the base of the equilateral triangle to the center of the small circle=1/3 the height of the triangle
the vertex of any side of the triangle( which touches the circumference of the bigger circle) to the center of the bigger circle (which also coincides with the center of the smaller circle)= 2/3 the height of the triangle
radius of smaller circle=1/3
radius of bigger circle=2/3
small circle area:bigger circle area=(1/3)^2:(2/3)^2=1:4

Here's a similar question to practice with: https://gre.myprepclub.com/forum/in-the-ab ... -9519.html

Cheers,
Brent

Many Many Many Thanks

huda
Ratio of two circles what?

Radius: 2:1
Area: 4:1
Circumference: 2:1

[NOTE: All the above ratios are bigger circle to smaller circle]

let S the length of the side of the triangle
so the height will be
H^2 +(S^2/4)=S^2
So H = sqr(3)*S/2
the area of the triangle will be =sqr (3) *S^2 /4
as the small circle attachments to the triangle can divide the area into three equal triangles
so the small triangle will be =sqr (3) *S^2 /12
so the radius of the small circle will be =2*(sqr (3) *S^2 /12)/S
so r = sqr (3)*S/6
so the radius of the big circle will be : height - radius of small
so R =2* sqr (3)*S/6
so the ration will be r to R is 1:4

Assuming they are looking for the ratio of the AREAS of the small circle to the big circle:

1. Draw a line from the center of the small circle to the center of one side of the triangle, such that it creates a 90-degree angle with the side of the triangle. This line is small radius = r. Area of small circle = pi*r^2

2. Draw another line from the center of the circle to one of the corners of the triangle immediately adjacent to the prior line. Since we know we can draw three of each of these types of lines, thus dividing the circle into 6 pie slices, and a circle has 360 degrees total, we now have an angle between these two lines at the center of the circle = 60 degrees. We have now created a 30-6o-90 triangle where the small side = r (as defined above) and the hypotenuse = 2r = the radius of the bigger circle. So the area of the bigger circle = pi * (2r)^2 = 4*pi*r^2.

3. (pi*r^2): (4*pi*r^2) = 1:4

Since the given triangle is equilateral,

we know, for Circumcircle: R=abc/4*area of triangle
and for incircle r=area of triangle/1/2(a+b+c)
Taking the area of triangle to be \sqrt{3}/4*a^2

Then on,
Calculating both R & r. Further, putting their areas in ratio, we'll get 1:4