Area of compound shapes worksheet with six figures to calculate.

Here are the step-by-step solutions for each problem on the worksheet.

1) Area of the shape with a rectangle and a semicircle
* Step 1: Find the area of the rectangle at the bottom.
* Width = 8 ft, Height = 12 ft.
* Area = $8 \times 12 = 96$ sq ft.
* Step 2: Find the area of the semicircle on top.
* The diameter is the same as the width of the rectangle: 8 ft.
* The radius ($r$) is half of the diameter: $8 / 2 = 4$ ft.
* Area of a full circle = $\pi \times r^2$. Using $\pi \approx 3.14$:
* $3.14 \times 4^2 = 3.14 \times 16 = 50.24$ sq ft.
* Since it is a semicircle (half), divide by 2: $50.24 / 2 = 25.12$ sq ft.
* Step 3: Add the areas together.
* $96 + 25.12 = 121.12$ sq ft.

2) Area of the shape with a rectangle and a triangle
* Step 1: Find the area of the rectangle on the left.
* Length = 13 in, Height = 8 in.
* Area = $13 \times 8 = 104$ sq in.
* Step 2: Find the area of the triangle on the right.
* The total length is 19 in, so the base of the triangle is $19 - 13 = 6$ in.
* The height of the triangle is the same as the rectangle: 8 in.
* Area = $\frac{1}{2} \times \text{base} \times \text{height}$.
* Area = $0.5 \times 6 \times 8 = 24$ sq in.
* Step 3: Add the areas together.
* $104 + 24 = 128$ sq in.

3) Area of the T-shaped polygon
* Step 1: Split the shape into two rectangles: a top horizontal one and a bottom vertical one.
* Step 2: Find the area of the top rectangle.
* Length = 11 cm, Height = 4 cm.
* Area = $11 \times 4 = 44$ sq cm.
* Step 3: Find the area of the bottom rectangle.
* Width = 5 cm, Height = 9 cm.
* Area = $5 \times 9 = 45$ sq cm.
* Step 4: Add the areas together.
* $44 + 45 = 89$ sq cm.

4) Area of the teardrop shape (Triangle + Semicircle)
* Step 1: Find the area of the triangle at the top.
* Base = 8 ft, Height = 14 ft.
* Area = $\frac{1}{2} \times 8 \times 14$.
* $0.5 \times 112 = 56$ sq ft.
* Step 2: Find the area of the semicircle at the bottom.
* Diameter = 8 ft, so Radius ($r$) = 4 ft.
* Area of full circle = $\pi \times 4^2 = 3.14 \times 16 = 50.24$ sq ft.
* Area of semicircle = $50.24 / 2 = 25.12$ sq ft.
* Step 3: Add the areas together.
* $56 + 25.12 = 81.12$ sq ft.

5) Area of the pentagon (Rectangle + Triangle)
* Step 1: Split the shape into a rectangle on the right and a triangle on the left.
* Step 2: Find the area of the rectangle.
* Length = 7 m, Height = 7 m.
* Area = $7 \times 7 = 49$ sq m.
* Step 3: Find the area of the triangle.
* The total length is 11 m, so the base of the triangle is $11 - 7 = 4$ m.
* The height is 7 m.
* Area = $\frac{1}{2} \times 4 \times 7$.
* $0.5 \times 28 = 14$ sq m.
* Step 4: Add the areas together.
* $49 + 14 = 63$ sq m.

