📏 SI Units & Conversions — T Level / Level 3 Engineering Maths

Every engineering measurement needs a unit. Using the wrong unit — or forgetting to convert — can cause catastrophic failure. From the NASA Mars orbiter crash (1999, caused by a metric/imperial mix-up!) to everyday workshop errors, units matter enormously.

T Level Core 4.1 7 SI base units Metric prefixes Imperial conversions Temperature scales

Two systemsMetric (SI) and Imperial — both still used in UK engineering

Metric (SI): mm, cm, m, km / g, kg, t

Imperial: inch, foot, yard, mile / oz, lb, ton

The metric system is used in science, engineering calculations and international trade. Imperial units are still common in construction, manufacturing and daily life in the UK — so engineers must handle both.

📌 In T Level exams you may be given a mix — always check the units in any question before calculating!

Why it mattersReal consequences of unit errors

NASA Mars orbiter (1999) — crashed because one team used imperial, another metric. $327 million lost.
Gimli Glider (1983) — Air Canada Boeing 767 ran out of fuel mid-flight: kg vs lbs confusion
Workshop errors — machining a part in inches when the drawing is in mm produces scrap
T Level exams — wrong units = wrong answer, even if the number is right

🔬SI Base Unitsthe 7 fundamentals

📏Prefixeskilo, milli, mega…

📐Metricwithin the system

🏴󠁧󠁢󠁥󠁮󠁧󠁿Imperialinches, feet, lbs

🔄Metric↔Imperialcross-system

🌡️Temperature°C, °F, K

⚙️Engineeringreal examples

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SI Base Units

The International System of Units (SI) has 7 base units. Every other unit in science and engineering is derived from these 7.

The 7 SI Base UnitsThese are the foundation of all measurement

📌 For T Level / Level 3 Engineering, you'll mostly use metre, kilogram, second and ampere. Kelvin appears in thermodynamics. Mole and candela are less common but you should know they exist.

Derived UnitsBuilt from the base units — the ones engineers use every day

Derived Unit	Symbol	What it measures	Made from…
Newton	N	Force	kg·m·s⁻²
Pascal	Pa	Pressure / Stress	N/m² = kg·m⁻¹·s⁻²
Joule	J	Energy / Work	N·m = kg·m²·s⁻²
Watt	W	Power	J/s = kg·m²·s⁻³
Volt	V	Voltage	W/A = kg·m²·s⁻³·A⁻¹
Ohm	Ω	Resistance	V/A = kg·m²·s⁻³·A⁻²
Hertz	Hz	Frequency	s⁻¹ (cycles per second)

💡 Memory trick for Newton: F = ma → force = mass × acceleration → kg × m/s² → N = kg·m·s⁻²

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SI Prefixes

Prefixes scale any SI unit up or down by powers of 10. Learn these and you can handle any measurement from nanometres to gigawatts!

The Prefix ScaleFrom nano (×10⁻⁹) all the way to Giga (×10⁹)

Full Prefix TableAll prefixes you need for T Level / Level 3 / Level 3 Engineering

Prefix	Symbol	Multiplier	Engineering Example
nano	n	× 10⁻⁹ = 0.000000001	45 nm — wavelength of UV light
micro	μ	× 10⁻⁶ = 0.000001	250 μm — human hair diameter
milli	m	× 10⁻³ = 0.001	3.5 mm — bolt thread pitch
centi	c	× 10⁻² = 0.01	2.54 cm = 1 inch
(base)	—	× 10⁰ = 1	1 m, 1 kg, 1 s
kilo	k	× 10³ = 1,000	4.5 kN — cable tension
Mega	M	× 10⁶ = 1,000,000	200 MPa — steel yield stress
Giga	G	× 10⁹ = 1,000,000,000	200 GPa — Young's modulus steel

Converting with prefixesAlways convert to base unit first

Replace the prefix with its multiplier (e.g. k = × 1000)
Multiply or divide to get the base unit
Then convert to the target prefix if needed

4.7 kN → 4.7 × 1000 = 4700 N

250 μm → 250 × 10⁻⁶ = 0.00025 m

0.85 MW → 0.85 × 10⁶ = 850,000 W

Worked ExamplesPrefix conversions step by step

Convert 3.2 km to mm

km→m: 3.2 × 1000 = 3200 m

m→mm: 3200 × 1000 = 3,200,000 mm ✅

Convert 47 mA to A

milli→base: 47 × 10⁻³

Answer: 0.047 A ✅

📏 Try it — SI Prefix ConverterAny prefix to any prefix

Value

From prefix

→

To prefix

👆Try: 4.7 kΩ → Ω · 47 mA → A · 2400 W → kW · 0.1 μF → pF

📐

Metric Conversions

Converting within the metric system — length, mass, area, volume and time. All based on powers of 10, so just multiply or divide.

