Math worksheet for calculating the area of irregular polygons with various geometric shapes and measurements.

Here are the step-by-step solutions for each problem on the worksheet. To find the area of these irregular shapes, we split them into simpler rectangles, calculate the area of each rectangle (Length × Width), and then add them together.

Problem 1
* Step 1: Split the shape into a tall rectangle on the left and a shorter one on the right.
* Step 2: Find the width of the left rectangle. The total width is 19 m, and the right part is 10 m wide. So, $19 - 10 = 9$ m.
* Step 3: Calculate the area of the left rectangle: $16 \text{ m} \times 9 \text{ m} = 144 \text{ m}^2$.
* Step 4: Calculate the area of the right rectangle: $8 \text{ m} \times 10 \text{ m} = 80 \text{ m}^2$.
* Step 5: Add them together: $144 + 80 = 224$.

Problem 2
* Step 1: Split the shape vertically.
* Step 2: Find the width of the left rectangle. Total width is 22 cm, right part is not given directly but we can see the top part is 10 cm. Let's look at the heights. The left side is 12 cm, right side is 5 cm.
* Alternative Split (Horizontal): Let's split it horizontally into a top rectangle and a bottom rectangle.
* Bottom rectangle height is 5 cm. The total width is 22 cm. Area = $22 \times 5 = 110 \text{ cm}^2$.
* Top rectangle sits on the left. Its height is $12 - 5 = 7$ cm. Its width is 10 cm. Area = $10 \times 7 = 70 \text{ cm}^2$.
* Total Area = $110 + 70 = 180$.
* Verification with Vertical Split:
* Left Rectangle: Width 10 cm, Height 12 cm. Area = $120 \text{ cm}^2$.
* Right Rectangle: Width is $22 - 10 = 12$ cm. Height is 5 cm. Area = $12 \times 5 = 60 \text{ cm}^2$.
* Total Area = $120 + 60 = 180$.

Problem 3
* Step 1: This looks like a big rectangle with a piece missing, or three vertical strips. Let's use three vertical strips.
* Step 2: Left Strip: Width 1 yd, Height 9 yd (same as right side). Area = $1 \times 9 = 9 \text{ yd}^2$.
* Step 3: Right Strip: Width 1 yd, Height 9 yd. Area = $1 \times 9 = 9 \text{ yd}^2$.
* Step 4: Middle Strip: The total width is 10 yd. The side strips take up $1 + 1 = 2$ yd. So the middle width is $10 - 2 = 8$ yd. The height of the middle section is given as 5 yd. Area = $8 \times 5 = 40 \text{ yd}^2$.
* Step 5: Total Area = $9 + 9 + 40 = 58$.

