OCR GCSE Mathematics: Units, compound measures, maps, bearings, perimeter, circumference and arc length (J560) – Revision Notes

These notes condense sections 10.01 (units and measurement) and 10.02 (perimeter calculations) of the OCR GCSE (9-1) Mathematics J560 specification, Version 2.1 (August 2026). All of it is for both Foundation and Higher tier; none of it is Higher tier only. For full explanations and longer worked examples, read the study guide. When you are ready to test yourself, use the practice questions. The OCR GCSE Mathematics hub, the checklist and the 10-minute diagnostics help you plan which parts to revisit.

10.01a Units: facts to recall

Type	Conversion
Length	1 km = 1000 m; 1 m = 100 cm; 1 cm = 10 mm
Area	1 m² = 10 000 cm²; 1 cm² = 100 mm²
Volume	1 m³ = 1 000 000 cm³; 1 cm³ = 1000 mm³
Capacity	1 litre = 1000 ml = 1000 cm³; 1 m³ = 1000 litres
Mass	1 tonne = 1000 kg; 1 kg = 1000 g
Time	1 min = 60 s; 1 h = 3600 s; 1 day = 24 h; 1 year = 365 days
Money	pounds and pence, dollars and cents, euros and cents; exchange rates by × or ÷

Method in steps for converting units

1. Write the link between the two units (for example 1 m² = 10 000 cm²).
2. Decide whether the new number should be bigger or smaller. Smaller units give bigger numbers.
3. Multiply or divide, then check the size is sensible.

Squared and cubed. If the length factor is k, the area factor is k² and the volume factor is k³. For m to cm, k = 100, so the area factor is 10 000.

Time. Minutes to hours: divide by 60. 4 h 45 min = 4.75 h. Decimal hours to minutes: multiply the decimal part by 60, so 1.4 h = 1 h 24 min. A 12-hour time after noon becomes a 24-hour time by adding 12: 7.25 pm = 19:25. Always count durations across midnight in two parts.

Exchange rates. If £1 buys a certain number of euros, multiply pounds by it to get euros and divide euros by it to get pounds. Check that the amount in the currency with the bigger numbers is the larger amount.

Algebraic contexts. Treat the letter as a number but keep its unit.

Starting measure	Converted measure
y cm	y/100 m
x litres	1000x ml
v m/s	3.6v km/h
t minutes	t/60 hours
a cm²	a/10 000 m²

Worked reminder: a rope of length 2p metres is cut into pieces of 25 cm. There are 200p cm, so 200p ÷ 25 = 8p pieces.

10.01b Compound units

Quantity	Relationship	Rearranged
Speed	distance over time	distance = speed × time; time = distance ÷ speed
Density	mass over volume	mass = density × volume; volume = mass ÷ density
Pressure	force over area	force = pressure × area; area = force ÷ pressure
Pay rate	pay ÷ time	pay = rate × time
Unit price	cost ÷ amount	compare costs of equal amounts

Memory aid. For speed, write D over S and T, with D on top. Cover the one you want and read off the rest.

Method in steps for converting a compound unit

1. Write the starting unit as a fraction, such as km/h.
2. Convert the top and the bottom separately.
3. Evaluate the new fraction.

Worked reminders:

280 m in 35 s: speed = 280 ÷ 35 = 8 m/s = 8 × 3.6 = 28.8 km/h
Density: 630 g in 150 cm³ → 630 ÷ 150 = 4.2 g/cm³ = 4200 kg/m³
Pressure: 360 N on 0.18 m² → 360 ÷ 0.18 = 2000 N/m² = 0.2 N/cm²
Pay: £8.50 per hour for 4.5 h → 8.50 × 4.5 = £38.25
Unit price: 600 g for £1.68 → 168 ÷ 6 = 28 p per 100 g; 900 g for £2.43 → 243 ÷ 9 = 27 p per 100 g

The 900 g pack is the better buy. Always state which one you mean.

Must-know distinctions

• Density in g/cm³ becomes kg/m³ by multiplying by 1000, so 1 g/cm³ = 1000 kg/m³.
• Speed in km/h and speed in m/s differ by 3.6.
• Pressure in N/m² and N/cm² differ by 10 000, because the area unit is on the bottom: N/cm² is the bigger pressure unit.

