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Circle gaps brainteaser

Ariel Marcy

26 April, 2023

What is the area of the orange star in the centre? The blue circles each have an area of 3 square centimetres, and the big square has sides that are 4 centimetres long.

Four circles in a square. The circles are each labelled 3 and the square side is labelled 4. There is a diamond shape in the middle made by the gap between the circles.

Scroll down or click for a hint, or the answer!

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Brainteaser hint

A good place to start is calculating the area of the big square. You can subtract the areas of the circles, and then try to work out what to do with what’s left!

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Brainteaser answer

There are lots of strange shapes in this puzzle! But with a bit of smart thinking, we can put them together.

We can start by calculating the area of the big square: 4 x 4 = 16 square centimetres.

Then, we can subtract the areas of the circles: 16 – 3 – 3 – 3 – 3 = 4 square centimetres.

What do we have left? We have the diamond in the middle, but we also have some other scraps too. There’s a rounded triangle shape on the top, bottom, left and right sides, and a smaller shape in each corner.

The key is combining these shapes. If we take the rounded triangles from the left and right sides and put them together, we get exactly the same diamond shape as in the middle!

We can do the same with the top and bottom rounded triangles.

And the 4 shapes in the corners will come together to make yet another diamond shape.

All the scraps together have an area of 4 square centimetres, and we can make exactly 4 diamond shapes using all the scraps. So each diamond must have an area of 1 square centimetre!

Categories:

Brainteaser, Maths

Tags:

mathematical sciences, number, problem solving

7 responses

Peter Perry

26 April, 2023

Of course, to touch the edges, the circles would need to have an area of 3.141592653… sq cm. As stated, there would be a gap of less than 0.05 cm between circles and edges, and about 0.09 cm between circles. If circles actually touch, then calculation becomes a little more complex, but still doable by considering the square made by joining circle centres.

Reply
1. David Shaw
  
  26 April, 2023
  
  Yup! Given the target demographic for these puzzles, I thought rounding the areas was the best course of action. The brainteaser initially only had the side length of the square, but the circle area formula is a bit hard for many of our readers.
  
  Feel free to adapt it back for your audiences!
  
  Reply
David Evans

28 April, 2023

I just cut the square up into quarters.
Each quarter was 2×2 = 4. Each quarter had 1 circle = 3.
Therefore the four equal parts (Scraps) around the circle had a combined area of 1. (4-3) = 1.
As the scrap from the middle star was one 1/4 of the star. The star had to have an area = 1.

Reply
1. David Shaw
  
  28 April, 2023
  
  That’s a great way to solve the question, David!
  
  Reply
Missy Lee

27 May, 2023

Love this one!

Reply
Warwick Watson

12 January, 2024

Does this shape have a name? I find it hard to believe that it wouldn’t, given how long this type of math has been done. The Greeks named anything they came across; as did the Egyptians. Any ideas?

Reply
1. David Shaw
  
  15 January, 2024
  
  Hi Warwick,
  I had a bit of a search around and found lots of similar shapes, but not this one.
  it looks similar to an astroid curve, but the construction is different: https://en.wikipedia.org/wiki/Astroid
  in construction, it’s similar to a circular triangle: https://en.wikipedia.org/wiki/Circular_triangle
  but it obviously has 4 sides. So maybe it’s a circular square?
  
  Cheers,
  David
  
  Reply