Estimation of physical quantities

CIE A-Level Physics· Unit 1: Physical quantities and units· 10 min read

1. Orders of magnitude and estimation basics★☆☆☆☆⏱ 3 min

📘 Definition

Order of magnitude estimation

Written as $\sim 1 0^{n}$

A rough approximation of a quantity's size, rounded to the nearest power of 10, used when exact values are not required

Example:

The height of an adult human is ~ 2 m, so its order of magnitude is $1 0^{0}$ m

For estimation questions in CIE exams, you only need to get the power of 10 correct. Factors of 2 or 3 do not change the order of magnitude, so you can round freely during calculations.

📐 Worked Example

Estimate the order of magnitude of the number of breaths an adult takes in one year.

1
Start with an estimate for breathing rate: typical resting rate is ~ 12 breaths per minute, which is ~ 0.2 breaths per second.
2
Calculate the number of seconds in one year:
3
$60 \times 60 \times 24 \times 365 \approx 3 \times 10^7$
4
Multiply rate by time to get total breaths: $0.2 \times 3 \times 1 0^{7} = 6 \times 1 0^{6}$ .
5
Round to nearest power of 10: $6 \times 1 0^{6}$ is closer to $1 0^{7}$ than $1 0^{6}$ , so the order of magnitude is $1 0^{7}$ breaths per year.

2. Common memorised estimates★★☆☆☆⏱ 4 min

CIE exams expect you to have memorised reasonable estimates for a range of common everyday and atomic-scale quantities. Most multiple-choice estimation questions test your knowledge of these standard values.

📐 Worked Example

What is the best estimate for the diameter of an atom?

1
Recall the standard estimate for atomic diameter: the size of a small atom like hydrogen is roughly 0.1 nanometers.
2
Convert to SI units (meters): 0.1 nm = $0.1 \times 1 0^{- 9}$ m = $1 \times 1 0^{- 10}$ m.
3
The order of magnitude is $1 0^{- 10}$ m, which is the correct answer.

3. Estimating derived quantities★★★☆☆⏱ 5 min

For quantities you haven't memorised, break the quantity down into simpler terms that you can estimate, then combine using the relevant physical formula. You only need to keep track of powers of 10.

📐 Worked Example

Estimate the order of magnitude of the mass of air in a small classroom.

1
Density of air $ρ = \frac{m}{V}$ , so mass $m = ρ V$ . We need estimates for density of air and volume of the classroom.
2
Density of air is approximately $1$ kg m $^{- 3}$ , so order of magnitude $1 0^{0}$ kg m $^{- 3}$ .
3
Estimate classroom dimensions: 4 m × 5 m × 3 m = 60 m³ ≈ 10² m³.
4
Multiply: $m = 1 \times 60 = 60$ kg, which rounds to order of magnitude $1 0^{2}$ kg.

4. Common Pitfalls

Wrong move:

Forgetting to convert all quantities to the same units before estimating

Why:

Exam questions intentionally use mixed units in options to test unit awareness, leading to incorrect order of magnitude

Correct move:

Always convert all values to SI units before calculating your estimate

Wrong move:

Wasting time trying to calculate an exact value instead of rounding early

Why:

Estimation questions only require the correct power of 10, exact values are never needed

Correct move:

Round all intermediate values to the nearest power of 10 early in your calculation to save time

Wrong move:

Mixing up the diameter of an atom and an atomic nucleus

Why:

A common mistake is to use 10⁻¹⁵ m for an atom or 10⁻¹⁰ m for a nucleus, which is five orders of magnitude wrong

Correct move:

Memorise: atom ~ 10⁻¹⁰ m, nucleus ~ 10⁻¹⁵ m

Wrong move:

Incorrectly rounding 70 kg (mass of adult) to order 10¹ kg

Why:

70 kg = 7 × 10¹ kg, which is closer to 10 × 10¹ = 10² kg than 1 × 10¹ kg

Correct move:

Always round the coefficient first before writing the final order of magnitude

5. Quick Reference Cheatsheet

Quantity	Typical order of magnitude (SI units)
Height of adult human	$1 0^{0}$ m
Mass of adult human	$1 0^{2}$ kg
Mass of an apple	$1 0^{- 1}$ kg
Mass of a car	$1 0^{3}$ kg
Speed of sound in air	$1 0^{2}$ m/s
Motorway car speed	$1 0^{1}$ m/s
Diameter of atom	$1 0^{- 10}$ m
Diameter of atomic nucleus	$1 0^{- 15}$ m
Wavelength of visible light	$1 0^{- 6}$ m
Mass of an electron	$1 0^{- 30}$ kg
Density of air	$1 0^{0}$ kg m $^{- 3}$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · Paper 1
MCQ: mass of adult human
2023 · Paper 1
MCQ: diameter of atom
2021 · Paper 2
Estimate KE of running child

What's Next

Estimation is a foundational skill that you will use throughout your A-Level Physics course, from checking that calculation answers are physically reasonable to planning practical experiments. It is one of the easiest topics to pick up marks for in Paper 1, so investing time to memorise the common values pays off in early exams. Estimation also links closely to unit analysis and dimensional checking, which help you verify your answers in longer calculation questions.