Microscopy

Lessons

Senior 4 · Biology

Microscopy

4 lessons

Lesson 3 of 4

Calculating magnification and measuring with a microscope

25 min

How long is a chloroplast?

An electron micrograph in a textbook shows a single chloroplast stretched across most of the page, 60 mm long. Under the picture is printed "×12 000". With those two numbers you can work out that the real chloroplast is only 5 µm long: two hundred of them end to end would just cover a millimetre. Every biologist who reads a micrograph, measures a cell or draws a specimen does this kind of calculation, and examiners ask for it in almost every paper.

In S1 you used the magnification formula with millimetres and micrometres. At A-level you must also work in nanometres and standard form, use scale bars, calibrate an eyepiece graticule, and give answers to a sensible number of significant figures.

Definition

Magnification ( $M$ ): how many times larger the image is than the actual object. It has no units: write it as ×12 000, never in µm.

Image size ( $I$ ): the size of the object as measured on a drawing, photograph or screen. Actual size ( $A$ ): the real size of the object.

Micrometre (µm): one thousandth of a millimetre, $10^{-6}\,\mathrm{m}$ . Nanometre (nm): one thousandth of a micrometre, $10^{-9}\,\mathrm{m}$ .

Eyepiece graticule: a glass disc with a scale of arbitrary divisions, placed in the eyepiece so that the scale appears over the specimen. Stage micrometer: a slide engraved with a scale of known divisions, often 0.01 mm (10 µm), used to calibrate the graticule.

Scale bar: a line on a micrograph labelled with the actual length it represents.

Units and the formula

Unit	Symbol	In metres	Typical object
millimetre	mm	$10^{-3}$	A large protoctist such as Amoeba (about 0.5 mm)
micrometre	µm	$10^{-6}$	Cells (10–100 µm), bacteria (1–5 µm), mitochondria (about 1 µm)
nanometre	nm	$10^{-9}$	Viruses (20–300 nm), ribosomes (about 25 nm), membranes (about 7 nm)

To change a larger unit into a smaller one, multiply by 1000 at each step: $1\,\mathrm{mm} = 1000\,\mu\mathrm{m} = 1\,000\,000\,\mathrm{nm}$ . To go the other way, divide by 1000.

M = \frac{I}{A} \qquad A = \frac{I}{M} \qquad I = A \times M

Before dividing, put $I$ and $A$ into the same unit. The usual method is to convert the image size, measured in millimetres with a ruler, into the unit you want the answer in (µm or nm).

Worked example 1: actual size from a magnification

The chloroplast in the textbook micrograph measures 60 mm, and the magnification is ×12 000. Find its actual length in µm.

$A = \frac{I}{M} = \frac{60\,\mathrm{mm}}{12\,000} = 0.005\,\mathrm{mm}$ .
Convert: $0.005\,\mathrm{mm} \times 1000 = 5\,\mu\mathrm{m}$ .
Check that the answer is sensible: chloroplasts are a few micrometres long, so 5 µm fits. A result of 5 mm or 5 nm would signal a slip in the units.

Worked example 2: using a scale bar

A scanning electron micrograph of a pollen grain has a scale bar labelled 10 µm, which measures 25 mm with a ruler. The pollen grain measures 70 mm across. Find (a) the magnification and (b) the actual diameter of the grain.

(a) Use the scale bar itself as the object: $I = 25\,\mathrm{mm} = 25\,000\,\mu\mathrm{m}$ , $A = 10\,\mu\mathrm{m}$ . $M = \frac{25\,000}{10} = 2500$ , so ×2500.
(b) $A = \frac{I}{M} = \frac{70\,\mathrm{mm}}{2500} = 0.028\,\mathrm{mm} = 28\,\mu\mathrm{m}$ .
Shortcut: the grain is $\frac{70}{25} = 2.8$ times as long as the scale bar, so it is $2.8 \times 10 = 28\,\mu\mathrm{m}$ .

Measuring under the microscope: calibrating a graticule

An eyepiece graticule lets you measure a cell while you look at it, but its divisions have no fixed size: what one division covers depends on the objective in use. So the graticule must be calibrated against a stage micrometer for every objective.

Put the stage micrometer on the stage and focus on its scale with the chosen objective.
Turn the eyepiece until the two scales are parallel, and move the stage until two lines coincide, for example the zeros.
Find another point, as far along as possible, where a graticule line exactly meets a micrometer line.
Calculate the length of one eyepiece unit (epu): $1\,\text{epu} = \frac{\text{length on stage micrometer}}{\text{number of eyepiece units}}$ .
Replace the micrometer with the specimen, count how many eyepiece units it covers, and multiply.

Worked example 3: measuring a guard cell

With the ×40 objective, 100 eyepiece units line up with 25 divisions of a stage micrometer whose divisions are 0.01 mm apart (Figure 1). A stomatal guard cell on a leaf peel spans 14 eyepiece units. (a) Calibrate the graticule. (b) Find the length of the guard cell. (c) What would one eyepiece unit be worth with the ×10 objective?

(a) 25 divisions $= 25 \times 0.01\,\mathrm{mm} = 0.25\,\mathrm{mm} = 250\,\mu\mathrm{m}$ . So $1\,\text{epu} = \frac{250\,\mu\mathrm{m}}{100} = 2.5\,\mu\mathrm{m}$ .
(b) $14 \times 2.5\,\mu\mathrm{m} = 35\,\mu\mathrm{m}$ .
(c) The ×10 objective magnifies 4 times less, so each eyepiece unit covers 4 times more of the specimen: $4 \times 2.5 = 10\,\mu\mathrm{m}$ . In practice you recalibrate rather than assume, because objectives are not exactly their stated power.

