Matemáticas
Scale Factor: k for Lengths, k Squared for Area, k Cubed for Volume
Multiply lengths by k and areas grow by k², volumes by k³. Why doubling a shape quadruples its area, plus fractional and negative scale factors.
Scale factor
A scale factor is the number by which every length in a shape is multiplied to produce an enlarged or reduced copy of the same shape.
- También llamado
- Linear scale factor
- Dónde lo encuentra el alumnado
- Grade 7 mathematics in enlargement work, and again at GCSE where area and volume scale factors are needed for similar shapes and solids.
La respuesta corta
A scale factor is the number every length in a shape is multiplied by to produce a similar copy of it. If lengths are multiplied by k, areas are multiplied by k squared and volumes by k cubed. Doubling every length therefore multiplies area by four and volume by eight.
Un ejemplo
k = 3 → lengths × 3, area × 9, volume × 27
A rectangle 3 cm by 2 cm has an area of 6 cm². Enlarge it by a scale factor of 2 and it becomes 6 cm by 4 cm, with an area of 24 cm² — four times the original, not twice. The reason is that both dimensions grew, so the doubling happened twice over. A cube behaves the same way in three directions: side 3 cm gives 27 cm³, side 6 cm gives 216 cm³, and 216 is eight times 27.
One factor, three different jobs
The linear scale factor k applies to anything measured in one direction: sides, heights, radii, perimeters, diagonals. Perimeter is included because it is a sum of lengths, which catches people out — a perimeter does not grow by k squared.
Anything measured in two directions grows by k², and anything measured in three by k³. Surface area is an area, so it takes k² even though it belongs to a solid.
- Lengths, perimeters, radii, heights — multiply by k
- Areas, including surface areas — multiply by k²
- Volumes and capacities — multiply by k³
- Angles — unchanged, because enlargement does not distort shape
Why doubling the lengths quadruples the area
Area is a product of two lengths. Double both of them and you have multiplied by two twice, which is four. Volume is a product of three lengths, so doubling all three multiplies by eight.
Working backwards is where care is needed. If two similar shapes have areas in the ratio 25 to 9, the linear scale factor is not 25/9 — it is the square root, 5/3. If two similar solids have volumes in the ratio 8 to 1, the lengths are in the ratio 2 to 1, from the cube root.
Factors below one, and factors below zero
A scale factor between 0 and 1 makes the shape smaller. GCSE still calls this an enlargement, which reads oddly the first time: an enlargement by a scale factor of ½ halves every length and leaves a quarter of the area.
A negative scale factor sends the image to the opposite side of the centre of enlargement and turns it upside down. A factor of −2 doubles every length and rotates the shape half a turn about that centre. The area factor is still 4, because squaring removes the sign.
Map scales are the same idea written as a ratio. A 1:25 000 map has a scale factor of 1/25 000, so 1 cm on the paper is 25 000 cm on the ground, which is 250 m. Area works on the square of that, which is why a small patch of map covers a surprising amount of country.
Preguntas frecuentes
If the scale factor is 3, what happens to the area?
It is multiplied by 9, because area involves two lengths and each of them tripled. Volume would be multiplied by 27. Angles do not change at all — an enlargement stretches a shape without bending it, which is what makes the original and the image similar.
How do you find the scale factor between two similar shapes?
Divide a length on the image by the matching length on the original, keeping the pair corresponding. If you are given areas instead, take the square root of their ratio; if given volumes, take the cube root. Using the raw area ratio as a length factor is the most common error on this topic.
Can a scale factor be a fraction?
Yes. A factor between 0 and 1 produces a smaller copy, and GCSE still describes it as an enlargement. A scale factor of ⅓ divides every length by three and leaves one ninth of the area. The word enlargement here names the transformation, not the direction of the change.
What does a negative scale factor do?
It places the image on the opposite side of the centre of enlargement and inverts it, which is the same as an enlargement followed by a half-turn about that centre. The sizes follow the absolute value, so −3 scales lengths by 3, areas by 9 and volumes by 27.
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