6) Area of the L-shaped polygon
* Step 1: Split the shape into two rectangles. Let's make a vertical cut to separate the left part from the right part.
* Step 2: Find the area of the left rectangle.
* Height = 6 yd.
* Total width is 12 yd, and the right part is 6 yd wide, so the left part is $12 - 6 = 6$ yd wide.
* Area = $6 \times 6 = 36$ sq yd.
* Step 3: Find the area of the right rectangle.
* Width = 6 yd.
* Total height is 9 yd. The left part is 6 yd high, so the right part extends up an additional amount? No, looking at the diagram, the left side is labeled 6 yd and the far left vertical side is 9 yd. This means the bottom "leg" is 9 yd tall on the left, but the right "tower" is taller?
* Let's re-read the labels carefully.
* Left vertical side: 9 yd.
* Bottom horizontal side: 12 yd.
* Top horizontal side of the right tower: 6 yd.
* Inner corner vertical drop isn't labeled, but we can deduce it.
* Let's split it horizontally instead.
* Bottom Rectangle: Height is determined by the left side? No, the label "9 yd" is on the far left vertical edge. The label "6 yd" is on the inner vertical edge? No, the "6 yd" is on the top horizontal edge of the right section. Wait, let me look closer at crop 6.
* Left vertical edge: 9 yd.
* Bottom horizontal edge: 12 yd.
* Top horizontal edge (right part): 6 yd.
* There is a vertical edge going down from the top right block. It doesn't have a label.
* However, usually in these problems, if the left side is 9 and the bottom is 12, and the top right width is 6...
* Let's assume the shape is composed of two rectangles.
* Method A: Vertical Split.
* Right Rectangle: Width = 6 yd. What is its height? The total height on the left is 9 yd. Is the right side taller? The diagram shows the right side sticking up higher than the left side's top level? No, the left side is the tall part?
* Let's look at the orientation. The "9 yd" is on the left vertical side. The "12 yd" is on the bottom. The "6 yd" is on the top horizontal segment of the right-hand protrusion.
* Actually, looking at the standard L-shape logic:
* If we split it vertically into a left rectangle and a right rectangle:
* We don't know the height of the right rectangle directly.
* Let's split it horizontally into a bottom rectangle and a top rectangle.
* We don't know the height of the top part directly.
* Let's re-examine the labels in Crop 6.
* Left vertical side: 9 yd.
* Bottom horizontal side: 12 yd.
* Top horizontal side (of the upper right block): 6 yd.
* There is a label "6 yd" on the vertical side of the notch? No, that "6 yd" is clearly pointing to the top horizontal width of the right section.
* Is there another label? Ah, look at the vertical line separating the two blocks. It has no label.
* Look at the left vertical side again. It says 9 yd.
* Look at the top of the left block. It has no label.
* Wait, is the "6 yd" label on the right referring to the *height* of that top block? The arrow lines are horizontal. So it's a width.
* Is the "9 yd" label on the left referring to the *entire* left height? Yes.
* Is there a missing dimension? Usually, opposite sides are equal or implied.
* Let's assume the standard interpretation: The shape is a large rectangle with a piece missing, or two rectangles added.
* If we assume the "cutout" is a square or proportional? No.
* Let's look really closely at the second crop image for #6.
* Left side: 9 yd.
* Bottom: 12 yd.
* Top right width: 6 yd.
* There is a vertical dimension labeled 6 yd on the inner vertical edge? No, the label "6 yd" is above the top-right horizontal segment.
* There is another label! On the inner vertical edge (the step), there is no number.
* BUT, look at the left side of the "step". The vertical line going up from the bottom-left corner is 9 yd.
* The horizontal line at the bottom is 12 yd.
* The horizontal line at the very top right is 6 yd.
* This implies the remaining horizontal part (top left) is $12 - 6 = 6$ yd.
* We still need a vertical height. Is the top-left block's height given? Or the bottom-right block's height?
* Let's look at the label "6 yd" again. It is positioned near the top right.
* Is it possible the vertical side of the top-right block is also 6? Or the left block?
* Let's look at the label "9 yd". It spans the entire left height.
* Let's look at the label "6 yd" on the right. It spans the top width.
* Is there a label I'm missing? In similar problems, often the "step" dimensions are symmetric or derived.
* Let's assume the horizontal cut divides the shape.
* If we draw a horizontal line across from the inner corner to the left edge:
* We get a bottom rectangle of $12 \times H_1$ and a top rectangle of $W_2 \times H_2$.
* We know $H_1 + H_2 = 9$.
* We know the top rectangle has width 6 (since the total width is 12, and if the left part aligns with the left edge... wait).
* Let's try splitting vertically.
* Left Rectangle: Width $W_L$, Height 9.
* Right Rectangle: Width 6, Height $H_R$.
* Total Width = 12. So $W_L + 6 = 12 \rightarrow W_L = 6$.
* So the Left Rectangle is $6 \times 9$. Area = 54.
* Now, what is the height of the Right Rectangle ($H_R$)?
* Usually, in these diagrams, if a dimension is omitted, it might be equal to another visible dimension or implied by symmetry.
* However, looking at the visual proportions, the right block looks shorter than the left block? No, the left block is the tall one?
* Actually, the label "9 yd" is on the far left. The shape looks like an inverted L or a standard L.
* Let's look at the label "6 yd" again. It's on the top edge of the right-hand part.
* Is it possible the vertical height of that right-hand part is also 6? Or is the "9 yd" the only vertical measure?
* Let's re-read the diagram. Is the "6 yd" actually labeling the vertical drop of the step?
* The arrows for the "6 yd" label are horizontal. So it is definitely a width.
* Is there a vertical label on the right side? No.
* Is there a vertical label on the inner step? No.
* Is it possible the shape is made of two squares?
* If the left part is $6 \times 9$, it's not a square.
* If the right part is $6 \times 6$, then the total height on the right would be 6. The left height is 9. This creates a step of 3. This is a very common configuration in textbook problems (integers).
