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An illustration shows an unfolded box net. It is comprised of a rectangle at the center, two squares extending into three trapezoids on the top and bottom, and a trapezoid on the left of the rectangle. The top side of the rectangle is labeled in three distinct sections: starting at the left, a solid line of 3 is drawn, followed by a dashed line of 3, and then a solid line of 6. The bottom of the rectangle is labeled in three distinct sections: a solid line of 6, a dashed line of 3, and a solid line of 3. The left width of the rectangle is drawn as a dashed line, labeled as 3, which becomes the base of a shaded trapezoid. The dashed lines on the top and bottom of the rectangle are extended into squares of side 3, sides drawn in dashed lines. The three outer sides of each square become the longer bases of shaded trapezoids. The top shorter side of the trapezoid is labeled as 1.2 and the perpendicular height is 1.

Decompose the box net into simple polygons to find how much cardboard in square inches is needed to make the cube. Assume that all angles that look like right angles are right angles. All polygons that look congruent are congruent. What is the area of the shape in square inches? Express your answer as a decimal. Do not round your answer.

(1 point)
square inches

1 answer

To determine the area of the box net, we will break down the box net into simpler shapes: rectangles and trapezoids.

Given the net structure:

  1. Dimensions of the Shapes:

    • The central rectangle has a top solid line of 3, a dashed line of 3, and a solid line of 6. The bottom side has a solid line of 6, a dashed line of 3, and a solid line of 3. So, the top and bottom measurements go as follows:
      • Top: 3 + 3 + 6 = 12
      • Bottom: 6 + 3 + 3 = 12 (confirming consistency)
    • The height of the rectangle is the left width labeled as a dashed line of 3.
    • There are squares on the top and bottom of the rectangle, which are 3x3 squares.

  2. Calculate the area of the rectangle:

    • Area of Rectangle = Length × Height
    • Length = 12, Height = 3
    • Area = 12 × 3 = 36 square inches.

  3. Calculate the area of the two squares:

    • Area of each square = side^2 = 3^2 = 9 square inches.
    • Since there are 2 squares: 2 × 9 = 18 square inches.

  4. Calculate the area of the trapezoids:

    • There are two trapezoids at the top and bottom, each represented by the following values:
    • The longer base of the trapezoid = 6 (the length of the rectangle)
    • The shorter base of the trapezoid = 1.2 (given)
    • The height of the trapezoid = 3 (the width of the rectangle)
    • Area of a trapezoid = (1/2) × (base1 + base2) × height
    • Area = (1/2) × (6 + 1.2) × 3 = (1/2) × 7.2 × 3 = (1/2) × 21.6 = 10.8 square inches.
    • Since there are two trapezoids: 2 × 10.8 = 21.6 square inches.

  5. Total area:

    • Now, let's sum the areas of all the individual components:
    • Area of Rectangle = 36 square inches
    • Area of Squares = 18 square inches
    • Area of Trapezoids = 21.6 square inches

Thus, total area = \[ 36 + 18 + 21.6 = 75.6 \]

Therefore, the total area of the shape in square inches is 75.6.