Tricky Composition

Geometry Level 1

The above shows a square inscribed in a circle, that is inscribed inside a larger square.

What fraction of the large square is shaded in red?

1 8 \frac18 1 6 \frac16 1 10 \frac1{10} 1 12 \frac1{12}

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Ajay Sambhriya by Ajay Sambhriya , 27, India

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3 solutions

Ajay Sambhriya
Aug 24, 2016

Relevant wiki: Length and Area - Composite Figures

13 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Wonderful! By rotating the inner square, we can see that the red region occupies only 1 2 × 1 4 = 1 8 \dfrac12\times\dfrac14 = \dfrac18 of the total area.

Sir ..please let me ...post solution to various problems

Ajay Sambhriya - 4 years, 9 months ago
Mick Martucci
Apr 24, 2017

if the radius of the circle is 1 then the area of the circle is π, the area of the smaller square is 2 and the area of the larger square is 4.

the area of the red corner of the square is (4 - π) / 4 and the area of the red circle segment is (π - 2) / 4

so the red area is (4 - π) / 4 + (π - 2) / 4 = (4 - π + π - 2) / 4 = 2 / 4 = 1 / 2

so the fractional area is 1 / 2 divided by 4 = 1/8

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Mike Holden
Nov 18, 2017

I totally agree with Ajay Sambhriya (Aug 24 2016). This is the best solution I can think of.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

thanks Mike

Ajay Sambhriya - 3 years, 6 months ago

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