Question 2
How many faces does this planar graph have?
Not quite. One way to get 3 is to count only the regions closed in by edges. The region outside the graph is a face too, which makes four. In 2021, 10.0% of Queensland students chose (A).
That’s it. Step through the hints below to see why. In 2021, 81.7% of Queensland students chose (B).
Not quite. 5 is the number of vertices, A, B, C, D and E. One way to get it is to count the dots instead of the regions. In 2021, 4.2% of Queensland students chose (C).
Not quite. 7 is the number of edges, counting the curve as one. One way to get it is to count the lines instead of the regions they enclose. In 2021, 3.4% of Queensland students chose (D).
Count what you can see first. The vertices are A, B, C, D and E, so $v=5$.
The edges are AE, ED, DB, BA, AC, CB and the curve from A to D, so $e=7$. The curve is one edge, however long it looks.
Euler’s formula is in the formula book. The graph is connected and drawn with no edges crossing, so it applies.
$$\begin{aligned}v+f-e&=2\\ 5+f-7&=2\\ f&=4\end{aligned}$$
Check by counting the regions:
- triangle ABC
- the rest of the square, bounded by A, C, B, D and E
- the region between the curve and the edges BA and BD
- the outside, everything around the graph
The outside is a face. Leave it out and you get 3, which is option (A).
$$\begin{aligned}v+f-e&=2\\ 5+f-7&=2\\ f&=4\end{aligned}$$
The graph has 4 faces: three enclosed regions and the outside. The 2021 subject report listed knowing what “planar graph” and “face” mean as a strength, and this question backs that up.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2021 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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