Problem 4
* Step 1: Split into three vertical rectangles.
* Step 2: Left Rectangle: Width 2 mm, Height 3 mm. Area = $2 \times 3 = 6 \text{ mm}^2$.
* Step 3: Right Rectangle: Width 2 mm, Height 2.5 mm. Area = $2 \times 2.5 = 5 \text{ mm}^2$.
* Step 4: Middle Rectangle: The total top width is 14 mm. The left part is 2 mm and the right part is 2 mm. So the middle width is $14 - 2 - 2 = 10$ mm. The height of the middle section connects the two sides. Looking at the diagram, the middle section's height isn't explicitly labeled on the inside, but usually, in these "step" problems, we assume the bottom is flat. Let's look closer. Actually, it's easier to view this as a large top rectangle and two small legs? No, let's stick to vertical splits.
* Wait, looking at the labels: Left height is 3mm. Right height is 2.5mm. The middle section connects them. Usually, this implies the middle height is determined by the geometry. However, a simpler interpretation for this level is often splitting it into a top horizontal bar and bottom blocks, or vice versa.
* Let's try splitting horizontally into a top rectangle and two bottom squares/rectangles? No.
* Let's re-read the diagram carefully. It shows a central block with width 14mm? No, the dimension line for 14mm spans the *entire* top width. The dimension lines for 2mm are the widths of the "legs".
* So, Left Leg: $2 \text{ mm} \times 3 \text{ mm} = 6 \text{ mm}^2$.
* Right Leg: $2 \text{ mm} \times 2.5 \text{ mm} = 5 \text{ mm}^2$.
* Middle Section: Width = $14 - 2 - 2 = 10$ mm. What is its height? The diagram doesn't explicitly give the middle height. However, often in these worksheets, if a height isn't given, it might be inferred or I am misinterpreting the split.
* Let's look at Problem 4 again. Is it possible the shape is composed of a top rectangle $14 \times \text{something}$? No.
* Let's assume the "middle" part has the same height as the shortest side? Unlikely.
* Let's look at the dashed lines. They suggest splitting into 3 vertical columns.
* Column 1 (Left): $2 \times 3 = 6$.
* Column 3 (Right): $2 \times 2.5 = 5$.
* Column 2 (Middle): Width is 10. Height? If we look at similar problems, sometimes the inner height is the average? No.
* Let's look really closely at crop 4. The label "14 mm" is for the top edge. The label "3 mm" is the left edge. The label "2.5 mm" is the right edge. The labels "2 mm" are the widths of the protruding parts.
* There is a missing height for the middle section. *However*, looking at the visual alignment, the top of the middle section is aligned with the tops of the side sections. The bottom of the middle section is higher than the bottoms of the side sections.
* Actually, usually these shapes are "U" shaped or inverted "U". This is an inverted U (a bridge).
* If it is a bridge, the area is Total Outer Box minus the empty space.
* Outer Box Width = 14. Outer Box Height = 3 (tallest side). Area = $14 \times 3 = 42$.
* Empty Space Width = 10. Empty Space Height = $3 - 2.5$? No, the right leg is 2.5 high. The left is 3 high. This implies the bottom is not flat? Or the top is not flat?
* The top line is straight. The bottom has steps.
* Okay, let's assume the standard interpretation: The shape consists of three vertical rectangles standing on a common baseline? No, the drawing shows the middle part is "floating" or higher up? No, the dashed lines go from top to bottom.
* Let's assume the middle height is the same as the right height (2.5) or left (3)?
* Let's look at the dashed lines again. They separate the shape into three distinct vertical rectangular regions.
* Region 1: $2 \times 3 = 6$.
* Region 3: $2 \times 2.5 = 5$.
* Region 2: Width 10. Height? In many such textbook problems, if the height isn't specified, it might be equal to one of the adjacent sides, or there's a typo. BUT, looking at the right side, the 2.5mm label is for the *entire* right vertical strip. The 3mm is for the *entire* left vertical strip.
* If the top is flat, the heights must be measured from the bottom. If the bottom is flat, the tops would be stepped. The drawing shows a flat top and stepped bottom.
* Therefore, the Left Rectangle is $2 \times 3$. The Right Rectangle is $2 \times 2.5$.
* The Middle Rectangle connects them. Since the top is flat, the "height" of the middle rectangle depends on where its bottom is. The drawing shows the bottom of the middle section is aligned with... nothing specific.
* *Correction*: Often in these specific "Math Monks" worksheets, the middle height is inferred to be the same as the shorter side if not specified, OR the shape is symmetric and I'm misreading the numbers. But 3 and 2.5 are different.
* Let's try another approach: Horizontal Slices.
* Slice 1 (Top): A rectangle of width 14 and height... unknown?
* Slice 2 (Bottom Left): $2 \times (3 - h_{mid})$.
* Let's look at the provided solution key logic for similar problems. Usually, all vertical segments share a base line unless drawn otherwise. Here, the dashed lines suggest the shape is composed of 3 adjacent rectangles.
* If we assume the "floor" is flat, then the middle height must be given. It is not.
* However, if we look at the right angle symbols or grid, maybe the middle height is 2mm? No.
* Let's assume the question implies the middle section has the same height as the *right* section (2.5mm) because the left one sticks down further? Or the left one (3mm)?
* Let's look at the visual proportions. The middle part looks slightly taller than the right part.
* Actually, there is a possibility that the "14 mm" refers only to the middle part? No, the arrows clearly span the whole width.
* Let's reconsider the shape. Is it possible the height of the middle part is 2 mm? (Matching the width labels?) No logic for that.
* Let's look at Problem 5 for a clue on style. Problem 5 has clear internal dimensions. Problem 4 is ambiguous.
* *Standard Assumption*: In cases where a central height is missing in a "bridge" shape with a flat top, and the sides are different, it's often a typo in the question or the middle height is assumed to be the average? No.
* Let's look at the dashed lines again. They define the areas.
* Left Area: $2 \times 3 = 6$.
* Right Area: $2 \times 2.5 = 5$.
* Middle Area: Width 10. If we assume the bottom of the middle section aligns with the bottom of the *shorter* leg (right leg), then the height is 2.5. Area = $10 \times 2.5 = 25$. Total = $6+5+25 = 36$.
* If we assume it aligns with the *longer* leg (left leg), height is 3. Area = $10 \times 3 = 30$. Total = $6+5+30 = 41$.
* Visually, the left leg extends lower. The right leg is higher. The middle section seems to align with the right leg's bottom? No, the right leg is the short one. The left leg is the long one. The middle section appears to have a bottom edge higher than the left leg's bottom. It looks aligned with the right leg's bottom.
* Let's proceed with Height = 2.5 mm for the middle section.
* Calculation: $(2 \times 3) + (10 \times 2.5) + (2 \times 2.5) = 6 + 25 + 5 = 36 \text{ mm}^2$.

