Extended. Double every length of a cube and it needs four times the paint but holds eight times the water. Areas scale by the square of the length factor; volumes by the cube.
If lengths are multiplied by \(k\), areas (including surface areas) are multiplied by \(k^2\) and volumes (and masses or capacities of the same material) by \(k^3\). Always get back to the LENGTH factor \(k\) first.
Fig. 1: enlarging a 1 cm cube by scale factor 2. Each face is made of 2² = 4 small squares and the cube of 2³ = 8 small cubes
Section 01
Three scale factors.
In the lab, enlarge a solid and watch three readouts: length, surface area and volume. Set the length scale factor to 2, then 3, then 0.5, and check that the area factor is always its square and the volume factor its cube.
Length factor
Area factor
Volume factor
\(k\)
\(k^2\)
\(k^3\)
2
4
8
3
9
27
\(\tfrac{1}{2}\)
\(\tfrac{1}{4}\)
\(\tfrac{1}{8}\)
Why: an area is length \(\times\) length, so it picks up \(k\) twice; a volume is length \(\times\) length \(\times\) length, so it picks up \(k\) three times.
In ratio form: if lengths are in the ratio \(a : b\), areas are \(a^2 : b^2\) and volumes \(a^3 : b^3\).
Worked example Area from a length ratio
Two similar triangles have corresponding sides in the ratio \(2 : 5\). The smaller triangle has area 12 cm2. Find the area of the larger.
Length factor \(k = \dfrac{5}{2} = 2.5\).
Area factor \(k^2 = 2.5^2 = 6.25\).
Larger area \(= 12 \times 6.25 = 75\) cm2.
Answer: 75 cm2
Examiner trap
Multiplying an area by the LENGTH factor is the most common error: \(12 \times 2.5 = 30\) cm2 is wrong. Areas need \(k^2\).
Section 02
Working backwards.
When you are given two areas or two volumes, take the square root or cube root to get the length factor first. From there you can reach any other factor.
Area ratio to length: \(k = \sqrt{\text{area factor}}\).
Volume ratio to length: \(k = \sqrt[3]{\text{volume factor}}\).
Area to volume (or back): go through \(k\). Square root, then cube; or cube root, then square.
Worked example Length from areas
Two similar shapes have areas 20 cm2 and 45 cm2. A side of the smaller shape is 8 cm. Find the matching side of the larger.
Area factor \(= 45 \div 20 = 2.25\).
Length factor \(k = \sqrt{2.25} = 1.5\).
Matching side \(= 8 \times 1.5 = 12\) cm.
Answer: 12 cm
Worked example Volume from surface areas
Two similar solids have surface areas in the ratio \(16 : 25\). The smaller has volume 128 cm3. Find the volume of the larger.
Area factor \(= \dfrac{25}{16}\), so \(k = \sqrt{\dfrac{25}{16}} = \dfrac{5}{4}\).
Never go straight from an area ratio to a volume ratio by multiplying by the area factor. Find \(k\) first: from areas 16 : 25 the volume factor is \((5/4)^3\), not \((25/16)\) and not \((25/16)^2\).
Stage 1 · Learn
Check your understanding
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Two similar shapes have length scale factor 3. What is the area scale factor?
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Two similar solids have length scale factor 2. What is the volume scale factor?
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The area scale factor between two similar shapes is 25. What is the length scale factor?
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The volume scale factor between two similar solids is 64. What is the area scale factor?
Section 03
Capacity, mass and paint.
In context, decide whether each quantity behaves like a length, an area or a volume. Capacity and mass (of the same material) go with volume; paint, label size and surface area go with area.
Worked example Similar bottles
Two similar bottles have heights 12 cm and 18 cm. The smaller bottle holds 400 ml. How much does the larger bottle hold?
Length factor \(k = 18 \div 12 = 1.5\).
Capacity is a volume, so multiply by \(k^3 = 1.5^3 = 3.375\).
\(400 \times 3.375 = 1350\) ml.
Answer: 1350 ml (1.35 litres)
Your turn
A model car is made to a scale of 1 : 20. The real car has a surface area of 12 m2. Find the surface area of the model in cm2.
Area factor \(= 20^2 = 400\), so the model has \(12 \div 400 = 0.03\) m2. \(1\) m2 \(= 10\,000\) cm2, so \(0.03 \times 10\,000 = 300\) cm2.
Examiner trap
Mass scales with volume only for solids made of the same material. And a map scale of 1 : 50 000 means AREAS shrink by \(50\,000^2\), not by 50 000.
Stage 2 · Exam
Exam-style questions
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Two similar cylinders have heights 4 cm and 10 cm. The smaller has volume 24 cm3. Find the volume of the larger.
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Two similar shapes have areas 36 cm2 and 81 cm2. A side of the smaller shape is 10 cm. Find the matching side of the larger shape.
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Two similar solids have volumes 54 cm3 and 250 cm3. The smaller has surface area 45 cm2. Find the surface area of the larger.
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Two similar statues are made of the same metal. Their heights are 15 cm and 45 cm. The smaller has mass 0.8 kg. Find the mass of the larger.
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Areas squared, volumes cubed.
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