Similar figures
Two figures are similar when one is an enlargement of the other. If you know two sides that correspond to each other, you know the scale factor — and then you can find all the other sides.
Your numbers
Figure 1 — the one you know
Figure 2 — the enlarged or reduced one
The side in figure 2 that is in the same place as a in figure 1.
More options: Volume of figure 1, unit
For similar solids: a box, a bucket, a model.
Answer
Scale factor k
2
Figure 2 is an enlargement of figure 1.
Side b′
8cm
Side c′
10cm
Area (k² = 4)
6 → 24cm²
Volume (k³ = 8)
10 → 80cm³
Figure 1 dashed, figure 2 solid — drawn to the same scale.
How to work it out
- 1k = a′ / a = 6 / 3 = 2The scale factor is how many times bigger figure 2 is. Always divide figure 2 by figure 1 — and use two sides that correspond to each other.
- 2b′ = k · b = 2 · 4 = 8 cm · c′ = k · c = 2 · 5 = 10 cmAll lengths in figure 2 are k times as long as in figure 1. The angles are the same.
- 3A₂ = k² · A₁ = 2² · 6 = 24 cm²An area has two directions — length and width — and both grow by k. Hence k². Typical mistake: multiplying the area by k.
- 4V₂ = k³ · V₁ = 2³ · 10 = 80 cm³Volume has three directions, so it grows by k³: twice the measurements gives eight times as much.
The formula
k = a′ / a
The scale factor: figure 2 divided by figure 1.
b′ = k · b
All lengths are multiplied by k.
A₂ = k² · A₁
Areas are multiplied by k squared.
V₂ = k³ · V₁
Volumes are multiplied by k cubed.
In depth: more numbers and “what if”
Length, area and volume at other scale factors
| k | Side b′ | Area | Volume |
|---|---|---|---|
| 0.5 | 2 cm | 1.5 cm² | 1.25 cm³ |
| 1 | 4 cm | 6 cm² | 10 cm³ |
| 2 | 8 cm | 24 cm² | 80 cm³ |
| 3 | 12 cm | 54 cm² | 270 cm³ |
With k = 2, lengths are multiplied by 2, areas by 4 and volumes by 8.
More: Geometry
- PythagorasFind the missing side, set out a right angle, check whether a room is square, and find the space diagonal.
- Area and perimeterRectangle, triangle, trapezium, circle and composite shapes — area and perimeter.
- Volume and surface areaBox, cylinder, cone, sphere, pyramid, prism, pipe and frustum — also in litres.
- The circleRadius, circumference and area from one measurement — plus sector, annulus, chord and rise.
- Right triangleSine, cosine and tangent: all sides and angles from two known values.
- AnglesThe third angle in a triangle and the angle sum of a polygon.
- Points in the coordinate systemDistance, midpoint and the line between two points — drawn in.
- Any triangleThe sine and cosine rules: find the rest from SSS, SAS, ASA or SSA.
- TransformationsReflect, rotate and translate a figure on a coordinate grid — before and after.