Problem 5
* Step 1: Split into three horizontal rectangles.
* Step 2: Top Rectangle: Width 7 yd, Height 3 yd. Area = $7 \times 3 = 21 \text{ yd}^2$.
* Step 3: Bottom Rectangle: Same dimensions as top (symmetric). Width 7 yd, Height 3 yd. Area = $21 \text{ yd}^2$.
* Step 4: Middle Rectangle: This is the tricky part. The total width is 7 yd. The "cutout" is 4 yd wide and 2 yd high.
* Wait, the shape is a C-block on its side? Or an E?
* Let's read the labels. Top width 7. Top height 3. Bottom height 3.
* The inner cutout has width 4 and height 2.
* Method: Calculate the area of the full bounding box and subtract the empty space.
* Full Height = $3 (\text{top}) + 2 (\text{middle gap}) + 3 (\text{bottom}) = 8$ yd.
* Full Width = 7 yd.
* Total Box Area = $7 \times 8 = 56 \text{ yd}^2$.
* Empty Space Area = $4 \text{ yd} \times 2 \text{ yd} = 8 \text{ yd}^2$.
* Shape Area = $56 - 8 = 48 \text{ yd}^2$.
* *Alternative Check*: Sum of parts.
* Top Bar: $7 \times 3 = 21$.
* Bottom Bar: $7 \times 3 = 21$.
* Middle Connector: The total width is 7. The empty space is 4 wide. So the solid part is $7 - 4 = 3$ yd wide. The height is 2 yd. Area = $3 \times 2 = 6$.
* Total = $21 + 21 + 6 = 48$. Matches.

