Covering One Triangle with Another - An Elegant Proof
Tuesday 5 May 2026 at 22:29
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Edited by Richard Walker, Tuesday 5 May 2026 at 22:55
Covering problems, which ask how a shape can be covered with other shapes, are part of what's called combinatorial mathematics. They often appear in recreational mathematics. have applications to real-world problems such as siting mobile phone mast to get adequate coverage, and are a subject of active current research.
Covering problems are often easy to state but even in simple cases the answers can be difficult to establish, because when you are arranging a bunch of shapes it's hard to be sure all the possibilities have been thought of.
One I thought of the other day and posted in this blog is
What is the smallest equilateral triangle that can cover every triangle whose longest side has length 1?
This is about as simple as it gets but it's not trivial. The first idea you might have, an equilateral triangle with sides of length 1, turns out not to be the answer; a bigger triangle is needed.
I haven't proved to my satisfaction what the smallest possible answer is but I can prove the following.
Any triangle whose longest side is 1 can be covered by an equilateral triangle of side length .
To see this consider Figures 1 and 2 below.
In Figure 1 AB is the longest side of the triangle we wish to cover, so its length is 1. Where can the third vertex of the triangle, call it C, be located?
If we draw circles of radius 1 centered at A and B then C must be in the lens-shaped region AXBY; if not, C would be more than 1 away from at least one of A and B , contradicting AB being the longest side.
From symmetry it is enough to just consider the shaded sector in Figure 1. In Figure 2 we see this sector is covered by equilateral triangle AX1B1, which therefore covers all three vertices of the triangle we want to cover and thus covers the whole of that triangle.
The side length of AX1B1 is , which completes the proof.
Covering One Triangle with Another - An Elegant Proof
Covering problems, which ask how a shape can be covered with other shapes, are part of what's called combinatorial mathematics. They often appear in recreational mathematics. have applications to real-world problems such as siting mobile phone mast to get adequate coverage, and are a subject of active current research.
Covering problems are often easy to state but even in simple cases the answers can be difficult to establish, because when you are arranging a bunch of shapes it's hard to be sure all the possibilities have been thought of.
One I thought of the other day and posted in this blog is
What is the smallest equilateral triangle that can cover every triangle whose longest side has length 1?
This is about as simple as it gets but it's not trivial. The first idea you might have, an equilateral triangle with sides of length 1, turns out not to be the answer; a bigger triangle is needed.
I haven't proved to my satisfaction what the smallest possible answer is but I can prove the following.
To see this consider Figures 1 and 2 below.
In Figure 1 AB is the longest side of the triangle we wish to cover, so its length is 1. Where can the third vertex of the triangle, call it C, be located?
If we draw circles of radius 1 centered at A and B then C must be in the lens-shaped region AXBY; if not, C would be more than 1 away from at least one of A and B , contradicting AB being the longest side.
From symmetry it is enough to just consider the shaded sector in Figure 1. In Figure 2 we see this sector is covered by equilateral triangle AX1B1, which therefore covers all three vertices of the triangle we want to cover and thus covers the whole of that triangle.
The side length of AX1B1 is , which completes the proof.