Triple the Area of a Circle

This is possibly the last Circle Challenge as the math / maths might be getting in the way of what are supported to be accurate design approaches to laying out and scaling circles using the power of Xara.

Can you accurately, without calculation, construct a circle from another so the new circle's area is three times larger?

Attachment 122539

At present I have two construction methods but both involve some in-between scaffolding.

Good thinking,
Acorn

Using a similar method to your twice the area...
Does it involve drawing a hexagon at the end, and then drawing a circle within the bounds of that hexagon?

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Originally Posted by Rik

Using a similar method to your twice the area...
Does it involve drawing a hexagon at the end, and then drawing a circle within the bounds of that hexagon?

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Rik, I don't know how to do that so have a go. I did try with hexagons but could scale up correctly.

Hint: my present approaches use one of the following constructors: (1) a triangle, (2) two circles & (3) one circle.

Acorn

So, does this method work.
I'm sure you don't want me to explain how?!
I'm sure you can duplicate what I've shown!

Rik, I just wish I could prove that mathematically, however it's beyond me! It does APPEAR to work though.

Attachment 122542

Quote:

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Originally Posted by ss-kalm

Rik, I just wish I could prove that mathematically, however it's beyond me! It does APPEAR to work though.

Attachment 122542

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Actually, if I now check the numbers, the exercise we did where we doubled the size of the circle, it is not accurate.
I drew a circle with a radius of 30. The area = 2827.43
The radius of the larger circle comes to 42.3. This makes the area = 5621.22
That's less than it should be.

Therefore, if we allow that, then my method of making the circle three times the area is close!!!
But, not accurate.
So, I don't know what Acorn thinks about all this?

Using a hexagon, I would take the following steps:

1. Circle of diameter 200px.
2. Clone circle, change to square and size to size 200px.
3. Clone square and size to side 200px (width 400px).
4. Clone hexagon, change to circle and inscribe inside hexagon.
5. Check area (94,300sqpx) is 3x original (31,430sqpx).

Attachment 122545

The proof would be that the diameter of the yellow circle is √3 that of the original, all using Pythagoras.

Acorn

ok I'm self confessed mathematically challenged - but can the yellow circle have a diameter that is the cube root of a smaller circle ? - or am I missing something again...

ok I misread I think... sorry

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Originally Posted by handrawn

ok I'm self confessed mathematically challenged - but can the yellow circle have a diameter that is the cube root of a smaller circle ? - or am I missing something again...
ok I misread I think... sorry

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Not need to apologise.

√3 ≃ 1.732051. This is the square root of three; it is not a cube root.
A circle with radius (r) 100px has an area of πr², which is 31,416sqpx.
A circle with radius (√3r) 173.2051px has an area of 3πr², which is 94,247sqpx. Xara's Area is approximate so anything between 94,200 and 94,300sqpx is good.

Acorn

thanks - I got as far as the increase in size of the new circle being 173.205% of the original [using {pi r squared:3(pi R squared)} so for r of 5 new R would be 8.66]

I meant square root of three - the brain must seen the 3 on the wrong side of the root symbol when I was typing :o

Here are four solutions.

1. Triangle Scaffold:
   
   1. Clone Red Circle.
   2. Change to Green Triangle, 0° rotation.
   3. Clone Red Circle.
   4. Change to Blue, place behind.
   5. Shift-Drag so edge touches top-right triangle apex.
   
   
2. Circle Scaffold:
   
   1. Clone Red Circle.
   2. Change to Green move centre horizontally to Red's right edge.
   3. Clone Red Circle.
   4. Change to Blue, centre on top intersection of Red/Green.
   5. Shift-Drag so edge touches bottom intersection of Red/Green.
   
   
3. Two Circle Scaffold:
   
   1. Clone Red Circle twice.
   2. Change to Green and position all three so each just touches.
   3. Clone Red Circle.
   4. Change to Blue, place behind.
   5. Shift-Drag so edge touches bottom intersection of Green/Green.
   
   
4. Square/Hexagon Scaffold:
   
   1. Clone Red Circle.
   2. Change to Green Square, 0° rotation, place behind.
   3. Clone Red Circle.
   4. Change to Cyan Hexagon, 0° rotation, place behind.
   5. Shift-Drag so edges match the Square.
   6. Clone Red Circle.
   7. Shift behind Green.
   8. Shift-Drag so edged touches Hexagon.

My solutions are here: Attachment 122562. The button open a layer that shows the math / maths behind the constructions.

Acorn

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Circle Scaffold:

1. Clone Red Circle.
2. Change to Green move centre horizontally to Red's right edge.
3. Clone Red Circle.
4. Change to Blue, centre on top intersection of Red/Green.
5. Shift-Drag so edge touches bottom intersection of Red/Green.

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2 Should be move centre horizontally to Red's centre not right edge Acorn?.

Quote:

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Originally Posted by Egg Bramhill

2 Should be move centre horizontally to Red's centre not right edge Acorn?.

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Egg, if I am moving the Green circle horizontally, I want its centre to touch Red's right edge.
I could have said move Green horizontally so its left edge touches Red's centre.

In the end, I wanted the centres to be a radius apart.

Acorn