Triple the Area of a Circle

**Acorn** · 02 November 2018, 03:50 PM

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?

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

Good thinking,
Acorn

**Rik** · 02 November 2018, 09:59 PM

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?

**Acorn** · 02 November 2018, 10:17 PM

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?

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

**Rik** · 02 November 2018, 10:43 PM

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!

**ss-kalm** · 02 November 2018, 11:49 PM

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

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**Rik** · 03 November 2018, 07:08 AM

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.

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?

**Acorn** · 03 November 2018, 10:27 AM

Using a hexagon, I would take the following steps:

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

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

Acorn

**handrawn** · 03 November 2018, 10:59 AM

~~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

**Acorn** · 03 November 2018, 01:38 PM

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

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

**handrawn** · 03 November 2018, 02:34 PM

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