A square is fitted into an equilateral triangle as shown. What fraction is shaded?

Solution

The shaded fraction is 2√3-3 which is approximately 0,46.

Draw the line segment across the middle and the one perpendicular to it through the square centre. Name half the square diagonal x and evaluate the upper right 30-60-90 triangle. The vertical side is √3 times the horizontal side.

Now if we set the triangle side to 1, the height is √3/2 and it is equal to x(1+√3). This determines x as (3-√3)/4 and the square surface as 2x²=(6-3√3)/4. Dividing this by the triangle surface √3/4 gives the desired fraction.