The net of a shape is made from two identical squares and four identical rectangles. Shade an area of the diagram below which can be represented by the expression 𝑎 multiplied by 𝑎 plus 𝑏.
Let’s begin by thinking about what we know about this net. The edges of a cuboid are equal distance and parallel to each other. So all four of these edges are also 𝑏 units. We are told that the cross section of the shape is a square, so we can repeat this process for the edges marked 𝑎 units. Next let’s find the area of each of the individual faces of the shape. The area of a rectangle is its base multiplied by its height. This rectangle is therefore 𝑎𝑏. The other four rectangles are all identical, so they also have an area of 𝑎𝑏. The squares have four equal sides, so the area of these are 𝑎 multiplied by 𝑎 or 𝑎 squared.
To decide which area of the diagram can be represented by the expression 𝑎 multiplied by 𝑎 plus 𝑏, let’s begin by expanding these brackets. We need to multiply each term inside the bracket by the 𝑎 on the outside. 𝑎 multiplied by 𝑎 is 𝑎 squared, and 𝑎 multiplied by 𝑏 is 𝑎𝑏. Since 𝑎 squared is the area of one square and 𝑎𝑏 is the area of one rectangle, this area represents one square and one rectangle, and we can shade that as shown.
Part b) Shade an area of the diagram below which can be represented by the expression two 𝑎 squared.
We can say that two 𝑎 squared is the same as 𝑎 squared plus another 𝑎 squared, and we said that 𝑎 squared is the area of one of the squares. So we have two squares that we need to shade, and we’re done.
Part c) Write an expression for the surface area of the cuboid that the net makes.
We said we had four rectangles which were identical, and the area of each rectangle is 𝑎𝑏. And we have two squares, the area for which is 𝑎 squared. So we have four multiplied by 𝑎𝑏 plus two multiplied by 𝑎 squared, which simplifies to four 𝑎𝑏 plus two 𝑎 squared. Remember, we can write these in any order. So we could say it’s two 𝑎 squared plus four 𝑎𝑏; it’s the same thing.