Lengthkm ↔ m ↔ cm ↔ mm — multiply going right, divide going left

💡 Going right (→) = getting smaller units = multiply. Going left (←) = getting bigger units = divide.

Mass conversions

t × 1000 → kg × 1000 → g × 1000 → mg

t ←÷1000 kg ←÷1000 g ←÷1000 mg

1 tonne = 1000 kg 1 kg = 1000 g 1 g = 1000 mg

Convert 4.25 kg to g

kg → g: × 1000

Answer: 4250 g ✅

Area & Volume conversions⚠️ Trickiest — square and cube the conversion factor!

Area: 1 m² = 100² cm² = 10,000 cm²

Area: 1 cm² = 10² mm² = 100 mm²

Vol: 1 m³ = 100³ cm³ = 1,000,000 cm³

Vol: 1 litre = 1000 cm³ = 1000 ml

⚠️ 1 m = 100 cm but 1 m² ≠ 100 cm² — it's 10,000 cm²! Always square or cube the factor!

Time conversionsSeconds, minutes, hours, days — NOT powers of 10!

📌 1 min = 60 s | 1 hour = 60 min = 3600 s | 1 day = 24 hours = 86,400 s

📐 Try it — Metric ConverterLength · Mass · Volume · Time

Value

From

→

To

👆Try: 4.25 kg → g · 2500 mm → m · 1 hour → s · 5 litres → cm³

🏴󠁧󠁢󠁥󠁮󠁧󠁿

Imperial Units

Imperial units are still widely used in UK engineering, construction and everyday life. You must be able to recognise them and convert to metric.

Imperial Length

Unit	Abbreviation	Relationship
inch	in or "	base of small lengths
foot	ft or '	1 ft = 12 inches
yard	yd	1 yd = 3 ft = 36 in
mile	mi	1 mile = 1760 yd = 5280 ft

💡 Memory: 12 inches in a foot (12 months, 12 hours — 12 is everywhere!)

Imperial Mass & Volume

Unit	Abbrev.	Relationship
ounce	oz	base of small mass
pound	lb	1 lb = 16 oz
stone	st	1 st = 14 lb
ton (imperial)	t	1 ton = 2240 lb
fluid ounce	fl oz	base of volume
pint	pt	1 pint = 20 fl oz
gallon	gal	1 gallon = 8 pints

⚠️ UK ton vs US ton: Imperial (UK) ton = 2240 lb. US "short ton" = 2000 lb. Metric tonne = 1000 kg ≈ 2205 lb. Always specify which "ton" you mean in engineering!

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Converting Between Metric & Imperial

The conversion factors you MUST know for T Level / Level 3 / Level 3 Engineering. Always write the factor as an equation, then multiply or divide.

MethodWrite: X units × conversion factor = Y units. Arrange to solve for what you need.

Visual ScaleHow imperial and metric lengths compare — shown as bars

Conversion Reference TableThe key conversion factors — memorise these!

Quantity	Imperial → Metric	Metric → Imperial
Length	1 inch = 2.54 cm	1 cm = 0.394 inches
Length	1 foot = 30.48 cm = 0.305 m	1 m = 3.281 feet
Length	1 mile = 1.609 km	1 km = 0.621 miles
Length	1 yard = 0.914 m	1 m = 1.094 yards
Mass	1 ounce (oz) = 28.35 g	1 g = 0.035 oz
Mass	1 pound (lb) = 453.6 g = 0.454 kg	1 kg = 2.205 lb
Mass	1 imperial ton = 1016 kg	1 tonne = 0.984 imp. ton
Volume	1 pint = 568 ml	1 litre = 1.76 pints
Volume	1 gallon = 4.546 litres	1 litre = 0.220 gallons

Convert 5 feet 8 inches to metres

Total inches: 5×12 + 8 = 68 inches

× 2.54: 68 × 2.54 = 172.72 cm

Answer: 1.727 m ✅

⚙️ Convert 3.5 kg to pounds (lbs)

Factor: 1 kg = 2.205 lb

Multiply: 3.5 × 2.205 = 7.7175

Answer: 7.72 lb (3 s.f.) ✅

🔄 Try it — Metric ↔ ImperialLength · Mass · Volume

Value

Convert

👆Try: 5 ft → m · 3.5 kg → lb · 26.2 miles → km (marathon) · 1 gallon → L

🌡️

Temperature Conversions

Three scales — Celsius (°C), Fahrenheit (°F) and Kelvin (K). Engineers use all three depending on the application.

