The two regular pentagons share a vertex and an edge. The side of the larger pentagon is twice that of the smaller one. Which triangle has larger area, red or blue? Or do they have the same area?
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2$\begingroup$ Almost identical to this recent hot network question: math.stackexchange.com/questions/5101157/… $\endgroup$Servaes– Servaes2025-10-13 21:44:27 +00:00Commented Oct 13 at 21:44
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$\begingroup$ @Servaes. Indeed, the underlying idea is the same: Cavalieri’s principle. $\endgroup$Pranay– Pranay2025-10-13 22:03:32 +00:00Commented Oct 13 at 22:03
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$\begingroup$ Nice. And somewhat easier than this pentagon HNQ from a month ago: math.stackexchange.com/q/5095643/207316 $\endgroup$PM 2Ring– PM 2Ring2025-10-14 00:01:50 +00:00Commented Oct 14 at 0:01
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3$\begingroup$ @PM2Ring. Yep, that was much harder. I have another pentagon problem which also seems harder at the moment. I’ll post it here or MSE depending on how it goes. $\endgroup$Pranay– Pranay2025-10-14 00:19:52 +00:00Commented Oct 14 at 0:19
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1 Answer
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The triangle areas are equal.
Consider any of the triangles. It is easy to see (and check) that we can shift its rightmost vertex parallelly to the opposite triangle side to the vertex of the small pentagon.
The shift keeps the triangle area, so the latter is equal to the area of a triangle whose sides are the side of the small pentagon and its diagonals.
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5$\begingroup$ Thanks for adding the figure. It looks great! $\endgroup$Pranay– Pranay2025-10-13 19:28:38 +00:00Commented Oct 13 at 19:28
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1$\begingroup$ @Pranay Sorry for not adding it earlier. I had to work with a paper and then a geometry seminar. $\endgroup$Alex Ravsky– Alex Ravsky2025-10-13 19:34:14 +00:00Commented Oct 13 at 19:34
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5$\begingroup$ @MathKeepsMeBusy Basically, it comes down to the formula for calculating the area of a triangle: ½b×h The figure-as-drawn involves moving corners in a manner that preserves the height, and thus the area. $\endgroup$Chronocidal– Chronocidal2025-10-13 19:56:36 +00:00Commented Oct 13 at 19:56
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2$\begingroup$ Neat. For me the parallelism of the blue/red line pairs wasn't obvious at first, but it follows from the fact that the diagonal of a pentagon is parallel to its opposite side (and the regularity of the shape in general). $\endgroup$Nuclear Hoagie– Nuclear Hoagie2025-10-13 20:28:29 +00:00Commented Oct 13 at 20:28
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5$\begingroup$ @z100. Yep, precisely what I had in mind. It’s worth noting that the statement about the relative sizes of the pentagons is completely irrelevant. It was a red herring. $\endgroup$Pranay– Pranay2025-10-13 20:52:03 +00:00Commented Oct 13 at 20:52