Question

given the two rectangles below. find the area of the shaded region.
6
6
8 2
answer attempt 1 out of 2
units² submit answer

Explanation:

Step1: Find the side length of the large square

The total height of the figure is \(6 + 6 = 12\), and the total width is \(8 + 2 = 10\)? Wait, no, actually, looking at the figure, the large rectangle (or square? Wait, no, let's re-examine. Wait, the shaded region can be found by calculating the area of the larger rectangle minus the area of the unshaded rectangle. Wait, the larger rectangle has a height of \(6 + 6 = 12\) and a width of \(8 + 2 = 10\)? No, wait, maybe the larger figure is a rectangle with length \(8 + 2 = 10\) and height \(6 + 6 = 12\), and the unshaded rectangle is \(8\times6\). Wait, no, let's check again.

Wait, the unshaded rectangle has dimensions \(8\) (width) and \(6\) (height). The larger rectangle (the one that includes both shaded and unshaded) has a width of \(8 + 2 = 10\) and a height of \(6 + 6 = 12\)? Wait, no, maybe the height of the larger rectangle is \(6 + 6 = 12\), and the width is \(8 + 2 = 10\), but the unshaded is \(8\times6\). Wait, no, perhaps another approach: the shaded region can be divided into two parts. One part is the top rectangle with dimensions \((8 + 2)\times6 = 10\times6 = 60\), and the other part is the right rectangle with dimensions \(2\times6 = 12\). Wait, no, let's see: the total height is \(6 + 6 = 12\), total width is \(8 + 2 = 10\). The unshaded is \(8\times6\). So area of large rectangle: \(10\times12 = 120\)? No, that can't be. Wait, maybe I'm overcomplicating.

Wait, the figure: the unshaded rectangle is \(8\) units wide and \(6\) units tall. The shaded region: above the unshaded is a rectangle of width \(8 + 2 = 10\) and height \(6\), and to the right of the unshaded is a rectangle of width \(2\) and height \(6\). So total shaded area: \((10\times6)+(2\times6)=60 + 12 = 72\). Wait, or another way: the large rectangle (the outer one) has length \(8 + 2 = 10\) and height \(6 + 6 = 12\), area \(10\times12 = 120\). The unshaded area is \(8\times6 = 48\). So shaded area is \(120 - 48 = 72\). Let's verify:

Step1: Calculate area of large rectangle

Length of large rectangle: \(8 + 2 = 10\) units. Height of large rectangle: \(6 + 6 = 12\) units. Area of large rectangle: \(10\times12 = 120\) square units.

Step2: Calculate area of unshaded rectangle

Unshaded rectangle has length \(8\) units and height \(6\) units. Area: \(8\times6 = 48\) square units.

Step3: Subtract unshaded area from large area

Shaded area = \(120 - 48 = 72\) square units.

Answer:

\(72\)