Rectangle inside a square

Geometry Level 3

A non-square rectangle is inscribed in a square so that each vertex of the rectangle is at the trisection point of the different sides of the square. Find the ratio of the area of the rectangle to the area of the square. Give your answer to three decimal places.


The answer is 0.444.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Let x x be the length of the side of the square. Divide each side into three equal parts. The length of each is 1 3 x \frac{1}{3}x . Draw the rectangle, each vertex must hit a trisection point and must lie on each side of the square as shown in the figure.

From the figure,

A r e c t a n g l e = A s q u a r e − 2 A 1 − 2 A 2 = x 2 − ( 2 3 x ) 2 − ( 1 3 x ) 2 = x 2 − 4 9 x 2 − 1 9 x 2 = 4 9 x 2 A_{rectangle} = A_{square} - 2A_{1} - 2A_{2} = x^2 - (\frac{2}{3}x)^2 - (\frac{1}{3}x)^2 = x^2 - \frac{4}{9}x^2 - \frac{1}{9}x^2 = \frac{4}{9}x^2

Solving for the ratio

A r e c t a n g l e A s q u a r e = 4 9 x 2 x 2 = 4 9 = 0.444 \frac{A_{rectangle}}{A_{square}} = \frac{\frac{4}{9}x^2}{x^2} = \frac{4}{9} = 0.444

5 Helpful 2 Interesting 2 Brilliant 3 Confused

@Marvin Kalngan Supposing the side of the square 3n will make the calculations easy! Btw nice pictorial representation!

Toshit Jain - 4 years, 2 months ago

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