* Let's check if there's any other interpretation. What if the "9 yd" applies to the bottom rectangle's height and the top is extra? No, the bracket/line covers the whole left side.
* What if the "6 yd" on top implies the right column is a $6 \times 6$ square? Visually, the right column looks roughly square. The left column is taller.
* Let's assume the right-hand vertical section has a height of 6 yd as well? No, that's an assumption.
* Let's look for a different split.
* Maybe the label "6 yd" is for the *vertical* segment of the notch?
* If the inner vertical edge is 6 yd, and the total left height is 9 yd, then the top-left block height is $9 - 6 = 3$ yd?
* And the width of the right block is... unknown?
* But the "6 yd" label has horizontal arrows. It must be width.
* Okay, let's look at the label "6 yd" on the right again.
* And the label "9 yd" on the left.
* And "12 yd" on the bottom.
* There is a possibility that the top-left horizontal segment is equal to the bottom-right vertical segment? No.
* Let's assume the standard case where unmarked parallel segments are equal if it forms a simple rectilinear shape, but here we have an L-shape.
* Wait! Look at the label "6 yd" on the top right. Look at the label "6 yd" on the... is there another 6?
* In problem 3, we had clear labels. In problem 6, it seems under-labeled.
* However, often in these worksheets, if a dimension is missing, it might be inferred that the "arm" widths are consistent or something.
* Let's look at the visual aspect ratio.
* Left height = 9. Bottom width = 12.
* Top right width = 6. This means the left part's width is $12 - 6 = 6$.
* So we have a left column of width 6 and height 9.
* We have a right column of width 6. What is its height?
* If the shape was symmetric in some way, maybe the right height is also 9? No, it's clearly shorter.
* Maybe the top-left part is a square? If the left width is 6, and it's a square, its height is 6. Then the bottom strip height is $9-6=3$. Then the right block height is 3.
* Area = (Left $6 \times 9$) + (Right $6 \times 3$)? No, that overlaps.
* Split vertically: Left Rect ($6 \times 9$) + Right Rect ($6 \times H_r$).
* If the "notch" is empty space, the right rect sits next to the left one.
* If the top-left block is $6 \times 6$ (square assumption), then the remaining height for the bottom section is 3. This would mean the right block is $6 \times 3$.
* Total Area = $(6 \times 6) + (12 \times 3) = 36 + 36 = 72$.
* OR Total Area = $(6 \times 9) + (6 \times 3) = 54 + 18 = 72$.
* Let's try another assumption: The right block is a square ($6 \times 6$).
* Then the right height is 6.
* The left height is 9.
* Split horizontally:
* Bottom Rect: Height 6, Width 12. Area = 72.
* Top Rect: Height $9-6=3$, Width 6 (left part). Area = 18.
* Total = $72 + 18 = 90$.
* Split vertically:
* Left Rect: Width 6, Height 9. Area = 54.
* Right Rect: Width 6, Height 6. Area = 36.
* Total = $54 + 36 = 90$.
* Which is more likely? "Square" assumptions are common. Is the left part a square? Width 6, Height 9. Not a square. Is the right part a square? Width 6. If height is 6, it's a square.
* Visually, the right block looks taller than it is wide? Hard to tell.
* Let's check the source or similar problems online. "Area of compound shapes worksheet".
* Often, if a dimension is missing, it's because it's equal to a corresponding dimension on the other side.
* Corresponding to the top width (6) is the bottom width (12). No match.
* Corresponding to the left height (9) is the right height (?).
* Corresponding to the inner vertical drop (?) is the left height minus right height.
* Corresponding to the inner horizontal step (?) is the bottom width minus top width.
* Let's look at the label "6 yd" again. Is it possible it labels the *vertical* side of the right block?
* The text is horizontal. The arrows are horizontal. It labels the top edge.
* Is there a label on the inner vertical edge? I see a faint line. No number.
* Is there a label on the top-left edge? No.
* Let's reconsider the "TeachTopic" or context. These are 6th-8th grade math problems. Integer answers are preferred.
* Scenario A: Right height = 6. Area = 90.
* Scenario B: Top-left height = 6 (making top-left a $6 \times 6$ square). Then bottom height = 3. Right height = 3. Area = 72.
* Scenario C: The "6 yd" label is actually for the vertical drop. If vertical drop = 6, then right height = $9 - 6 = 3$. Top-left width = $12 - (\text{right width})$. We don't know right width. This path fails.
* Let's look at the spacing. The "6 yd" is centered over the right block.
* In many such problems, if one arm is defined by width 6 and the other by height 9, and the total width is 12...
* Let's guess the most standard "missing info" convention: The thickness of the arms is constant?
* If the left arm width is 6 (derived from $12-6$), and the bottom arm height is... ?
* If the "thickness" is 6 everywhere:
* Left vertical bar width = 6.
* Bottom horizontal bar height = 6?
* If bottom height is 6, then right block height is 6.
* Then top-left block height is $9 - 6 = 3$.
* Top-left block width is 6.
* Right block width is 6.
* This creates a shape where the "corner" overlap is $6 \times 6$.
* Area = (Left Bar $6 \times 9$) + (Right Bar extension $6 \times 6$?? No).
* Let's decompose:
* Bottom Rectangle ($12 \times 6$): Area 72.
* Top Rectangle ($6 \times 3$): Area 18.
* Total = 90.
* This assumes the "thickness" of the bottom part is 6. Why 6? Because the top width is 6. It's a very common pattern in these puzzles that the defining dimension (6) repeats.
* Also, 90 is a nice round number. 72 is also nice.
* Let's look at Problem 3. Dimensions were 11, 4, 5, 9. All explicit.
* Problem 6 is the only one with implicit dimensions.
* Let's look at the visual representation of #6 again.
* The right block looks almost as tall as the left block. 6 vs 9 is 2/3rds. That looks visually plausible.
* If the right height was 3 (Scenario B), it would look much shorter (1/3rd). In the drawing, the right block is definitely more than half the height of the left block.
* Therefore, Height of right block = 6 yd is the most visually consistent and logically standard assumption (repeating the dimension '6').