10.01c Maps, bearings and scale drawings

Scale. On a 1 : n map, 1 cm stands for n cm. Real length = map length × n. Map length = real length ÷ n. Convert to sensible units at the end: divide cm by 100 for m and by 100 000 for km.

Scale 1 : 2000, map length 8 cm → 8 × 2000 = 16 000 cm = 160 m

Bearings. Measured clockwise from north, with three figures. Method in steps:

1. Mark the point you start from. The bearing is “of B from A”, so start at A.
2. Draw a north line at that point.
3. Measure clockwise to the line to B and write at least three figures, such as 047°.

Back bearing. Add or subtract 180° so that the answer is between 000° and 360°. A bearing of 305° from X to Y gives a bearing of 125° from Y to X. This works because the north lines at the two points are parallel.

Scale drawings. Choose a scale that makes the drawing a sensible size, such as 1 cm to 2 km. Draw each north line, measure angles with a protractor, draw lengths to scale, then measure the answer and convert back. Label the scale on the drawing.

10.02 Perimeter, circumference and arc length

Rectilinear shapes. Find the missing sides: on a staircase outline, the horizontal steps add up to the full width and the vertical steps to the full height. Then add every side. A shape with a corner notch removed has the same perimeter as the outer rectangle.

Circles.

Item	Formula
Circumference	C = πd = 2πr
Arc length	(angle/360) × 2πr
Perimeter of a sector	arc length + 2r
Perimeter of a semicircle	πr + 2r

Leave answers in terms of π if the question says so, or if there is no calculator. If you round, say how.

Method in steps for composite perimeters

1. Sketch the outline and mark every given length.
2. Split it into straight edges and arcs.
3. Find any missing lengths, such as a radius from a diameter.
4. Include only the outside edge and add everything.

Worked reminder (semicircle on one side of a square of side 8 cm): the outside edge is three straight sides of 8 cm and an arc of ½ × π × 8 = 4π. The perimeter is 24 + 4π = 36.6 cm (3 s.f.). The shared side is inside, so it is not counted.

Quick self-test

1. Convert 5.2 km to metres.
2. Convert 0.045 m² to cm².
3. Convert 3600 ml to litres.
4. Write 7.25 pm in the 24-hour clock.
5. Change 18 m/s to km/h.
6. A metal has density 19.3 g/cm³. Find the mass of 40 cm³ of it.
7. A force of 60 N acts on 0.015 m². Find the pressure in N/m².
8. On a 1 : 50 000 map two towns are 3.2 cm apart. How far apart are they in km?
9. The bearing of P from Q is 047°. Find the bearing of Q from P.
10. Find the circumference of a circle of diameter 7 cm, to 3 significant figures.
11. A sector has radius 10 cm and angle 36°. Give its arc length in terms of π and its perimeter to 3 significant figures.
12. Find the perimeter of a semicircle of radius 5 cm, including the straight edge, to 3 significant figures.

Answers

1. 5200 m
2. 450 cm²
3. 3.6 litres
4. 19:25
5. 18 × 3.6 = 64.8 km/h
6. 19.3 × 40 = 772 g
7. 60 ÷ 0.015 = 4000 N/m²
8. 3.2 × 50 000 = 160 000 cm = 1.6 km
9. 047° + 180° = 227°
10. 7π = 22.0 cm
11. Arc = 36/360 × 20π = 2π cm. Perimeter = 20 + 2π = 26.3 cm.
12. 5π + 10 = 25.7 cm

Where marks are usually lost

• Using 100 or 1000 for area and volume conversions instead of 10 000 and 1 000 000.
• Writing 4 h 45 min as 4.45 h.
• Dividing density or speed without converting both parts to matching units first.
• Mixing up which place the bearing is “from”.
• Writing a bearing with one or two figures.
• Stopping at the arc when the question wants the whole perimeter of the sector.
• Using d in the formula 2πr, or r in πd.
• Rounding early and then giving a final answer that is out by a unit in the last figure.
• Leaving off units, or giving the wrong one, such as cm for an area.
• Not saying which pack is the better buy after a unit-price comparison.

Official syllabus

OCR GCSE (9-1) Mathematics J560 specification, Version 2.1 (August 2026), sections 10.01 and 10.02, published by OCR.