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Similarity

Similar shapes: areas and volumes

Similarity Paper 1 + 2 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Why squares and cubes
  4. Triangles with the same height
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 4.4: areas and volumes of similar shapes
  • 4.4Use: lengths × \(k\), areas × \(k^{2}\), volumes × \(k^{3}\).
  • 4.4Work back from an area or volume ratio to the length scale factor.
  • 4.4Use triangles with the same height: area ratio = base ratio.

Before you start: Similar shapes: lengths.

Formula bank

Length, area, volumeLearn itlength × \(k\); area × \(k^{2}\); volume × \(k^{3}\)k = the LENGTH scale factor.Use it when similar shapes or solids.
Back to kLearn it\(k=\sqrt{\text{area ratio}}\) or \(k=\sqrt[3]{\text{volume ratio}}\)Use it when given two areas or two volumes.
Same heightLearn ittriangles with the same height: areas in the ratio of their basesUse it when triangles sharing a vertex and a base line.

Why squares and cubes

Lengths × 3 gives area × 913area 1area 9
Sides × 3 → area × 9.

Area multiplies two lengths, each × k, so area × \(k^{2}\). Volume multiplies three lengths: × \(k^{3}\). Mass of the same material follows volume.

General methodHow to answer any similar area/volume question

  1. Find the LENGTH scale factor k (from two matching lengths, or by square-rooting an area ratio, or cube-rooting a volume ratio).
  2. Area wanted: multiply by k². Volume (or capacity, mass) wanted: multiply by k³. Length wanted: × k.
  3. Use the same units for both shapes when finding k.

Example Two similar cylinders have heights 6 cm and 9 cm. The small one has volume 160 cm³. Find the large one's volume.

k = 1.5, then k³.

  1. \(k=9\div6=1.5\)
  2. \(160\times1.5^{3}\)

Answer 540 cm³

Example 2 Two similar jugs hold 250 ml and 54 ml. The large one has surface area 400 cm². Find the small one's.

Cube root for k, then square it.

  1. \(k^{3}=\tfrac{54}{250}=0.216\Rightarrow k=0.6\)
  2. \(400\times0.6^{2}\)

Answer 144 cm²

Try it yourself

1
Lengths × 2; small area 5 cm². Large area (cm²)?
Show answer

Answer 20

2
Volumes 8 : 27. Ratio of lengths: small ÷ large as a fraction?
Show answer

Answer 2/3

3
Every length of a model increases by 10%. By what % does its volume increase?
Show answer

Answer 33.1

Triangles with the same height

Triangles with the same heightABDC4610?h
ABD and ADC share the height h from A.

Area = ½ × base × h with the same h, so the areas are in the ratio of the bases: 4 : 6. If area ABD = 10 cm², area ADC = 15 cm².

Try it yourself

1
D is on BC with BD = 4, DC = 6; area ABD = 10 cm². Area ABC (cm²)?
Show answer

Answer 25

Ways they test it

Type 1 Volume from lengths

"Two similar vases have heights 15 cm and 25 cm. The smaller holds 540 ml. How much does the larger hold?" (3 marks)

Spot it two lengths and a volume.

  1. k = 25/15 = 5/3.
  2. × k³.

For example \(540\times\left(\tfrac53\right)^{3}\)

    Answer 2500 ml

    Type 2 Length from volumes or areas

    "Similar bottles hold 250 ml and 54 ml. The larger is 20 cm tall. Height of the smaller?" (3 marks)

    Spot it two volumes and one length.

    1. Cube root of the volume ratio.
    2. × k.

    For example \(k=0.6\)

      Answer 12 cm

      Type 3 Scale model

      "A 12 cm model of a 4.8 m car. Model windscreen 6 cm². Real windscreen in m²?" (3 marks)

      Spot it a model and a real object.

      1. Same units for k (480 ÷ 12 = 40).
      2. Area × k², convert units.

      For example \(6\times40^{2}=9600\) cm²

        Answer 0.96 m²

        Type 4 Same height

        "Area of triangle ABD is 10 cm², BD : DC = 4 : 6. Find area ABC." (2 marks)

        Spot it a point on the base of a triangle.

        1. Areas in the ratio of the bases.

        Mistakes to avoid

        • ✗ volume × k✓ volume × k³Three dimensions.
        • ✗ k = area ratio✓ k = √(area ratio)Go back to lengths first.
        • ✗ mixing cm and m in k✓ same unitsk must be a pure number.

        Summary

        • Length × k, area × k², volume × k³.
        • From areas: k = √ratio; from volumes: k = ∛ratio.
        • Triangles with the same height: area ratio = base ratio.

        If you see …

        • two volumes and a lengthcube-root the volume ratio.
        • a model and the real thingsame units, then k² or k³.

        Practice

        Practice: easy, medium, hard

        Easy

        1
        Lengths × 3. By what number is the area multiplied?
        Show answer

        Answer 9

        2
        Lengths × 3. By what number is the volume multiplied?
        Show answer

        Answer 27

        3
        Areas 16 and 25 (similar shapes). Length scale factor small → large?
        Show answer

        Answer 1.25

        Medium

        1
        Two similar cylinders, heights 6 cm and 9 cm; the small surface area is 80 cm². Large surface area (cm²)?
        Guide me

        What is it asking? an area

        What do we know? k = 1.5

        Plan × k²

        Show working
        1. 80 × 2.25 = 180

        Answer 180

        2
        Similar shapes have areas 20 cm² and 45 cm². The small one is 8 cm long. Length of the large (cm)?
        Guide me

        What is it asking? a length

        What do we know? k² = 2.25

        Plan k = 1.5

        Show working
        1. 8 × 1.5 = 12

        Answer 12

        3
        Similar bottles hold 250 ml and 54 ml. The larger is 20 cm tall. Height of the smaller (cm)?
        Guide me

        What is it asking? a length

        What do we know? k³ = 0.216

        Plan k = 0.6

        Show working
        1. 20 × 0.6 = 12

        Answer 12

        Hard

        1
        Two similar vases have heights 15 cm and 25 cm. The smaller holds 540 ml. How much does the larger hold (ml)?
        Guide me

        What is it asking? a volume

        What do we know? k = 5/3

        Plan × k³

        Show working
        1. \(540\times\tfrac{125}{27}=2500\)

        Answer 2500

        PartAnswer / workingMarksGuidance
        k = 5/3B1
        × k³M1
        2500A1
        Total: 3 marks
        2
        A 12 cm model of a 4.8 m car has a 6 cm² windscreen. Area of the real windscreen in m²?
        Guide me

        What is it asking? an area in m²

        What do we know? k = 480 ÷ 12 = 40

        Plan 6 × 1600 cm², ÷ 10 000

        Show working
        1. 9600 cm²
        2. 0.96 m²

        Answer 0.96

        PartAnswer / workingMarksGuidance
        k = 40B1
        9600 cm²M1
        0.96 m²A1
        Total: 3 marks
        3
        Every length of a model is increased by 10%. By what % does its area increase?
        Guide me

        What is it asking? a percentage

        What do we know? k = 1.1

        Plan k² = 1.21

        Show working
        1. 21% increase

        Answer 21

        PartAnswer / workingMarksGuidance
        1.21M1
        21%A1
        Total: 2 marks

        Can I do this?

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

        Quiz

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