Calculate Fabric Pieces for Lining Wooden Box

This solution calculates the approximate number of square fabric pieces required to line the inside of a wooden box, including the floor and four walls, accounting for extra fabric needed for cuts and overlaps.

Box Dimensions and Fabric Size

First, ensure all measurements are in the same unit. We'll use centimeters (cm).

• Inner Box Length (L): 2.5 m = 250 cm
• Inner Box Width (W): 1.8 m = 180 cm
• Inner Box Height (H): 1.2 m = 120 cm
• Fabric piece side length (s): 50 cm

Calculate Total Inner Surface Area

The fabric needs to cover the inner floor and the four inner side walls.

• Area of the inner floor: $A_{floor} = L \times W = 250 \text{ cm} \times 180 \text{ cm} = 45000 \text{ cm}^2$
• Area of the two longer inner walls: $A_{long\_walls} = 2 \times (L \times H) = 2 \times (250 \text{ cm} \times 120 \text{ cm}) = 60000 \text{ cm}^2$
• Area of the two shorter inner walls: $A_{short\_walls} = 2 \times (W \times H) = 2 \times (180 \text{ cm} \times 120 \text{ cm}) = 43200 \text{ cm}^2$
• Total inner surface area to be lined: $A_{total} = A_{floor} + A_{long\_walls} + A_{short\_walls} = 45000 + 60000 + 43200 = 148200 \text{ cm}^2$

Calculate Required Fabric Area

Calculate the area of one fabric piece and the total area needed, including the 15% extra.

• Area of one fabric piece: $A_{piece} = s^2 = (50 \text{ cm})^2 = 2500 \text{ cm}^2$
• The total area needed is the inner surface area plus 15% for cuts and overlaps.
• Total required fabric area: $A_{required} = A_{total} \times (1 + 0.15) = 148200 \text{ cm}^2 \times 1.15 = 170430 \text{ cm}^2$

Determine the Number of Fabric Pieces

Divide the total required fabric area by the area of a single piece to find the number of pieces needed. Since you can only buy whole pieces, round up to the nearest whole number.

• Number of pieces = $\frac{A_{required}}{A_{piece}} = \frac{170430 \text{ cm}^2}{2500 \text{ cm}^2} \approx 68.172$
• Rounding up to the nearest whole number gives 69.

Therefore, approximately 69 fabric pieces should be purchased.