In the figure shown, suppose that 𝐴𝐵𝐶𝐷 is a parallelogram with 𝐴𝐵 equals 6.41 and 𝐸𝐹 equals 3.82. Find the area of the shaded region rounded to the nearest hundredth.
Looking at the diagram, we can see that we have a triangle 𝐴𝐵𝐸 within the parallelogram 𝐴𝐵𝐶𝐷. And the area outside the triangle, but within the parallelogram, is the shaded area we’re looking to calculate. We’ve been given the lengths of two lines, so let’s add these measurements to the diagram. We can see that in order to find the shaded area, we need to find the total area of the parallelogram and then subtract the area of the triangle.
Let’s look at the area of the parallelogram first of all. The area of a parallelogram is found by multiplying its base by its perpendicular height. In this question, those measurements are 6.41 and 3.82, so we multiply these together to give the area of the parallelogram.
Now let’s think about the area of the triangle 𝐴𝐵𝐸. The area of a triangle is found by multiplying its base by its perpendicular height and then dividing by two. The base and the perpendicular height of the triangle are the same as the base and perpendicular height of the parallelogram, so we have 6.41 multiplied by 3.82, and then we’re halving it. Evaluating each of these areas gives 24.4862 minus 12.2431. This gives a value of 12.2431. The question asked us to find the area to the nearest hundredth, so we need to round this answer.
And so we have that the area of the shaded region, to the nearest hundredth, is 12.24. And there are no units for this as we weren’t given any units in the question; it would just be general square units.
Now you may actually have spotted a shortcut that could have been taken in the working out. When we found the area of the parallelogram, we multiplied the base by the height. The area of the triangle with the base times height divided by two. And then we subtracted. As the base and the height of the two shapes are the same, the area of the triangle is in fact exactly half of the area of the parallelogram. Therefore, once we’d worked out our calculation for the area of the parallelogram, we could’ve just halved it in order to find the remaining shaded area. This would’ve been a slightly quicker way to arrive at our answer.