Write an expression for the area of the shaded region in the shape below.
We have a diagram of a rectangle with three shapes cut out of it, and the remaining area is what we’re being asked to calculate. The dimensions of the rectangle and the cut-out shapes are given in terms of the letter 𝑤, which is why we’re asked to write an expression for the area rather than calculate it. Let’s think about how to approach this question.
In order to find the shaded area, we first need to find the area of the larger rectangle. We then need to subtract the areas of each of the cut-out shapes. So we have four areas, for which we need to find expressions all together. Let’s look at the area of the larger rectangle first of all. To find the area of a rectangle, we multiply the length by the width. In this case, the measurements are 13 𝑤 and 11 𝑤 plus 13. The result of multiplying these two expressions together is 13 𝑤 into 11 𝑤 plus 13. And we’ll expand this bracket later on.
Now let’s look at the areas of the cut-out shapes. There is a rectangle at the top of the diagram, which has dimensions of two and three 𝑤. Its area is therefore two multiplied by three 𝑤, which is six 𝑤. The other two cut-out shapes are two identical squares, both with sides of length three 𝑤. The areas of each of these squares are found by multiplying three 𝑤 by three 𝑤, which gives a result of nine 𝑤 squared.
Now, let’s substitute these expressions for the area of the cut-out shapes into our expression for the shaded area. We have 13 𝑤 multiplied by 11 𝑤 plus 13 as before minus six 𝑤 and then minus nine 𝑤 squared minus nine 𝑤 squared again. Remember there were two of these squares that we need to subtract. Next, we need to simplify our expression.
So we’ll begin by expanding the brackets. The result of expanding this bracket is 143 𝑤 squared plus 169 𝑤. We’re still subtracting six 𝑤 and simplifying minus nine 𝑤 squared minus another nine 𝑤 squared. We now have minus 18 𝑤 squared.
The final step is to simplify our expression by grouping the like terms. 143 𝑤 squared minus 18 𝑤 squared gives 125 𝑤 squared and positive 169 𝑤 minus six 𝑤 gives positive 163 𝑤. And so we have our expression for the area of the shaded region: 125 𝑤 squared plus 163 𝑤.