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I was wondering if it's possible to create these relatively basic shapes from the primary geometric shapes (i.e. polygons and ellipses - particularly circles, squares, and triangles).

You don't necessarily have to use Illustrator, that's just what I use (pathfinder, shape builder, boolean operations, etc.). It's probably more related to geometry and tech drawing with a straight edge & compass than anything.

enter image description here

These were done relatively freehand and by tracing curves with the pencil & pen/Bézier curve tools. None came out perfectly symmetrical, so I split them down the middle, and paired the best half of each one with its mirror image.

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When you draw two overlaping circles and a line, a simple shield shape (similar to a gothic arc) emerges in the overlap:

If you move the circles closer to each other, you'll get a narrower shield:

If you instead move the line upwards, you can extend the circle segments on the sides:

If you combine three circles, you can get a circle segment at the top:

And if you throw in a rectangle and some additional circles, you can start building more complex shapes:

When you have build a geometry you like, you can use Shape Builder Tool to combine the shapes like this:

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    Protip: When you made the shield, hit ctrl+x to cut the shield then do a ctrl+a and delete. Its faster. – joojaa Nov 24 '19 at 19:08
  • @joojaa, that's right, but it looks strange on a gif 😊. – Wolff Nov 24 '19 at 20:06
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Creating one half of a shield, and then duplicating and reflecting is probably the simplest way to get a perfectly symmetrical shape.

You could certainly start by using geometric shapes such as the Ellipse tool and Rectangle tool.

Here's an example shown below where I drew some ellipses and a rectangle to create the basic shape (left), then used the Shape Builder tool Shift+M to create the desired shape (middle), then deleted the other parts by holding down Alt and clicking on them.

Then I tweaked the Bézier curves using the Direct Selection tool A, then made a copy of the shape and reflected it (left), and moved it into position using Smart Guides to help align everything.

enter image description here

Finally, the two shapes can then be united using the Shape Builder tool by clicking and dragging it across both shapes.

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