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Mensuration

Mensuration

Units, perimeter and area

2D Basic Shapes Core + Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. Converting units
  4. Areas of 2D shapes
  5. Compound shapes
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 5.1, 5.2, 5.5: units, perimeter and area
  • 5.1Convert lengths, areas and volumes between units, and volume ↔ capacity.
  • 5.2Find the perimeter and area of a rectangle, triangle, parallelogram and trapezium.
  • 5.5Find areas of compound shapes, and work backwards to a missing length.

Next: Circles, arcs and sectors.

Formula bank

RectangleLearn it\(A=lw\)Use it when rectangles and squares.
TriangleOn the formula sheet\(A=\tfrac12bh\)h is the PERPENDICULAR height.Use it when triangles.
ParallelogramLearn it\(A=bh\)perpendicular height, not the slanting side.Use it when parallelograms.
TrapeziumLearn it\(A=\tfrac12(a+b)h\)a and b are the parallel sides.Use it when trapezia.
Unit conversionsLearn it1 m² = 100² cm² = 10 000 cm²; 1 m³ = 100³ cm³ = 1 000 000 cm³; 1 cm³ = 1 ml; 1000 cm³ = 1 litre; 1 m³ = 1000 litresUse it when changing area or volume units.

Converting units

1 m = 100 cm, but a square metre is 100 cm by 100 cm, so 1 m² = 10 000 cm². A cubic metre is 100 × 100 × 100 = 1 000 000 cm³. Capacity: 1 cm³ = 1 ml, so 1 litre = 1000 cm³ and 1 m³ = 1000 litres.

General methodHow to convert an area or volume

  1. Find the length conversion (e.g. 1 m = 100 cm).
  2. Area: square it (× or ÷ 100² = 10 000). Volume: cube it (× or ÷ 100³ = 1 000 000).
  3. Going to a bigger unit (cm² → m²): divide. Going to a smaller unit: multiply.
  4. Capacity: cm³ = ml; ÷ 1000 for litres.

Example Convert 650 cm² to m², and 0.024 m³ to cm³.

Areas use 100², volumes 100³.

  1. \(650\div10\,000=0.065\)
  2. \(0.024\times1\,000\,000=24\,000\)

Answer 0.065 m² and 24 000 cm³

Try it yourself

1
Convert 3 m² to cm².
Show answer

Answer 30 000

2
Convert 3.5 litres to cm³.
Show answer

Answer 3500

3
Convert 45 000 cm² to m².
Show answer

Answer 4.5

Areas of 2D shapes

Trapeziumb = 10 cma = 6 cmh = 4
Trapezium: ½ × (6 + 10) × 4 = 32 cm².

Every area formula uses the perpendicular height - the distance at right angles to the base, not a slanting side. Perimeter is the distance all the way round (add the sides).

General methodHow to find the area of a 2D shape

  1. Name the shape and pick its formula.
  2. Find the base and the perpendicular height (or the two parallel sides for a trapezium).
  3. Substitute; give units squared (cm², m²).
  4. Missing length: put the area into the formula and solve.

Example A trapezium has area 52 cm² and parallel sides 5 cm and 8 cm. Find its height.

Put the area into the formula and solve for h.

  1. \(\tfrac12(5+8)h=52\)
  2. \(6.5h=52\)

Answer h = 8 cm

Try it yourself

1
Area of a parallelogram with base 9 cm and perpendicular height 5 cm (cm²)?
Show answer

Answer 45

2
Area of a triangle with base 12 cm and height 7 cm (cm²)?
Show answer

Answer 42

3
A triangle has area 30 cm² and base 12 cm. Height (cm)?
Show answer

Answer 5

Compound shapes

Compound shape10634
Split the L-shape, or take a corner away.

Area: \(10\times6-4\times3=48\) cm² (big rectangle minus the missing corner) - or split into two rectangles. Perimeter: the L has the same perimeter as the 10 by 6 rectangle: \(2(10+6)=32\) cm.