Three Temperature ScalesKey reference points shown visually

°C ↔ °F Formulas

°F = (°C × 9/5) + 32

°C = (°F − 32) × 5/9

Convert 25°C to °F

Formula: °F = (25 × 9/5) + 32

Calculate: (25 × 1.8) + 32 = 45 + 32

Answer: 77°F ✅

°C ↔ Kelvin

K = °C + 273.15

°C = K − 273.15

📌 Kelvin is used in thermodynamics formulas (e.g. ideal gas law: PV = nRT). Temperature differences are the same in K and °C — only the zero point shifts.

Convert 350°C to Kelvin

Formula: K = 350 + 273.15

Answer: 623.15 K ✅

🌡️ Try it — Temperature Converter°C · °F · K

Value

From

→

To

👆Try: 25°C → °F · 350°C → K · 150°C → K (gas law) · 212°F → °C

💡 Quick memory: °C to K — just add 273. K to °C — just subtract 273. The 0.15 is only needed for high-precision work.

⚙️

Applications

Where this topic is used in engineering, manufacturing, maintenance and daily life.

🔧Mechanical: Force N→kN. Pressure Pa→MPa. Torque N·m for bolts.
⚡Electrical: mV→V, mA→A, kΩ→Ω. Unit conversion in circuit analysis.
🏗️Construction: mm→m→km. mm²→m². cm³→litres→m³.
🌡️Daily Life: °C↔K. mph↔m/s. kWh on electricity bill.

⚙️

Units in Engineering

Every engineering calculation involves units. Here's where you'll actually need each conversion skill in your T Level and on the job.

🔩 Precision Machining

Drawings may mix inches and mm. CNC programs work in mm. A 3/8" drill bit = 3/8 × 25.4 = 9.525 mm. Thread pitch M8 = 1.25 mm = 1.25 × 0.0394 = 0.049 inches.

3/8" drill bit:
3 ÷ 8 = 0.375 inches
0.375 × 25.4 = 9.525 mm

M8 pitch = 1.25 mm
= 1.25 ÷ 25.4 = 0.0492"

⚡ Electrical Engineering

Voltage, current and resistance use SI prefixes constantly. A 4.7 kΩ resistor = 4700 Ω. A 100 μF capacitor = 0.0001 F.

4.7 kΩ = 4.7 × 10³ = 4700 Ω
47 mA = 47 × 10⁻³ = 0.047 A
V = IR = 0.047 × 4700
= 220.9 V

🏗️ Structural Engineering

Forces in kN, stresses in MPa or N/mm² (same thing!), Young's modulus in GPa. Unit consistency in stress calculations is critical.

Stress = Force ÷ Area
F = 4.5 kN = 4500 N
A = 300 mm²
σ = 4500 ÷ 300 = 15 N/mm²
= 15 MPa

🌡️ Thermodynamics

Heat engines, refrigerators and gas laws all use Kelvin. Thermal expansion coefficients use 1/°C or 1/K (same difference in change).

Ideal gas: PV = nRT
T must be in KELVIN
T = 150°C + 273 = 423 K

Expansion: ΔL = L × α × ΔT
ΔT same in °C or K

🔧 T Level Exam Style — Unit Consistency Check

Problem: Calculate the pressure (Pa) on a piston. Force = 2.4 kN, area = 0.008 m².

Convert: 2.4 kN = 2.4 × 1000 = 2400 N

Formula: P = F ÷ A = 2400 ÷ 0.008

Answer: P = 300,000 Pa = 300 kPa = 0.3 MPa ✅

⚙️ Try it — Engineering Unit ConverterForce · Pressure · Power · Speed

Value

Convert

👆Try: 2.4 kN → N · 300000 Pa → kPa · 70 mph → m/s · 1 kWh → J

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Quick Fire Quiz

10 questions, instant feedback.

