Area of Polygons Worksheets - Math Monks - Free Printable

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 smaller rectangles, calculate the area of each rectangle (Length × Width), and then add them all together.

1. Shape 1
* Split: Divide the shape vertically into a left rectangle and a right rectangle.
* Left Rectangle: The height is 16 m. The total width is 19 m and the right part is 10 m, so the left width is $19 - 10 = 9$ m.
* Area = $16 \text{ m} \times 9 \text{ m} = 144 \text{ m}^2$.
* Right Rectangle: The width is 10 m and the height is 8 m.
* Area = $10 \text{ m} \times 8 \text{ m} = 80 \text{ m}^2$.
* Total Area: $144 + 80 = 224 \text{ m}^2$.

2. Shape 2
* Split: Divide the shape vertically into a left tall rectangle and a right short rectangle.
* Left Rectangle: The width is 10 cm and the height is 12 cm.
* Area = $10 \text{ cm} \times 12 \text{ cm} = 120 \text{ cm}^2$.
* Right Rectangle: The width is 22 cm. The total height on the left is 12 cm, and the bottom section height is 5 cm. Looking at the diagram, the right block sits on the same baseline. Its height is given as 5 cm.
* Area = $22 \text{ cm} \times 5 \text{ cm} = 110 \text{ cm}^2$.
* Total Area: $120 + 110 = 230 \text{ cm}^2$.

3. Shape 3
* Split: This looks like a large outer rectangle with a piece missing from the middle top, or three vertical rectangles. Let's use three vertical rectangles: Left, Middle, Right.
* Left Rectangle: Width is 1 yd. Height is not explicitly labeled on the outside, but the right side is 9 yd. Assuming symmetry or looking at the inner label "5 yd" which indicates the depth of the cutout. Let's look closer. The total width is 10 yd. The left leg is 1 yd wide. The right leg is 1 yd wide. This leaves the middle gap width as $10 - 1 - 1 = 8$ yd.
* The height of the outer legs is 9 yd. The inner vertical line says 5 yd. This usually means the empty space goes down 5 yards from the top. So the solid part in the middle has a height of $9 - 5 = 4$ yd.
* Let's try splitting it horizontally instead.
* Bottom Rectangle: Spans the full width of 10 yd. What is its height? The total height is 9 yd. The cutout depth is 5 yd. So the bottom solid part is $9 - 5 = 4$ yd high.
* Area = $10 \text{ yd} \times 4 \text{ yd} = 40 \text{ yd}^2$.
* Top Left Rectangle: Width 1 yd, Height 5 yd.
* Area = $1 \text{ yd} \times 5 \text{ yd} = 5 \text{ yd}^2$.
* Top Right Rectangle: Width 1 yd, Height 5 yd.
* Area = $1 \text{ yd} \times 5 \text{ yd} = 5 \text{ yd}^2$.
* Total Area: $40 + 5 + 5 = 50 \text{ yd}^2$.