General methodHow to find the area of a compound shape

  1. Split it into rectangles, triangles and trapezia - OR start with a big shape and subtract the missing piece.
  2. Work out any missing lengths from the ones given.
  3. Find each area and add (or subtract).

Try it yourself

1
A rectangle 12 cm by 8 cm has a right-angled triangle with legs 12 cm and 3 cm cut off one long side. Area left (cm²)?
Show answer

Answer 78

Ways they test it

Type 1 Unit conversion

"Convert 780 cm² into m²." (1 mark)

Spot it two different area or volume units.

  1. Square (area) or cube (volume) the length conversion.

Type 2 Area of a shape

"Find the area of the trapezium." (2 marks)

Spot it a labelled shape.

  1. Right formula, perpendicular height.

Type 3 Missing length

"The area is 52 cm². Find h." (2 marks)

Spot it the area is given.

  1. Formula = area; solve.

Type 4 Compound shape

"Find the shaded area" / "the perimeter of the shape". (3 marks)

Spot it an L-shape or a shape with a piece removed.

  1. Split or subtract.
  2. Perimeter: add only the outside edges.

Mistakes to avoid

  • ✗ 650 cm² = 6.5 m²✓ 0.065 m²Areas convert by 100², not 100.
  • ✗ Using the slant side as the height✓ the perpendicular heightHeight is at right angles to the base.
  • ✗ Trapezium: a × b × h✓ \(\tfrac12(a+b)h\)Add the parallel sides.

Summary

  • Areas convert by the length factor squared; volumes by it cubed.
  • 1 cm³ = 1 ml; 1000 cm³ = 1 litre; 1 m³ = 1000 litres.
  • Triangle ½bh, parallelogram bh, trapezium ½(a + b)h - perpendicular heights.
  • Compound shapes: split, or big shape minus the missing piece.

If you see …

  • cm² ↔ m²÷ or × 10 000.
  • an area given and a length missingformula = area, then solve.

Practice

Practice: easy, medium, hard

Easy

1
Convert 650 cm² to m².
Show answer

Answer 0.065

2
Convert 0.024 m³ to cm³.
Show answer

Answer 24 000

3
Area of a trapezium with parallel sides 6 cm and 10 cm, height 4 cm (cm²)?
Show answer

Answer 32

Medium

1
Convert 2.4 m³ to litres.
Guide me

What is it asking? litres

What do we know? 1 m³ = 1000 litres

Plan × 1000

Show working
  1. 2400 litres

Answer 2400

2
A trapezium has area 52 cm² and parallel sides 5 cm and 8 cm. Height (cm)?
Guide me

What is it asking? a length

What do we know? \(\tfrac12(13)h=52\)

Plan solve

Show working
  1. \(6.5h=52\)
  2. h = 8

Answer 8

3
An L-shape is a 10 cm by 6 cm rectangle with a 4 cm by 3 cm corner removed. Area (cm²)?
Guide me

What is it asking? an area

What do we know? big rectangle

Plan minus the corner

Show working
  1. 60 − 12 = 48

Answer 48

Hard

1
Convert 5 km² to m².
Guide me

What is it asking? m²

What do we know? 1 km = 1000 m

Plan × 1000²

Show working
  1. 5 × 1 000 000

Answer 5 000 000

PartAnswer / workingMarksGuidance
× 1000²M1
5 000 000A1
Total: 2 marks
2
A trapezium has parallel sides 7 cm and 13 cm and area 60 cm². Find its height (cm).
Guide me

What is it asking? a length

What do we know? \(\tfrac12(20)h=60\)

Plan 10h = 60

Show working
  1. h = 6

Answer 6

PartAnswer / workingMarksGuidance
\(\tfrac12(7+13)h=60\)M1
6A1
Total: 2 marks
3
Perimeter of the L-shape (10 by 6 with a 4 by 3 corner removed), in cm?
Guide me

What is it asking? a perimeter

What do we know? same as the full rectangle

Plan 2(10 + 6)

Show working
  1. 32 cm

Answer 32

PartAnswer / workingMarksGuidance
32B1
Total: 1 mark

Can I do this?

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

Quiz

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