⚡ Quick Blitz

Score: 0Streak: 0 🔥Q 1/10

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Practice Questions

15 questions in 3 tiers — T Level / Level 3 Engineering style with full working shown.

🟢 Tier 1 — Foundations

Q1

State the SI unit and symbol for: (a) force (b) pressure (c) power

[3]

(a) Newton (N)
(b) Pascal (Pa)
(c) Watt (W)

Q2

Convert: (a) 4.5 km to m (b) 850 mm to m (c) 2.3 cm to mm

[3]

(a) 4.5 × 1000 = 4500 m
(b) 850 ÷ 1000 = 0.85 m
(c) 2.3 × 10 = 23 mm

Q3

Convert: (a) 3.2 kg to g (b) 750 g to kg

[2]

(a) 3.2 × 1000 = 3200 g
(b) 750 ÷ 1000 = 0.75 kg

Q4

Write the prefix and symbol for: (a) × 10⁻³ (b) × 10³ (c) × 10⁶

[3]

(a) milli, m
(b) kilo, k
(c) Mega, M

Q5

Convert 25°C to (a) Fahrenheit (b) Kelvin

[3]

(a) °F = (25 × 9/5) + 32 = 45 + 32 = 77°F
(b) K = 25 + 273.15 = 298.15 K

🟡 Tier 2 — Core Skills

Q6

Convert 47 mA to amperes, then to μA.

[3]

47 mA = 47 × 10⁻³ = 0.047 A
0.047 A = 0.047 × 10⁶ μA = 47,000 μA

Q7

A component is 3.5 inches long. Convert to mm. (1 inch = 25.4 mm)

[2]

3.5 × 25.4 = 88.9 mm ✅

Q8

Convert 5 feet 4 inches to centimetres. (1 inch = 2.54 cm)

[3]

Total inches = 5×12 + 4 = 64 inches
64 × 2.54 = 162.56 cm ✅

Q9

Convert an area of 2.5 m² to cm².

[3]

Remember: 1 m = 100 cm, so 1 m² = 100² cm²

1 m² = 10,000 cm²
2.5 × 10,000 = 25,000 cm² ✅

Q10

Convert 350°F to Celsius.

[2]

°C = (350 − 32) × 5/9 = 318 × 5/9 = 1590/9 = 176.7°C ✅

🔴 Tier 3 — T Level Challenge

Q11

Calculate the stress on a bolt. Force = 8.5 kN, cross-sectional area = 50 mm². Express the answer in (a) N/mm² (b) MPa (c) Pa in standard form.

[5]

F = 8.5 kN = 8500 N
σ = F/A = 8500 ÷ 50 = 170 N/mm²
(a) 170 N/mm²
(b) 170 MPa (1 N/mm² = 1 MPa)
(c) 170 × 10⁶ Pa = 1.70 × 10⁸ Pa ✅

Q12

An engineer in the USA specifies a shaft diameter of 1.375 inches. The UK workshop works in mm. Convert to mm to 2 decimal places. (1 inch = 25.4 mm)

[3]

1.375 �

Q13

A steel bar is heated from 20°C to 680°C. The coefficient of linear thermal expansion is 12 × 10⁻⁶ /K. The bar is 2.4 m long. Calculate the extension in mm.

[4]

ΔT = 680 − 20 = 660 K
ΔL = L × α × ΔT
= 2.4 × 12×10⁻⁶ × 660
= 0.019008 m
= 19.0 mm ✅

Q14

A fuel tank holds 45 gallons. Calculate its capacity in (a) litres and (b) cm³. (1 gallon = 4.546 litres)

[4]

(a) 45 × 4.546 = 204.57 litres
(b) 204.57 × 1000 = 204,570 cm³ ✅

Q15

An electric motor outputs 2.2 kW at 1450 rpm. Calculate the shaft torque using T = P / ω where ω = 2πn/60. Show all unit conversions.

[5]

P = 2.2 kW = 2200 W
ω = 2π × 1450 ÷ 60 = 151.84 rad/s
T = 2200 ÷ 151.84
= 14.49 N·m ✅

🎯 Score guide: Q1–5 = foundations (14 marks) · Q6–10 = core skills (13 marks) · Q11–15 = T Level challenge (21 marks)