Ratio of areas

Geometry Level 1

As shown in the diagram to the right, extending the sides of a regular hexagon forms a six-pointed star. A red circle is inscribed in the hexagon, and a blue circle is circumscribed around the star.

What is the ratio of blue circle's area to the red circle's area?

3:2 5:2 4:1 6:1

A Former Brilliant Member by A Former Brilliant Member

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2 solutions

Consider the diagram above. By Pythagorean theorem, the radius of the red circle is

r = 6 2 − 3 2 = 3 3 r=\sqrt{6^2-3^2}=3\sqrt{3}

By symmetry, the radius of the blue circle is 2 r = 6 3 2r=6\sqrt{3} .

Finally, the ratio of the area of the blue circle to the area of the red circle, which is also the ratio of the squares of their radii is

( 6 3 ) 2 ( 3 3 ) 2 = 108 27 = 4 1 \dfrac{(6\sqrt{3})^2}{(3\sqrt{3})^2}=\dfrac{108}{27}=\dfrac{4}{1} .

7 Helpful 0 Interesting 0 Brilliant 0 Confused

This is in fact ratio of the area of circumcircle to the area of the incircle for an equilateral triangle of any size. Their radius ratio is 2:1.

Maria Kozlowska - 3 years, 8 months ago

From where? 6 and 3 came.

Hashaam Elahi - 3 years, 6 months ago
Chew-Seong Cheong
Oct 20, 2017

The regular six-pointed star is made up of 12 equilateral triangles as shown in the figure above. If the radius of the red circle is r r , from the figure, we can see that the radius of the blue circle is 2 r 2r . The ratio of their areas is thus A b l u e : A r e d = π ( 2 r ) 2 : π r 2 = 4 : 1 A_{\color{#3D99F6}blue} : A_{\color{#D61F06}red} = \pi (2r)^2 : \pi r^2 = \boxed{4:1}

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