Significant figures and sensible answers

Give an answer to the same number of significant figures as the least precise measurement: a ruler reading of 25 mm (2 s.f.) does not justify an answer of 28.0000 µm.
Large magnifications are often written in standard form: ×12 000 = $\times 1.2 \times 10^{4}$ .
Always check the result against known sizes (Table above). If a "cell" comes out at 3 mm or 3 nm, a unit conversion is wrong.
The magnification of a drawing is found the same way: measure the drawing ( $I$ ), measure the specimen ( $A$ ) with the graticule, and divide.

D Biology Classroom: "Measuring cells: Calibrate Eyepiece graticule, Magnification, Resolution". Follow the calibration calculation step by step and compare it with Worked example 3.

Warning

Common mistakes

Dividing millimetres by micrometres. Convert both measurements to the same unit first.
Giving magnification with units ("×400 µm") or as a fraction. It is a plain number of times.
Using the graticule without calibrating it, or using the calibration from one objective with another.
Measuring the scale bar label instead of the bar: measure the length of the drawn line with a ruler.

Tip

Memory hook. "I AM": Image = Actual × Magnification. Cover the quantity you want: $A = \frac{I}{M}$ , $M = \frac{I}{A}$ . And "×1000 each step down": mm → µm → nm.

✦

Practice: magnification and measurement

Convert (a) 0.35 mm to µm; (b) 2.4 µm to nm; (c) 850 nm to µm.
A mitochondrion 1.5 µm long appears 45 mm long in a micrograph. Calculate the magnification.
A ribosome is 25 nm across. How large would it appear in a micrograph at ×200 000? Give your answer in mm.
A scale bar of 0.5 µm measures 20 mm on a micrograph. A nucleus measures 120 mm. Find its actual diameter.
With a ×10 objective, 40 eyepiece units equal 0.4 mm on a stage micrometer. Calculate one eyepiece unit in µm, and the length of a cell that covers 9 eyepiece units.
A student draws a 35 µm guard cell 70 mm long. Calculate the magnification of the drawing.
Two students measure the same cell: one gets 38 µm with a calibrated graticule, the other 52 µm using the field of view at ×100. Evaluate which result is more reliable and why.

Show answersHide answers

(a) $0.35 \times 1000 = 350\,\mu\mathrm{m}$ ; (b) $2.4 \times 1000 = 2400\,\mathrm{nm}$ ; (c) $\frac{850}{1000} = 0.85\,\mu\mathrm{m}$ .
$45\,\mathrm{mm} = 45\,000\,\mu\mathrm{m}$ ; $M = \frac{45\,000}{1.5} = 30\,000$ , so ×30 000.
$I = 25 \times 200\,000 = 5\,000\,000\,\mathrm{nm} = 5000\,\mu\mathrm{m} = 5\,\mathrm{mm}$ .
$M = \frac{20\,\mathrm{mm}}{0.5\,\mu\mathrm{m}} = \frac{20\,000}{0.5} = 40\,000$ ; $A = \frac{120\,\mathrm{mm}}{40\,000} = 0.003\,\mathrm{mm} = 3\,\mu\mathrm{m}$ .
$0.4\,\mathrm{mm} = 400\,\mu\mathrm{m}$ ; $1\,\text{epu} = \frac{400}{40} = 10\,\mu\mathrm{m}$ ; cell $= 9 \times 10 = 90\,\mu\mathrm{m}$ .
$70\,\mathrm{mm} = 70\,000\,\mu\mathrm{m}$ ; $M = \frac{70\,000}{35} = 2000$ , so ×2000.
The graticule result (38 µm) is more reliable. It was measured directly against a scale calibrated for that objective, with small divisions. The field-of-view method relies on an estimated field diameter and on counting how many cells fit across it, so it can only give a rough estimate. Repeating the graticule measurement on several cells and taking a mean would improve it further.

Quick check

Key points

$M = \frac{I}{A}$ ; put image and actual size in the same unit first; magnification has no units.
1 mm = 1000 µm; 1 µm = 1000 nm.
A scale bar gives the magnification: measure the bar and divide by the length it represents.
Calibrate the eyepiece graticule against a stage micrometer for each objective: 1 epu = micrometer length ÷ number of eyepiece units.

Sources and credits

2.1.2: Studying Cells, from Biology 2e by OpenStax (Mary Ann Clark, Matthew Douglas, Jung Choi), via LibreTexts Biology, licensed under CC BY 4.0. Used for units, magnification and the sizes of cells and their parts. Adapted by Ganzaa.
Figure 1, the worked examples (the micrograph and cell measurements are illustrative), practice and quick check: written by Ganzaa.
Photos: RIT RAJARSHI, CC BY 4.0, via Wikimedia Commons.
Video: "A-Level Biology - Measuring cells : Calibrate Eyepiece graticule, Magnification, Resolution", D Biology Classroom, embedded from YouTube.

1How long is a chloroplast?

2Units and the formula

3Measuring under the microscope: calibrating a graticule

4Significant figures and sensible answers

How long is a chloroplast?

Units and the formula

Measuring under the microscope: calibrating a graticule

Significant figures and sensible answers