Decompose the box net into simple polygons to find how much cardboard and 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?

Answer 1

To find the area of the shape in square inches, we need to calculate the area of each simple polygon and then add them up.

1. There are 6 identical squares for the faces of the cube. Each square has sides of 3 inches (since each side length of the cube is 3 inches). The area of one square is 3 inches x 3 inches = 9 square inches. So, the total area of all 6 squares is 6 x 9 square inches = 54 square inches.

2. There are 4 identical rectangles around the sides of the cube. Each rectangle has a length of 3 inches (the same as the side length of the cube) and a width of 6 inches (twice the side length of the cube). The area of one rectangle is 3 inches x 6 inches = 18 square inches. So, the total area of all 4 rectangles is 4 x 18 square inches = 72 square inches.

3. Finally, there are 4 identical triangles on the top and bottom of the cube. Each triangle has a base of 3 inches and a height of 3 inches (since it is a right triangle with legs equal to the side length of the cube). The area of one triangle is 0.5 x 3 inches x 3 inches = 4.5 square inches. So, the total area of all 4 triangles is 4 x 4.5 square inches = 18 square inches.

Adding up the areas of all the simple polygons gives us the total area of the shape:

54 square inches (6 squares) + 72 square inches (4 rectangles) + 18 square inches (4 triangles) = 144 square inches

Therefore, the area of the shape in square inches is 144 square inches.