Question

what is the total area of this shape?
6 in.
2 in.
2 in.
5 in.
3 in.
8 in.
a 36 square inches
b 40 square inches
c 46 square inches
d 54 square inches

Explanation:

Step1: Divide the shape into two rectangles

We can split the L - shaped figure into two rectangles. One rectangle has dimensions \(6\) in (length) and \(5\) in (height)? Wait, no, let's do it correctly. Let's take the lower rectangle: length is \(8\) in and height is \(3\) in. The upper rectangle: length is \(6\) in and height is \(2\) in (since the total height is \(5\) in, and the lower part is \(3\) in, so \(5 - 3=2\) in). Or another way: the big rectangle if we consider the missing part. Wait, the area of the composite shape can be calculated as the area of the large rectangle (if we extend the left side) minus the area of the missing small square? Wait, no. Let's use the method of splitting into two rectangles.

First rectangle: width \(8\) in, height \(3\) in. Area \(A_1 = 8\times3=24\) square inches.

Second rectangle: width \(6\) in, height \(2\) in (because the top part has height \(5 - 3 = 2\) in). Area \(A_2=6\times2 = 12\) square inches? Wait, no, that's not right. Wait, the left - most part: the horizontal length of the upper rectangle. Wait, the total length of the bottom rectangle is \(8\) in, and the upper rectangle has length \(6\) in, but the left - hand side of the upper rectangle is indented by \(2\) in (since the bottom rectangle has a width of \(8\) in, and the upper rectangle has a width of \(6\) in, so the indent is \(8 - 6=2\) in). Wait, maybe a better way: the shape can be seen as a rectangle of length \(8\) in and height \(5\) in minus a rectangle of length \(2\) in (the indent) and height \(2\) in (the vertical indent). Wait, the large rectangle (if we fill the indent) would have dimensions \(8\) in (length) and \(5\) in (height), area \(A_{large}=8\times5 = 40\) square inches. The missing part is a rectangle with length \(2\) in (since \(8 - 6 = 2\)) and height \(2\) in (since \(5 - 3=2\))? Wait, no, the missing part is a square? Wait, the indent on the left - top: the horizontal length of the indent is \(2\) in (because the bottom rectangle is \(8\) in long, and the top rectangle is \(6\) in long, so \(8 - 6 = 2\) in) and the vertical length of the indent is \(2\) in (because the bottom rectangle is \(3\) in tall, and the top is \(2\) in tall, so \(5-3 = 2\) in). So the area of the missing part is \(2\times2 = 4\) square inches. Then the area of the composite shape is \(A = 8\times5-2\times2=40 - 4=36\)? No, that's not matching. Wait, I think I made a mistake.

Wait, let's split the shape into two rectangles properly. Let's look at the vertical sides. The total height is \(5\) in. The lower rectangle has height \(3\) in, so the upper rectangle has height \(5 - 3=2\) in. The lower rectangle: length is \(8\) in, height \(3\) in, area \(8\times3 = 24\). The upper rectangle: length is \(6\) in (since the horizontal length from the right is \(6\) in), height is \(2\) in, area \(6\times2=12\). Then total area is \(24 + 12=36\)? But wait, another way: the shape can be considered as a rectangle with length \(6\) in and height \(5\) in plus a rectangle with length \(2\) in ( \(8 - 6\)) and height \(3\) in (the lower part). So \(6\times5=30\), \(2\times3 = 6\), total \(30 + 6=36\). Wait, but let's check the answer options. Option A is 36, B is 40, C is 46, D is 54.

Wait, maybe my first approach was wrong. Let's calculate again. The figure:

- The bottom rectangle: width = \(8\) in, height = \(3\) in. Area = \(8\times3 = 24\) in².

- The top rectangle: width = \(6\) in, height = \(2\) in (because \(5 - 3=2\) in). Area = \(6\times2=12\) in².

- Wait, no, the top rectangle's width: is it \(6\) in? Wait, the horizontal length of the top rectangle: from the right, it's \(6\) in, and the left - hand side of the top rectangle is at \(2\) in from the left (since the bottom rectangle is \(8\) in wide, and \(8 - 6 = 2\) in). So the top rectangle is \(6\) in (width) and \(2\) in (height). The bottom rectangle is \(8\) in (width) and \(3\) in (height). But wait, the bottom rectangle's width is \(8\) in, but the part under the top rectangle is \(6\) in, so the remaining part of the bottom rectangle (the left - most part) is \(8 - 6 = 2\) in wide and \(3\) in tall. So area of left - most bottom rectangle: \(2\times3 = 6\) in². Area of top rectangle: \(6\times5\)? No, no. Wait, I think I messed up the height. The total height is \(5\) in. The bottom part is \(3\) in, so the top part is \(5 - 3=2\) in. The top part's width: from the right, it's \(6\) in, so the area of the top part is \(6\times2 = 12\) in². The bottom part: the width is \(8\) in, height \(3\) in, area \(8\times3 = 24\) in². Then total area is \(24+12 = 36\) in². So the answer should be A. 36 square inches.

Wait, but let's check with the other method: area of the large rectangle (if we fill the indent) is \(8\times5 = 40\) in². The indent is a rectangle with width \(2\) in ( \(8 - 6\)) and height \(2\) in ( \(5 - 3\)), so area of indent is \(2\times2 = 4\) in². Then the area of the shape is \(40 - 4=36\) in². Yes, that matches. So the correct answer is A.

Answer:

A. 36 square inches