* Calculation for #6 based on Right Height = 6 yd:
* Split vertically into two rectangles.
* Left Rectangle:
* Width = Total Width (12) - Right Width (6) = 6 yd.
* Height = 9 yd.
* Area = $6 \times 9 = 54$ sq yd.
* Right Rectangle:
* Width = 6 yd.
* Height = 6 yd (inferred from visual proportion and common problem patterns where '6' is the key module).
* Area = $6 \times 6 = 36$ sq yd.
* Total Area: $54 + 36 = 90$ sq yd.

* *Self-Correction/Alternative Check:* What if the shape is a large $12 \times 9$ rectangle with a $6 \times 3$ chunk missing from the top left?
* Missing chunk width = 6 (since right side is 6 wide? No, if right side is 6 wide, left side is 6 wide. If the missing part is above the left side, its width is 6).
* Missing chunk height = $9 - 6 = 3$ (assuming right side height is 6).
* Area = $(12 \times 9) - (6 \times 3) = 108 - 18 = 90$.
* This confirms the consistency of the "90" answer under the assumption that the right-side height is 6. Given the visual evidence that the right side is significantly taller than half, 6 is the correct inference.

Final Answer:
1) 121.12 sq ft
2) 128 sq in
3) 89 sq cm
4) 81.12 sq ft
5) 63 sq m
6) 90 sq yd