Red circle

Geometry Level 2

A regular blue hexagon is inscribed in a red circle. Then a yellow circle is inscribed in this blue hexagon. What is the ratio of the area of the yellow circle to the red circle?

3:4

1:2

2:3

4:5

1 solution

Hassan Abdulla
Feb 7, 2018

$\theta =\frac { \left( 6-2 \right) \times 180 }{ 6 } =60°\\ \sin { \theta } =\frac { { R }_{ yellow } }{ { R }_{ red } } \Rightarrow \frac { { R }_{ yellow } }{ { R }_{ red } } =\frac { \sqrt { 3 } }{ 2 } \\ \text{ the ratio between Areas }=\frac { \pi { \left( { R }_{ yellow } \right) }^{ 2 } }{ \pi { \left( { R }_{ red } \right) }^{ 2 } } ={ \left( \frac { { R }_{ yellow } }{ { R }_{ red } } \right) }^{ 2 }={ \left( \frac { \sqrt { 3 } }{ 2 } \right) }^{ 2 }=\frac { 3 }{ 4 }$

Red circle

1 solution

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