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Mensuration

Mensuration

Prisms and cylinders

3D Basic Shape Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Volume of a prism
  4. Surface area
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 5.4: prisms and cylinders
  • 5.4Find the volume and surface area of a cuboid, any prism and a cylinder.
  • 5.4Give answers in terms of \(\pi\) and convert to capacity (ml, litres).
  • 5.4Work backwards to a missing length.

Before you start: Circles and nets. Next: Pyramids, cones and spheres.

Formula bank

PrismOn the formula sheet\(V=\) area of cross-section \(\times\) lengthany solid with the same cross-section all the way through.Use it when prisms, cuboids, cylinders.
CuboidLearn it\(V=lwh\); surface area \(=2(lw+lh+wh)\)Use it when boxes.
CylinderOn the formula sheet\(V=\pi r^{2}h\); curved surface area \(=2\pi rh\)total surface area \(=2\pi rh+2\pi r^{2}\) (closed).Use it when cylinders.
CapacityLearn it1 cm³ = 1 ml; 1000 cm³ = 1 litreUse it when 'how many litres'.

Volume of a prism

Triangular prismcross-sectionlength
A prism has the same cross-section all along its length.
Cylinderrh
A cylinder is a prism with a circular cross-section.

Volume = (area of the end) × (length). Cuboid: \(5\times4\times3=60\) cm³. Cylinder radius 3, height 10: \(\pi\times9\times10=90\pi\approx282.74\) cm³ (= 282.74 ml).

General methodHow to find the volume of any prism

  1. Find the area of the cross-section (the end shape): rectangle, triangle, trapezium, circle…
  2. Multiply by the length (the distance between the two ends).
  3. Units cubed (cm³); for capacity 1 cm³ = 1 ml.
  4. Missing length: volume ÷ cross-section area.

Example A prism has a right-angled triangle cross-section with legs 6 cm and 8 cm, and length 12 cm. Find its volume.

Triangle area × length.

  1. cross-section \(\tfrac12\times6\times8=24\) cm²
  2. \(24\times12\)

Answer 288 cm³

Try it yourself

1
Volume of a cuboid 5 cm × 4 cm × 3 cm (cm³)?
Show answer

Answer 60

2
Volume of a cylinder, radius 3 cm, height 10 cm, in terms of π. Type the number in front of π.
Show answer

Answer 90

3
A cylinder of radius 4 cm has volume 500 cm³. Height (2 d.p.)?
Hint
\(500\div(16\pi)\).
Show answer

Answer 9.95

Surface area

General methodHow to find the surface area of a prism or cylinder

  1. Picture the net: two ends + the faces round the sides.
  2. Prism: 2 × end area + perimeter of the end × length.
  3. Cylinder: curved surface \(2\pi rh\) + the two circles \(2\pi r^{2}\) (leave out a lid if it is open).
  4. Add; units squared.

Example Find the total surface area of a closed cylinder of radius 3 cm and height 10 cm.

Curved part + two circles.

  1. curved \(2\pi\times3\times10=60\pi\)
  2. ends \(2\times9\pi=18\pi\)

Answer \(78\pi\approx245.04\) cm²

Try it yourself

1
Surface area of a cuboid 5 cm × 4 cm × 3 cm (cm²)?
Show answer

Answer 94

2
Surface area of the triangular prism (legs 6 and 8, hypotenuse 10, length 12) in cm²?
Hint
2 × 24 + (6 + 8 + 10) × 12.
Show answer

Answer 336

Ways they test it

Type 1 Volume

"Calculate the volume of the cylinder." (2 marks)

Spot it a prism or cylinder with lengths.

  1. Cross-section area × length.

Type 2 Surface area

"Calculate the total surface area of the closed cylinder." (3 marks)

Spot it 'surface area'.

  1. List the faces (think net).
  2. Add them.

Type 3 Capacity and missing lengths

"The tank holds 30 litres. Find its height." (3 marks)

Spot it litres or a volume given.

  1. Litres → cm³ (× 1000).
  2. ÷ cross-section area.

For example 40 cm × 30 cm base, 30 litres

  1. 30 000 ÷ 1200

Answer 25 cm

Mistakes to avoid

  • ✗ Cylinder volume \(2\pi rh\)✓ \(\pi r^{2}h\)\(2\pi rh\) is the curved AREA.
  • ✗ Forgetting the two ends✓ add both circles (or end faces)Unless it is open.
  • ✗ cm³ ÷ 100 for litres✓ ÷ 10001000 cm³ = 1 litre.

Summary

  • Prism volume = cross-section area × length.
  • Cylinder: \(V=\pi r^{2}h\), curved area \(2\pi rh\).
  • Surface area = add every face (think of the net).
  • 1 cm³ = 1 ml; 1000 cm³ = 1 litre.

If you see …

  • "volume" of a prismarea of the end × length.
  • "open" containerleave out the missing face.

Practice

Practice: easy, medium, hard

Easy

1
Volume of a cube of edge 3 cm (cm³)?
Show answer

Answer 27

2
A prism has cross-section 9 cm² and length 6 cm. Volume (cm³)?
Show answer

Answer 54

3
Curved surface area of a cylinder, radius 3 cm, height 10 cm, in terms of π. Type the number in front of π.
Show answer

Answer 60

Medium

1
Volume of a cylinder, radius 4 cm, height 10 cm (2 d.p.)?
Guide me

What is it asking? a volume

What do we know? \(\pi\times16\times10\)

Plan calculator

Show working
  1. \(160\pi=502.65\)

Answer 502.65

2
Total surface area of a closed cylinder, radius 4 cm, height 10 cm (2 d.p.)?
Guide me

What is it asking? an area

What do we know? \(2\pi(4)(10)+2\pi(16)\)

Plan add

Show working
  1. \(80\pi+32\pi=112\pi\)

Answer 301.59

3
A tank 40 cm by 30 cm by 25 cm. Capacity in litres?
Guide me

What is it asking? litres

What do we know? volume in cm³

Plan ÷ 1000

Show working
  1. 30 000 cm³ = 30 litres

Answer 30

Hard

1
A cylinder of radius 4 cm holds 500 cm³. Find its height (2 d.p.).
Guide me

What is it asking? a height

What do we know? \(\pi\times16\times h=500\)

Plan divide

Show working
  1. \(h=500\div16\pi=9.95\)

Answer 9.95

PartAnswer / workingMarksGuidance
\(16\pi h=500\)M1
9.95A1
Total: 2 marks
2
A triangular prism: right-angled cross-section with legs 6 cm and 8 cm, length 12 cm. Total surface area (cm²)?
Guide me

What is it asking? an area

What do we know? hypotenuse 10 (Pythagoras)

Plan 2 ends + three rectangles

Show working
  1. 2 × 24 = 48
  2. (6 + 8 + 10) × 12 = 288
  3. 336

Answer 336

PartAnswer / workingMarksGuidance
hypotenuse 10B1
48 + 288M1
336A1
Total: 3 marks
3
An open cylinder (no lid), radius 3 cm, height 10 cm. Outside surface area in terms of π. Type the number in front of π.
Guide me

What is it asking? an area

What do we know? curved \(60\pi\)

Plan one base \(9\pi\)

Show working
  1. \(60\pi+9\pi=69\pi\)

Answer 69

PartAnswer / workingMarksGuidance
\(60\pi+9\pi\)M1
\(69\pi\)A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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