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I have a square with sides that are all the same length. I want to place an equilateral rectangle inside but whilst left to right the triangle is the same length, the triangle is shorter top to bottom.

I can't quite get my head around why an equilateral with all sides and angles being the same would not fit perfectly in a square. What am I missing/how can this be achieved please?

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    Possible Duplicate at another Stack -- math.stackexchange.com/questions/59616/… -- It's essentially a math problem.
    – Scott
    Jul 29 at 6:49
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    This is a basic property of circles and rotation. There is in fact no way of making a square with side length of x and a equilateral triangle with side x so that the corners of the triangle hit the square. If the triangles sides are longer than x then you can but the triangle is rotated by 15 degrees.
    – joojaa
    Jul 29 at 6:54

1 Answer 1

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It will fit inside a square. Here's one possible method using Illustrator. Before you start make sure Smart Guides and Snap to Point are enabled.

  1. Draw a square, rotate it 45 degrees, and drop horizontal and vertical guides to intersect with the top vertex of the square

  2. Draw an equilateral triangle using the Polygon tool. Click and drag it to intersect the top corner with the guides

  3. Click and drag the mid point of the bottom of the triangle, while holding down Shift as you click and drag until crosses over the square

  4. Click and drag, and drop a horizontal guide until it intersects where the lines cross

  5. Click and drag the mid point of the bottom of the triangle, while holding down Shift as you click and drag, until it intersects with the last guide you dropped.

This isn't really as complicated as it sounds. Here's an animation of the process

enter image description here

  1. If you want you can select all and rotate 45 degrees again.
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  • Thanks for your reply. This is the problem I am facing: i.imgur.com/E9xpSYd.png
    – Shaun Hill
    Jul 30 at 7:10
  • @ShaunHill Yeah, it won't fit that way. It's not possible geometrically, only in the orientation shown in my answer.
    – Billy Kerr
    Jul 30 at 8:36
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    @ShaunHill, have a look at this: i.imgur.com/yjhn74z.gif
    – Wolff
    Jul 30 at 8:51
  • @Wolff yeah precisely!
    – Billy Kerr
    Jul 30 at 9:01
  • Thank you for confirming, my head doesn't work well with things that don't seem logical. It's hard to understand why something that is equal in length on all sides wouldn't fit inside a square.
    – Shaun Hill
    Jul 30 at 9:30

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