4. Shape 4
* Split: Divide into three vertical rectangles: Left, Middle, Right.
* Left Rectangle: Width 2 mm, Height 3 mm.
* Area = $2 \text{ mm} \times 3 \text{ mm} = 6 \text{ mm}^2$.
* Right Rectangle: Width 2 mm, Height 2.5 mm.
* Area = $2 \text{ mm} \times 2.5 \text{ mm} = 5 \text{ mm}^2$.
* Middle Rectangle: We need its width and height.
* Total width is 14 mm. Left is 2 mm, Right is 2 mm. Middle width = $14 - 2 - 2 = 10$ mm.
* The height of the middle section connects the bottoms of the left and right pieces? No, looking at the diagram, the top edge is flat across the left and middle, but the right is lower? Or is the bottom flat?
* Let's look at the labels again. Left height 3mm. Right height 2.5mm. The dashed lines suggest we should split it into three distinct blocks sitting on a common baseline? No, the shape is contiguous.
* Let's assume the bottom is flat. Then the middle height is determined by the connections. Actually, usually these diagrams imply the "steps".
* Let's try splitting horizontally.
* Top strip: Height is not uniform.
* Let's go back to vertical splits assuming they sit on a flat bottom line.
* Left Block: $2 \times 3 = 6$.
* Right Block: $2 \times 2.5 = 5$.
* Middle Block: Width 10. What is its height? The diagram shows the top of the middle block aligns with the top of the left block? No, there is a step down. The label "14 mm" is the total width. The label "3 mm" is the left height. The label "2.5 mm" is the right height. There is no height label for the middle section directly. However, often in these problems, if not specified, the middle might align with one side. But wait, look at the dashed lines. They extend from the corners.
* Alternative interpretation: It's a large rectangle with chunks removed? No.
* Let's look at the shape again. It looks like a "U" shape but uneven.
* Let's assume the standard convention: The vertical dashed lines separate the shape into 3 rectangles.
* Rectangle 1 (Left): $2 \text{ mm} \times 3 \text{ mm} = 6 \text{ mm}^2$.
* Rectangle 3 (Right): $2 \text{ mm} \times 2.5 \text{ mm} = 5 \text{ mm}^2$.
* Rectangle 2 (Middle): Width is $14 - 2 - 2 = 10 \text{ mm}$. What is the height? The diagram doesn't explicitly give the middle height. However, looking at the geometry, the top of the middle section seems lower than the left (3mm) and higher than the right (2.5mm)? Or does it align with the right?
* Let's re-read the diagram carefully. Ah, the "3 mm" label is for the left vertical edge. The "2.5 mm" is for the right vertical edge. The top horizontal line of the middle section is NOT aligned with the left. It steps down. Is it aligned with the right? It looks like it steps down from left to middle, then stays flat to right? No, the right is a separate block.
* Let's look at Problem 4 again. Maybe the middle height is the same as the right? Or the left?
* Actually, usually, if a dimension is missing, it can be inferred. Is it possible the total height is 3mm everywhere except the right? No.
* Let's look at the dashed lines. They drop down from the inner corners. This implies we are calculating the area of the three vertical strips.
* Strip 1: $2 \times 3 = 6$.
* Strip 3: $2 \times 2.5 = 5$.
* Strip 2: Width 10. Height? If we assume the top of the middle part aligns with the top of the right part (2.5mm), then Area = $10 \times 2.5 = 25$. Total = $6+5+25 = 36$.
* If we assume the top of the middle part aligns with the left part (3mm), then Area = $10 \times 3 = 30$. Total = $6+5+30 = 41$.
* Let's look at the visual proportions. The middle top looks lower than the left top. It looks level with the right top. Let's assume the middle height is 2.5 mm.
* Wait, there is another possibility. Is the "14 mm" just the top width? Yes.
* Let's check if there's a different split. Horizontal split?
* Bottom rectangle of height 2.5 mm across the whole width 14 mm? Area = $14 \times 2.5 = 35$.
* Plus the extra bit on the top left. The left is 3 mm high, so the extra bit is $3 - 2.5 = 0.5$ mm high. Width is 2 mm. Area = $2 \times 0.5 = 1$.
* Total Area = $35 + 1 = 36 \text{ mm}^2$.
* This interpretation (that the right and middle share the same height of 2.5, and only the left sticks up) fits the visual "step" pattern best.
* Let's double check. If the middle was 3mm high, the right would be the only low one. The drawing shows the left is highest, and the middle/right look lower. The line between middle and right is solid, implying they are separate blocks or there is a change. But the top edge of the middle and right seem collinear in many such textbook problems unless marked otherwise. However, looking closely at crop 4, the top of the middle section and the top of the right section are NOT at the same level? Actually, the right block is labeled 2.5mm. The left is 3mm. The middle has no label.
* Let's look at the dashed lines again. The dashed line on the left drops from the inner corner of the "L" shape formed by the left block. The dashed line on the right drops from the inner corner of the right block. This confirms the vertical split method.
* Without an explicit height for the middle, we must infer. In similar problems, if the top surface is stepped, and only two heights are given for the ends, the middle often matches one of them. Visually, the middle top is lower than the left. It appears level with the right. I will proceed with Height = 2.5 mm for the middle.
* Calculation:
* Left: $2 \times 3 = 6$
* Middle: $10 \times 2.5 = 25$
* Right: $2 \times 2.5 = 5$
* Total: $6 + 25 + 5 = 36 \text{ mm}^2$.

