Question

find the area of the figure. (sides meet at right angles.)

Explanation:

Step1: Divide the figure into two rectangles

We can split the L - shaped figure into two rectangles. One rectangle has dimensions \(4\mathrm{cm}\times4\mathrm{cm}\) (the bottom part) and the other has dimensions \((4 - 2)\mathrm{cm}\times3\mathrm{cm}\) (the top part). First, calculate the area of the bottom rectangle: \(A_1=4\times4 = 16\mathrm{cm}^2\). Then, calculate the area of the top rectangle. The length of the top rectangle is \(4 - 2=2\mathrm{cm}\) and the width is \(3\mathrm{cm}\), so \(A_2 = 2\times3=6\mathrm{cm}^2\).

Step2: Sum the areas of the two rectangles

To find the total area of the figure, we add the areas of the two rectangles together. \(A = A_1+A_2=16 + 6=22\mathrm{cm}^2\)? Wait, no, maybe another way. Wait, let's re - examine the figure. Wait, the height of the big rectangle part: the total height is \(7\mathrm{cm}\), and the bottom part is \(4\mathrm{cm}\), so the top part's height is \(7 - 4 = 3\mathrm{cm}\). The length of the big rectangle (the base) is \(4\mathrm{cm}\), and the length of the small rectangle (the part that is indented) is \(4 - 2=2\mathrm{cm}\). Wait, maybe a better way: the figure can be considered as a large rectangle with length \(4\mathrm{cm}\) and height \(7\mathrm{cm}\) minus the area of the indented rectangle. The indented rectangle has length \(2\mathrm{cm}\) and height \(7 - 4=3\mathrm{cm}\)? No, wait, the indented part: the horizontal length is \(2\mathrm{cm}\) and the vertical length is \(3\mathrm{cm}\) (since \(7 - 4 = 3\)). Wait, let's do it correctly. Let's split the figure into two rectangles:

First rectangle: width \(4\mathrm{cm}\), height \(4\mathrm{cm}\), area \(A_1 = 4\times4=16\mathrm{cm}^2\).

Second rectangle: width \((4 - 2)\mathrm{cm}=2\mathrm{cm}\), height \(3\mathrm{cm}\), area \(A_2=2\times3 = 6\mathrm{cm}^2\).

Wait, but \(16+6 = 22\)? No, that can't be. Wait, maybe I made a mistake. Wait, the total height is \(7\mathrm{cm}\), and the bottom rectangle is \(4\mathrm{cm}\) tall. The top rectangle: the width is \(4 - 2 = 2\mathrm{cm}\) and the height is \(3\mathrm{cm}\) (because \(7-4 = 3\)). But also, the left - most part: wait, the figure has a horizontal length of \(4\mathrm{cm}\), vertical height of \(7\mathrm{cm}\), with a notch of \(2\mathrm{cm}\) (horizontal) and \(3\mathrm{cm}\) (vertical). Wait, another approach: the area of the figure is equal to the area of a rectangle with length \(4\mathrm{cm}\) and height \(7\mathrm{cm}\) minus the area of the rectangle that is missing. The missing rectangle has length \(2\mathrm{cm}\) and height \(3\mathrm{cm}\) (since \(7 - 4=3\)). So the area of the large rectangle is \(4\times7 = 28\mathrm{cm}^2\), and the area of the missing rectangle is \(2\times3 = 6\mathrm{cm}^2\). Then the area of the figure is \(28-6 = 22\mathrm{cm}^2\)? Wait, no, wait, let's check the dimensions again. Wait, the horizontal length of the figure is \(4\mathrm{cm}\), vertical height is \(7\mathrm{cm}\). The indented part: the horizontal length is \(2\mathrm{cm}\) (because from the left, there is a \(4\mathrm{cm}\) part and then a \(2\mathrm{cm}\) indent? Wait, no, the figure's base is \(4\mathrm{cm}\), the left part is \(4\mathrm{cm}\) tall, and the right part is \(7\mathrm{cm}\) tall. The horizontal indent is \(2\mathrm{cm}\) (so the length of the top rectangle is \(4 - 2=2\mathrm{cm}\)) and the vertical indent is \(3\mathrm{cm}\) (since \(7 - 4 = 3\)). So when we split into two rectangles:

Rectangle 1: \(4\mathrm{cm}\) (width) \(\times4\mathrm{cm}\) (height) = \(16\mathrm{cm}^2\)

Rectangle 2: \(2\mathrm{cm}\) (width) \(\times3\mathrm{cm}\) (height) = \(6\mathrm{cm}^2\)

Total area: \(16 + 6=22\mathrm{cm}^2\)? Wait, but let's check with another method. Let's use the formula for the area of a composite figure with right angles. The figure can also be seen as a rectangle of length \(4\mathrm{cm}\) and height \(7\mathrm{cm}\) minus a rectangle of length \(2\mathrm{cm}\) and height \(3\mathrm{cm}\). The area of the large rectangle is \(4\times7 = 28\), the area of the small rectangle is \(2\times3 = 6\), so \(28-6 = 22\). Wait, but maybe I misread the figure. Wait, the vertical side is \(7\mathrm{cm}\), the bottom side is \(4\mathrm{cm}\), the left side is \(4\mathrm{cm}\), the indent is \(2\mathrm{cm}\) (horizontal) and \(3\mathrm{cm}\) (vertical). Yes, so the area is \(4\times7-2\times3=28 - 6 = 22\mathrm{cm}^2\). Wait, but let's do it by splitting into two rectangles:

First rectangle: width \(4\mathrm{cm}\), height \(4\mathrm{cm}\) (area \(16\)).

Second rectangle: width \((4 - 2)\mathrm{cm}=2\mathrm{cm}\), height \(3\mathrm{cm}\) (area \(6\)).

Total area \(16 + 6 = 22\mathrm{cm}^2\).

Answer:

\(22\mathrm{cm}^2\)