Maximizing Picture Frame Area with Minimal Siding

Eddie estimated that he would need at least 1 meter (100 cm) of wooden sidings to make a picture frame. The challenge lies in determining the dimensions of the largest frame that can be made with this constraint. Let's explore the options:

Square Frame

A square is a candidate for maximizing the enclosed area for a given perimeter. Since the perimeter of the square must be 100 cm, each side would be:

100 cm / 4 25 cm

The area of the square would be:

25 cm x 25 cm 625 cm2

Rectangular Frame

A rectangular shape is another option, but since a square is a special case of a rectangle, we can compare the area of a square with a rectangle that has the same perimeter. For simplicity, let's assume the rectangle has the same dimensions as the square:

Rectangular frame: 25 cm x 25 cm

Area: 25 cm x 25 cm 625 cm2

Pentagonal Frame

A pentagon is a fifth-sided polygon. The perimeter is 100 cm, so each side would be:

100 cm / 5 20 cm

To find the area of the pentagon, we can use the formula for the area of a regular pentagon:

Area of pentagon 5(1/2) * 20 cm * 13.76 cm (height)

Area of pentagon 5 * 10 cm * 13.76 cm 688.19 cm2

Conclusion

The pentagon provides the largest area for the given perimeter of 100 cm. Here is a summary of the dimensions:

Square: 25 cm x 25 cm, Area: 625 cm2 Rectangular: 25 cm x 25 cm, Area: 625 cm2 Pentagon: Sides 20 cm each, Area: 688.19 cm2

Conclusion: The pentagon provides the largest frame area with minimal siding, making it the optimal choice for maximizing the enclosed space within the given constraint of 1 meter of wooden sidings.

For more information on the geometric properties of polygons and their applications in real-world scenarios, explore additional resources on geometry and practical design.

Additional Reading

Explore the Properties of Polygons

Understand the Area Formulas for Various Polygons