5. Shape 5
* Split: This is a C-shape or bracket shape. We can split it into three horizontal rectangles: Top, Middle (the connecting vertical part?), No, let's do vertical slices.
* Vertical Slice 1 (Left Back): The total height is $3 + 2 + 3 = 8$ yd? No, the labels are for the segments.
* Top arm height: 3 yd.
* Gap height: 2 yd.
* Bottom arm height: 3 yd.
* Total Height = $3 + 2 + 3 = 8$ yd.
* The width of the vertical back spine is not explicitly given as a single number, but we have "7 yd" for the top width and "4 yd" for the inner cutout width.
* So, the thickness of the left vertical spine = Total Width - Inner Width = $7 - 4 = 3$ yd.
* Method 1: Subtraction
* Imagine a large bounding box of $7 \text{ yd} \times 8 \text{ yd}$. Area = 56 sq yd.
* Subtract the empty space in the middle right.
* The empty space has width 4 yd. Its height is the gap height, which is 2 yd.
* Wait, is the "4 yd" label the width of the empty space? Yes, it's inside the gap.
* Is the "2 yd" label the height of the empty space? Yes, it's next to the vertical gap.
* So, Area of void = $4 \text{ yd} \times 2 \text{ yd} = 8 \text{ yd}^2$.
* Total Area = Area of Big Box - Area of Void?
* Let's check the dimensions of the big box.
* Width = 7 yd.
* Height = $3 (\text{top}) + 2 (\text{gap}) + 3 (\text{bottom}) = 8$ yd.
* Big Box Area = $7 \times 8 = 56 \text{ yd}^2$.
* The void is on the right side. Does it go all the way to the edge? Yes, the shape is open on the right.
* So, Area = $56 - 8 = 48 \text{ yd}^2$.
* Method 2: Addition (Verification)
* Split into Left Vertical Bar, Top Horizontal Arm, Bottom Horizontal Arm.
* Left Vertical Bar: Width = $7 - 4 = 3$ yd. Height = 8 yd. Area = $3 \times 8 = 24 \text{ yd}^2$.
* Remaining Top Arm: Width = 4 yd. Height = 3 yd. Area = $4 \times 3 = 12 \text{ yd}^2$.
* Remaining Bottom Arm: Width = 4 yd. Height = 3 yd. Area = $4 \times 3 = 12 \text{ yd}^2$.
* Total = $24 + 12 + 12 = 48 \text{ yd}^2$.
* Both methods match.

6. Shape 6
* Split: L-shape. Split vertically into a left rectangle and a right rectangle.
* Left Rectangle:
* Width = 5 ft? No, "5 ft" is the total width at the bottom.
* Height = 2.5 ft.
* Wait, let's look at the labels.
* Bottom total width = 5 ft.
* Left height = 2.5 ft.
* Top part of the right leg extends up. The label "8.8 ft" is the total height of the right side? Or just the top segment? The arrow/line spans the entire right vertical edge. So Total Height = 8.8 ft.
* Label "3.8 ft" is the width of the top horizontal segment of the right leg? Or the inner horizontal segment? It is placed on the inner horizontal shelf. So the width of the "cutout" or the left part's width?
* Let's parse the geometry.
* It's an L-shape lying on its back? No, standard L.
* Total Width at bottom = 5 ft.
* Total Height at right = 8.8 ft.
* Height of the left "foot" = 2.5 ft.
* Length of the inner horizontal step = 3.8 ft.
* We can split this into two rectangles: A bottom horizontal one and a top vertical one.
* Rectangle 1 (Bottom):
* Height = 2.5 ft.
* Width = 5 ft.
* Area = $5 \times 2.5 = 12.5 \text{ ft}^2$.
* Rectangle 2 (Top Right):
* This sits on top of the bottom rectangle.
* Height = Total Height - Bottom Height = $8.8 - 2.5 = 6.3$ ft.
* Width = ?
* We know the total width is 5 ft. We know the inner horizontal segment is 3.8 ft. This 3.8 ft represents the width of the empty space to the left of the top tower? Or the width of the bottom part extending to the left?
* Looking at the diagram, the "3.8 ft" is labeling the horizontal surface of the lower block that is exposed. This means the width of the left part is 3.8 ft?
* If the left part width is 3.8 ft, then the width of the right tower is $5 - 3.8 = 1.2$ ft.
* Let's verify this interpretation. The label 3.8 ft is above the left section. The label 5 ft is the total bottom width. So yes, the left section is 3.8 ft wide.
* So, Right Tower Width = $5 - 3.8 = 1.2$ ft.
* Right Tower Height = 6.3 ft.
* Area = $1.2 \times 6.3$.
* $1.2 \times 6 = 7.2$. $1.2 \times 0.3 = 0.36$. Total = 7.56 ft².
* Total Area: $12.5 + 7.56 = 20.06 \text{ ft}^2$.
* *Alternative Split:* Vertical split.
* Left Rectangle: Width 3.8 ft, Height 2.5 ft. Area = $3.8 \times 2.5 = 9.5 \text{ ft}^2$.
* Right Rectangle: Width $(5 - 3.8) = 1.2$ ft. Height 8.8 ft. Area = $1.2 \times 8.8$.
* $1.2 \times 8.8 = 10.56 \text{ ft}^2$.
* Total Area = $9.5 + 10.56 = 20.06 \text{ ft}^2$.
* Matches.

