中共有120株雏菊，平均每平方米12株。若田野面积为500 m²，则估算总数为12 × 500 = 6000株。
7. Rate of Photosynthesis | 光合作用速率
Photosynthesis rate can be measured by the oxygen produced or the time taken for a leaf disc to rise in water. A common laboratory formula is:
光合作用速率可用产生的氧气量或叶圆片上浮所需时间来衡量。常见的实验室公式为：
Rate = Volume of O₂ produced ÷ Time
Alternatively, if using the floating disc method, rate can be expressed as 1 ÷ time taken for discs to float. This works because photosynthesising discs release O₂ bubbles, making them buoyant.
若使用叶圆片上浮法，速率可表示为1 ÷ 上浮所需时间。原理是进行光合作用的叶圆片释放氧气气泡，使其浮起。
For pondweed experiments, collect oxygen in a capillary tube or syringe and divide the volume by the minutes of exposure to light. Graph plotting rate against light intensity or CO₂ concentration often reveals a limiting factor.
水草实验中，用毛细管或注射器收集氧气，将体积除以照光时间。以速率对光照强度或CO₂浓度作图常可揭示限制因子。
8. Rate of Transpiration | 蒸腾速率
A potometer estimates transpiration rate by measuring water uptake. The movement of an air bubble in the capillary tube indicates the volume of water taken up.
蒸腾计通过测量吸水量来估算蒸腾速率。毛细管中气泡的移动表示吸入的水量。
Rate = Distance moved by bubble ÷ Time
Alternatively, if the capillary tube is calibrated, use volume absorbed per time. Rate is often expressed in mm min⁻¹ or cm³ min⁻¹.
若毛细管标有刻度，也可使用单位时间的吸水体积。速率常以mm min⁻¹或cm³ min⁻¹表示。
Factors such as light, humidity, temperature and wind speed affect the rate. Always ensure the shoot is cut under water to prevent air locks, and allow time for acclimatisation before recording.
光照、湿度、温度和风速等因素都会影响蒸腾速率。确保在水下剪切枝条以防气栓，并在记录前让植物适应一段时间。
9. Energy Content of Food | 食物能量含量
The energy stored in food can be estimated by burning a sample and using the heat released to warm water. The calculation relies on the specific heat capacity of water.
储存于食物中的能量可通过燃烧样品并用水吸收释放的热量来估算，这依赖于水的比热容。
Energy per gram = (Temperature rise × Volume of water × 4.2) ÷ Mass of food
Temperature rise is in °C, volume of water in cm³ (equivalent to g), 4.2 J g⁻¹ °C⁻¹ is the specific heat capacity of water, and mass of food is in g. The result is in joules per gram (J g⁻¹).
温度升高以°C为单位，水的体积以cm³计（相当于克），4.2 J g⁻¹ °C⁻¹是水的比热容，食物质量以克计。计算结果为每克焦耳数（J g⁻¹）。
If 1.5 g of a crisp raises the temperature of 20 cm³ water by 18 °C, the energy content is (18 × 20 × 4.2) ÷ 1.5 = 1512 ÷ 1.5 = 1008 J g⁻¹. This is an underestimate because heat is lost to the surroundings.
如果1.5克薯片使20 cm³的水温升高了18 °C，则能量含量为(18 × 20 × 4.2) ÷ 1.5 = 1512 ÷ 1.5 = 1008 J g⁻¹。这一数值偏低，因为有热量散失到环境中。
10. Body Mass Index (BMI) | 身体质量指数
BMI is a screening tool that compares mass to height to assess whether a person is underweight, healthy, overweight or obese.
BMI是一种筛查工具，通过比较体重与身高来评估一个人是否体重过轻、健康、超重或肥胖。
BMI = Mass (kg) ÷ Height² (m²)
Always use kilograms for mass and metres for height. A BMI below 18.5 is classed as underweight, 18.5–24.9 is healthy, 25–29.9 is overweight, and 30 or above is obese.
体重用千克，身高用米。BMI低于18.5属体重过轻，18.5–24.9为健康，25–29.9为超重，30及以上为肥胖。
For example, a person weighing 70 kg and 1.75 m tall has a BMI of 70 ÷ (1.75)² = 70 ÷ 3.0625 ≈ 22.9 kg m⁻², which is in the healthy range. BMI does not distinguish between muscle and fat, so it has limitations for athletes.
例如，一个体重70 kg、身高1.75 m的人，BMI为70 ÷ (1.75)² = 70 ÷ 3.0625 ≈ 22.9 kg m⁻²，属于健康范围。BMI不能区分肌肉和脂肪，因此对运动员有局限性。
11. Rate of Enzyme-Controlled Reactions | 酶控反应速率
Enzyme activity is often monitored by measuring the rate at which product appears or substrate disappears. The general formula is:
酶活性通常通过测定产物出现或底物消失的速率来监测。通用公式为：
Rate = Amount of product formed ÷ Time
In the starch–amylase investigation, the time taken for iodine to stop turning blue-black is recorded. Rate can then be expressed as 1 ÷ time (s⁻¹). As temperature or pH changes, the rate changes accordingly.
在淀粉-淀粉酶实验中，记录碘液不再变蓝黑所需的时间。速率可表示为1 ÷ 时间（s⁻¹）。随着温度或pH的变化，速率也会相应改变。
Always specify the unit of rate, e.g. cm³ O₂ min⁻¹ for catalase, or absorbance units min⁻¹ for a colorimeter. The initial rate is usually the most reliable because substrate concentration is not yet limiting.
务必标明速率单位，如过氧化氢酶实验用cm³ O₂ min⁻¹，使用比色计则用吸光度单位min⁻¹。初始速率通常最可靠，因为此时底物浓度还未成为限制因素。
12. Surface Area to Volume Ratio | 表面积与体积比
The surface area to volume ratio (SA:V) is fundamental to understanding transport in organisms. It influences rates of diffusion, heat exchange and osmosis.
表面积与体积比（SA:V）是理解生物体运输的基础，它影响扩散速率、热交换和渗透作用。
SA:V = Surface area ÷ Volume
For a cube of side length s, surface area = 6s² and volume = s³, so SA:V = 6 ÷ s. This illustrates that as an object gets larger, its SA:V decreases, making diffusion less efficient.
对于边长为s的立方体，表面积 = 6s²，体积 = s³，因此SA:V = 6 ÷ s。这表明物体越大，SA:V越小
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