The question is asking to calculate the area of a hexagonal park $ABCDEF$ by splitting it into triangles and trapeziums. The given measurements are:

• $FP = 6 \, \text{cm}, PL = 1.5 \, \text{cm}, CM = 4.5 \, \text{cm}, MD = 2.25 \, \text{cm}$
• $EN = 9 \, \text{cm}, LN = 6 \, \text{cm}, AP = 4.5 \, \text{cm}, BL = 6 \, \text{cm}, NM = 1.5 \, \text{cm}$

Solution:

1. Break the hexagon into parts:

   • $\triangle CML, \triangle ENL, \triangle PLF$ (triangles)
   • $\text{Trapeziums: } \text{APLD and BMLD}$

2. Calculate areas of triangles: For each triangle, the area formula is: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

   • $\triangle PLF: \text{base } FP = 6 \, \text{cm}, \text{ height } PL = 1.5 \, \text{cm}$ $\text{Area of } \triangle PLF = \frac{1}{2} \times 6 \times 1.5 = 4.5 \, \text{cm}^2$

   • $\triangle CML: \text{base } CM = 4.5 \, \text{cm}, \text{ height } MD = 2.25 \, \text{cm}$ $\text{Area of } \triangle CML = \frac{1}{2} \times 4.5 \times 2.25 = 5.0625 \, \text{cm}^2$

   • $\triangle ENL: \text{base } EN = 9 \, \text{cm}, \text{ height } LN = 6 \, \text{cm}$ $\text{Area of } \triangle ENL = \frac{1}{2} \times 9 \times 6 = 27 \, \text{cm}^2$

3. Calculate areas of trapeziums: The area of a trapezium is given by: $\text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$

   • $\text{Trapezium } APLD: \text{parallel sides } AP = 4.5 \, \text{cm}, BL = 6 \, \text{cm}, \text{height } PL = 1.5 \, \text{cm}$ $\text{Area of } APLD = \frac{1}{2} \times (4.5 + 6) \times 1.5 = \frac{1}{2} \times 10.5 \times 1.5 = 7.875 \, \text{cm}^2$

   • $\text{Trapezium } BMLD: \text{parallel sides } BL = 6 \, \text{cm}, NM = 1.5 \, \text{cm}, \text{height } MD = 2.25 \, \text{cm}$ $\text{Area of } BMLD = \frac{1}{2} \times (6 + 1.5) \times 2.25 = \frac{1}{2} \times 7.5 \times 2.25 = 8.4375 \, \text{cm}^2$

4. Sum up all the areas: $\text{Total Area} = 4.5 + 5.0625 + 27 + 7.875 + 8.4375 = 52.875 \, \text{cm}^2$

Final Answer:

The area of the hexagonal park is $52.875 \, \text{cm}^2$.

Would you like to discuss details or have further questions?

5 Related Questions:

1. How do you find the area of a hexagon without dividing it into simpler shapes?
2. Can the properties of symmetry in hexagons simplify such calculations?
3. How does this formula change if the hexagon is regular?
4. Why are trapeziums useful in geometry for irregular polygons?
5. What are other methods to calculate areas of composite shapes?

Tip:

Always verify the units of each dimension before performing area calculations to ensure consistency.