PQRSTU is a regular hexagon whose diagonals meet at point at O. Find the ratio of area of quadrilateral PQOU to the area of hexagon PQRSTU.

Solution

In a regular hexagon, all sides are equal, and all internal angles are 120°. The diagonals of a regular hexagon divide it into six equilateral triangles.

Let’s denote the side length of the hexagon as \(a\).

Area of the Hexagon PQRSTU:

The area of an equilateral triangle with side length \(a\) is given by:
\[ \text{Area of equilateral triangle} = \frac{\sqrt{3}}{4}a^2 \]

Since the hexagon is made up of six equilateral triangles, the area of the hexagon is:
\[ \text{Area of hexagon} = 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2 \]

Area of Quadrilateral PQOU:

Quadrilateral PQOU is made up of two equilateral triangles, POQ and UOQ. Therefore, the area of quadrilateral PQOU is:
\[ \text{Area of quadrilateral PQOU} = 2 \times \frac{\sqrt{3}}{4}a^2 = \frac{\sqrt{3}}{2}a^2 \]

Ratio of Areas:

The ratio of the area of quadrilateral PQOU to the area of hexagon PQRSTU is:
\[ \text{Ratio} = \frac{\text{Area of quadrilateral PQOU}}{\text{Area of hexagon}} = \frac{\frac{\sqrt{3}}{2}a^2}{\frac{3\sqrt{3}}{2}a^2} = \frac{1}{3} \]

Conclusion

The ratio of the area of quadrilateral PQOU to the area of hexagon PQRSTU is 1:3.