I'm in a rush and kind of brain-fried, but I've been meaning to take a stab at this.
If the triangle is equilateral, then each of the blue regions is equal to the equivalently-shaped white regions. The blue area is therefore 1/3rd of the area of the larger circle, excluding the area of the orange circle.
The question then becomes, how big is the orange circle? If it is exactly 1/4 the size of the larger circle, then it will be exactly the same size as the blue area. If it is larger than 1/4 of the large circle, it will be bigger than the blue area, and if smaller, then smaller. :P
Okay, here's where I'm running into difficulty. How can I tell the size of the smaller circle? Let's call the radius of the larger triangle 2. That makes the area of the larger circle 4pi. That means that we can figure out the area of the triangle by splitting it up into six triangles which use the radius as their hypotenuse. Those six triangles will be 30-60-90 triangles, since they'll have 90 degree angles along the actual sides of the large triangle, and 60 degree angles around the center of the circle (360/6 being of course sixty). I had to look the formula up, but if the hypotenuse is 2 then the other sides are 1 and √3. We multiply them to get the area of the mini triangle of 1√3/2. That makes the total area of the triangle 6√3/2 or 3√3. The triangle is therefore about 2/3rds the total area of the larger circle.
* I actually started with radius 1 but changed it later for ease of calculating.
There is probably more math I can do to figure out the size of the inner circle but I don't know it offhand. :/ If the inner circle is a similar proportion to the triangle as the triangle was to the outer circle, then it is 4/9thsish-sized, which is way more than 1/4 the size of the full circle. Even if it's half of the triangle, it's more than 1/4 the size of the full circle. So I'm going to guess that the orange circle is bigger than the blue parts. But I haven't proved anything and I could believe the circle is more like 1/3 of the triangle and my brain hurts, so I'm going to stop typing and see what the answer is. :)
And my brain hurts. So I'm going to give up and see what the answer is.
no subject
If the triangle is equilateral, then each of the blue regions is equal to the equivalently-shaped white regions. The blue area is therefore 1/3rd of the area of the larger circle, excluding the area of the orange circle.
The question then becomes, how big is the orange circle? If it is exactly 1/4 the size of the larger circle, then it will be exactly the same size as the blue area. If it is larger than 1/4 of the large circle, it will be bigger than the blue area, and if smaller, then smaller. :P
Okay, here's where I'm running into difficulty. How can I tell the size of the smaller circle? Let's call the radius of the larger triangle 2. That makes the area of the larger circle 4pi. That means that we can figure out the area of the triangle by splitting it up into six triangles which use the radius as their hypotenuse. Those six triangles will be 30-60-90 triangles, since they'll have 90 degree angles along the actual sides of the large triangle, and 60 degree angles around the center of the circle (360/6 being of course sixty). I had to look the formula up, but if the hypotenuse is 2 then the other sides are 1 and √3. We multiply them to get the area of the mini triangle of 1√3/2. That makes the total area of the triangle 6√3/2 or 3√3. The triangle is therefore about 2/3rds the total area of the larger circle.
* I actually started with radius 1 but changed it later for ease of calculating.
There is probably more math I can do to figure out the size of the inner circle but I don't know it offhand. :/ If the inner circle is a similar proportion to the triangle as the triangle was to the outer circle, then it is 4/9thsish-sized, which is way more than 1/4 the size of the full circle. Even if it's half of the triangle, it's more than 1/4 the size of the full circle. So I'm going to guess that the orange circle is bigger than the blue parts. But I haven't proved anything and I could believe the circle is more like 1/3 of the triangle and my brain hurts, so I'm going to stop typing and see what the answer is. :)
And my brain hurts. So I'm going to give up and see what the answer is.