Question

find the area of the shape. (sides meet at right angles.) 3 ft 2 ft 2 ft 1 ft 2 ft 2 ft 2 ft 6 ft square feet

Explanation:

Step1: Divide the shape into two rectangles

We can split the composite shape into a top rectangle and a bottom rectangle. The bottom rectangle has length 6 ft and height 2 ft. The top rectangle has length 3 ft (since 6 - 1 - 2 = 3? Wait, no, looking at the diagram, the top part's width: the left indent is 1 ft, right indent is 2 ft, so total length of top rectangle is 6 - 1 - 2 = 3 ft? Wait, no, the top rectangle's width is 3 ft (given as 3 ft in the diagram), height 2 ft. Wait, maybe better to calculate the area of the bottom rectangle and the top rectangle.

Bottom rectangle: length = 6 ft, height = 2 ft. Area = length × height = 6 × 2 = 12 sq ft.

Top rectangle: length = 3 ft, height = 2 ft. Area = 3 × 2 = 6 sq ft. Wait, but wait, is there another part? Wait, no, maybe I made a mistake. Wait, the total height? Wait, the bottom part is 2 ft, top part is 2 ft, but wait, the left side: 2 ft (top) + 2 ft (bottom) = 4 ft? Wait, no, the diagram shows: left side has 2 ft (top) and 2 ft (bottom). Wait, maybe another way: the shape can be considered as a large rectangle minus the missing parts, but splitting into two rectangles is easier.

Wait, let's re-examine the diagram. The bottom rectangle: length 6 ft, height 2 ft. Then the top rectangle: the horizontal length is 3 ft (since from the left, after 1 ft indent, the top part starts, and on the right, after 2 ft indent, so 6 - 1 - 2 = 3 ft), and height 2 ft. Then, also, the left vertical part: 1 ft (the indent) and height 2 ft? Wait, no, maybe the shape is composed of three rectangles? Wait, no, let's count the lengths.

Alternative approach: The total area can be calculated by adding the area of the bottom rectangle (6x2) and the area of the top rectangle (3x2) and the area of the left small rectangle? Wait, no, the left side: the bottom part is 6x2, the top part is 3x2, and the left part: 1 ft (width) and 2 ft (height)? Wait, no, the left indent is 1 ft, so the left vertical strip: width 1 ft, height 2 ft? Wait, no, maybe I messed up.

Wait, let's look at the coordinates. Let's assume the bottom rectangle is from (0,0) to (6,2). Then the top rectangle is from (1,2) to (4,4) (since left indent is 1 ft, right indent is 2 ft, so 6 - 2 = 4, so from x=1 to x=4, which is 3 ft, and y=2 to y=4, height 2 ft). Then the area of bottom rectangle: 6*2=12. Top rectangle: 3*2=6. Then the left vertical strip: from (0,2) to (1,4), width 1 ft, height 2 ft. Area 1*2=2. Wait, now total area: 12 + 6 + 2 = 20? Wait, that can't be. Wait, no, maybe I'm overcomplicating.

Wait, let's use the split into two rectangles: the bottom rectangle (6 ft length, 2 ft height) and the top rectangle (3 ft length, 2 ft height) plus the left rectangle (1 ft length, 2 ft height). Wait, 6*2=12, 3*2=6, 1*2=2. 12+6+2=20. Wait, but let's check the total length. The bottom rectangle is 6 ft, top rectangle is 3 ft, left rectangle is 1 ft. 1+3+2=6 (since right indent is 2 ft). Yes, that adds up. So three rectangles? Wait, no, maybe two: the bottom rectangle (6x2) and the top rectangle ( (1+3)x2? No, 1+3=4, 4x2=8? No, this is confusing.

Wait, let's look at the given dimensions:

- Bottom horizontal length: 6 ft, bottom vertical height: 2 ft.

- Top horizontal length: 3 ft (given as 3 ft), top vertical height: 2 ft.

- Left vertical indent: 1 ft, right vertical indent: 2 ft.

So the shape can be divided into:

1. Bottom rectangle: 6 ft (length) × 2 ft (height) = 12 sq ft.

2. Top rectangle: 3 ft (length) × 2 ft (height) = 6 sq ft.

3. Left rectangle: 1 ft (width) × 2 ft (height) = 2 sq ft.

Wait, but 1+3+2=6, which matches the bottom length. So total area: 12 + 6 + 2 = 20? Wait, no, that can't be. Wait, maybe the left rectangle is not needed. Wait, maybe the top rectangle is 3 ft (length) and 2 ft (height), and the bottom rectangle is 6 ft (length) and 2 ft (height), and the middle part? No, maybe I made a mistake.

Wait, another way: the total height is 2 (bottom) + 2 (top) = 4 ft. The total length is 6 ft. But there are two indents: left 1 ft, right 2 ft. So the area of the large rectangle (6x4) would be 24 sq ft. Then subtract the area of the two indented rectangles: left indent: 1 ft (width) × 2 ft (height) = 2 sq ft. Right indent: 2 ft (width) × 2 ft (height) = 4 sq ft. So total area = 24 - 2 - 4 = 18? Wait, that's different.

Wait, let's calculate the large rectangle: length 6 ft, height 2 (bottom) + 2 (top) = 4 ft. Area = 6*4=24.

Now, the missing parts:

- Left missing rectangle: width 1 ft, height 2 ft (the bottom part? No, the top part's left indent: 1 ft width, 2 ft height (since top height is 2 ft).

- Right missing rectangle: width 2 ft, height 2 ft (top height is 2 ft).

So area of missing parts: 1*2 + 2*2 = 2 + 4 = 6.

Thus, total area = 24 - 6 = 18 sq ft.

Ah, that makes sense. Let's verify:

Large rectangle: 6x4=24.

Subtract left indent (1x2=2) and right indent (2x2=4). 24 - 2 -4=18.

Yes, that works. Let's check with splitting into two rectangles:

Bottom rectangle: 6x2=12.

Top rectangle: (6 -1 -2)x2=3x2=6.

Wait, 12 +6=18. Oh! I see, I made a mistake earlier. The top rectangle's length is 6 -1 -2=3 ft, height 2 ft. So area 3x2=6. Then bottom rectangle is 6x2=12. Total area 12+6=18.

Yes, that's correct. So the two rectangles: bottom (6x2) and top (3x2). 6*2=12, 3*2=6, sum is 18.

Step2: Calculate the area of each rectangle and sum them

Bottom rectangle area: $6 \times 2 = 12$ square feet.

Top rectangle area: $3 \times 2 = 6$ square feet.

Total area: $12 + 6 = 18$ square feet.

Answer:

18