A regular hexagon with its circumscribed and inscribed circles, what is the ratio of the area of the circles?

Question

Please advise, thanks.

Hi Dr D, can you explain why the ratio is 4:3 please? thanks.

Answer 1

If R is the radius of the circumscribed circle, you can calculate the radius of the inscribed circle by drawing a triangle between the circle center and one side of the hexagon, bisecting the side, and solving the triangle (the bisector will pass through the circle center). This will permit an easy solution to the problem.

Answer 2

4/3

The radius, R of the circumscribed triangle is the "diagonal". The radius, r of the inscribed circle is the distance from the center to the midpoint of an edge.
These form a 30-60-90 triange with hypothenuse R, and
r = R*sqrt(3) / 2

So R/r = 2 / sqrt(3)
R^2 / r^2 = 4 / 3 = ratio of areas

Answer 3

A cirumcircle /A inscribed circle = 4/3
A circumincircle = pi R^2
r = Rsqrt(3)/2
So area incircle = pi3R^2/4
So there ratio is 4/3

Answer 4

(cos(30))^2=3/4
or 4/3