Problem 6
* Step 1: Split vertically into a left rectangle and a right rectangle.
* Step 2: Left Rectangle: Width 5 ft? No, the bottom label "5 ft" spans the whole bottom? Or just the left part?
* Looking at the dimension lines: The "5 ft" is under the left section. The "3.8 ft" is above the left section's right neighbor? No.
* Let's trace the lines.
* Bottom-left width: 5 ft.
* Inner horizontal step: 3.8 ft.
* Left vertical height: 2.5 ft.
* Right vertical height: 8.8 ft.
* This implies an L-shape.
* Left Part (Bottom): Width 5 ft, Height 2.5 ft. Area = $5 \times 2.5 = 12.5 \text{ ft}^2$.
* Right Part (Vertical Tower): We need its width and height.
* Height: The total height on the right is 8.8 ft. The left part is 2.5 ft high. So the part sticking up is $8.8 - 2.5 = 6.3$ ft? Or is 8.8 the total height? Yes, usually outer labels are total.
* Width: The label "3.8 ft" is on the horizontal shelf. This usually indicates the width of that segment. If the left block is 5 ft wide, and the next segment is 3.8 ft wide... wait.
* Let's look at the shape again. It's an L shape rotated.
* Vertical Split:
* Left Rectangle: Width 5 ft. Height 2.5 ft. Area = $12.5$.
* Right Rectangle: Height 8.8 ft. What is the width? The label "3.8 ft" is positioned over the horizontal segment connecting the left block to the right tower. This implies the width of the right tower is NOT 3.8, but rather the distance between them? No, they are connected.
* Standard interpretation: The horizontal dimension "3.8 ft" applies to the width of the right-hand vertical column? Or the length of the horizontal arm?
* Let's assume the shape is composed of a bottom horizontal rectangle and a right vertical rectangle.
* Bottom Rectangle: Height 2.5 ft. Width = $5 + 3.8 = 8.8$ ft? No, the 5 ft label ends at the corner. The 3.8 ft label starts at the corner and goes to the right edge. So the total width is $5 + 3.8 = 8.8$ ft.
* So, Bottom Rectangle Area: Width 8.8 ft, Height 2.5 ft. Area = $8.8 \times 2.5 = 22 \text{ ft}^2$.
* Top Rectangle (sitting on the right side): Width 3.8 ft. Height = Total Height (8.8) - Bottom Height (2.5) = 6.3 ft. Area = $3.8 \times 6.3 = 23.94 \text{ ft}^2$.
* Total Area = $22 + 23.94 = 45.94 \text{ ft}^2$.
* *Alternative Split (Vertical)*:
* Left Rectangle: Width 5 ft, Height 2.5 ft. Area = $12.5$.
* Right Rectangle: Width 3.8 ft, Height 8.8 ft. Area = $3.8 \times 8.8 = 33.44$.
* Total = $12.5 + 33.44 = 45.94$. Matches.

Problem 7
* Step 1: Split into a large left rectangle and a smaller right rectangle? Or a big bottom rectangle and a top one?
* Step 2: Let's use a vertical split.
* Left Rectangle: Width is not explicitly given for just the left part. The top label "16 km" covers the top edge. The bottom label "20 km" covers the total width.
* So, the right "notch" has a width of $20 - 16 = 4$ km.
* The left rectangle has width 16 km. Height 10 km. Area = $16 \times 10 = 160 \text{ km}^2$.
* The right rectangle has width 4 km. Height 5 km. Area = $4 \times 5 = 20 \text{ km}^2$.
* Total Area = $160 + 20 = 180 \text{ km}^2$.
* *Check with Horizontal Split*:
* Bottom Rectangle: Width 20 km. Height 5 km (from the right side label). Area = $20 \times 5 = 100 \text{ km}^2$.
* Top Rectangle: Width 16 km. Height = Total Height (10) - Bottom Height (5) = 5 km. Area = $16 \times 5 = 80 \text{ km}^2$.
* Total = $100 + 80 = 180 \text{ km}^2$. Matches.

Problem 8
* Step 1: This is a T-shape or a table shape. Split into a top horizontal bar and two legs? Or a top bar and a middle block?
* Step 2: Let's split it into a top rectangle and two bottom legs.
* Top Rectangle: Width 22 mm. Height 2.5 mm. Area = $22 \times 2.5 = 55 \text{ mm}^2$.
* Legs: There are two legs.
* Left Leg: Width 4 mm. Height 6 mm. Area = $4 \times 6 = 24 \text{ mm}^2$.
* Right Leg: Width 4 mm. Height 6 mm. Area = $4 \times 6 = 24 \text{ mm}^2$.
* Total Area = $55 + 24 + 24 = 103 \text{ mm}^2$.
* *Wait*, let's check the connection. Are the legs attached to the bottom of the 2.5mm bar? Yes.
* Is there a middle gap? The total width is 22. The legs are 4 and 4. $4+4=8$. The gap is $22-8=14$. The shape shows a gap. So yes, just two legs.
* Calculation: $55 + 24 + 24 = 103$.

Final Answer:
1. 224 m²
2. 180 cm²
3. 58 yd²
4. 36 mm²
5. 48 yd²
6. 45.94 ft²
7. 180 km²
8. 103 mm²