In the figure below the area of the red hexagon is exactly 3 square units. What is the area of the blue hexagon?

This question was not easy but we hope you got it right!

The thing is that the correct answer can be obtained in a number of ways.

The easiest one is to decompose the inner hexagon into six triangles (equilateral ones) to determine the radius of the circle. Then, use that radius to determine the area of the larger hexagon.

There is also another way - you can decompose the inner hexagon into a series of eighteen triangles identical to the six blue triangles on each face of the hexagon.

Either way, you get four square units as the area.

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What are your thoughts on this subject?

56 Comments

Stevie G

No formulae to work this out you can only do it by decomposing. Poor explanation that I am sure many of the site members will not understand. You have also not given a link to prove your answer.

Mark Gavin

This answer is incorrect. The area of the larger hexagon is 8 divided by square root of 3, which is approximately 4.6188. Decompose the smaller hexagon into six equilateral triangles. Their areas must each be 1/2 to make the total of 3. An equilateral triangle with area 1/2 has sides 2 over root 3 by the cosine rule. The larger hexagon consists of the inner hexagon plus six isoceles triangles with base of 2 over root 3 and opposite angle 120 degrees. Use of cosines gives the length of "half a side" of the larger hexagon as 2/3, so the full length is 4/3. The larger hexagon consists of six equilateral trinagles. Their base is 4/3 and their perpendicular height is the length of the smaller hexagon, 2 over root 3. Hence their area is the product of these, 4 over (3 times root 3). The total area is six times this, which is 6 x 4 = 24 over (3 root 3), or 8 over root 3, which is 4.6188.

howardm

Happy to see a challenging question here, but this isn't exactly "trivia."

Stephen Forest

Just did a quick estimate

hweishaupt

Not Trivia. MATH! 😵‍💫

sanman

True not a sure fire trivia question, but yet a real live head scratcher. Sure got the the folk all fire up talking about it. Don't recall this many comments on many other questions.