7. Shape 7
* Split: L-shape. Split vertically into a left large rectangle and a right small rectangle.
* Left Rectangle:
* Width? Total bottom width is 20 km. Top width is 16 km. The right part sticks out? No, the left part is wider.
* Looking at the shape: The top edge is 16 km. The bottom edge is 20 km. The left edge is 10 km. The right edge of the lower part is 5 km.
* This implies the shape is composed of a top block and a bottom block, or left and right.
* Let's split horizontally.
* Top Rectangle:
* Width = 16 km.
* Height = ? Total left height is 10 km. The right lower height is 5 km. Assuming the bottom is flat, the top block sits on the bottom block?
* If we split horizontally at the level of the right shoulder (5 km high):
* Bottom Rectangle: Spans the full width? No, the shape widens at the bottom.
* Let's look at the coordinates.
* Left side height = 10 km.
* Bottom width = 20 km.
* Top width = 16 km.
* Right lower vertical side = 5 km.
* This suggests the "notch" is at the top right.
* So, we have a main body of height 5 km?
* Let's split into two vertical rectangles.
* Left Rectangle:
* Height = 10 km.
* Width = 16 km? No, the top width is 16 km. Since the left side is straight up, the width of this left column is 16 km.
* Area = $16 \text{ km} \times 10 \text{ km} = 160 \text{ km}^2$.
* Wait, if the left part is 16 km wide and 10 km high, what is the rest?
* Total bottom width is 20 km. So there is an extra piece on the right?
* $20 - 16 = 4$ km.
* This extra piece on the right has width 4 km. What is its height?
* The label "5 km" is on the far right vertical edge. So the height of this right extension is 5 km.
* Area = $4 \text{ km} \times 5 \text{ km} = 20 \text{ km}^2$.
* Total Area = $160 + 20 = 180 \text{ km}^2$.
* Let's double check this interpretation.
* Does the diagram show the left part being the tall one? Yes.
* Does the top width correspond to the tall part? Yes, 16 km.
* Does the bottom width correspond to the total width? Yes, 20 km.
* Does the right short side correspond to the short part? Yes, 5 km.
* So, Shape = (Left Tall Rect) + (Right Short Rect).
* Left Rect: $16 \times 10 = 160$.
* Right Rect: Width is $20 - 16 = 4$. Height is 5. Area = $4 \times 5 = 20$.
* Total = 180.

8. Shape 8
* Split: This is a T-shape upside down? Or a bridge? It's a wide top bar with two legs.
* Top Rectangle:
* Width = 22 mm.
* Height = 2.5 mm.
* Area = $22 \times 2.5$.
* $22 \times 2 = 44$. $22 \times 0.5 = 11$. Total = 55 mm².
* Legs:
* There are two legs.
* Left Leg Width = 4 mm.
* Right Leg Width = 4 mm.
* What is the height of the legs?
* The label "6 mm" is on the outer vertical edge of the left leg. So Height = 6 mm.
* Assuming symmetry, Right Leg Height = 6 mm.
* Area of Left Leg = $4 \text{ mm} \times 6 \text{ mm} = 24 \text{ mm}^2$.
* Area of Right Leg = $4 \text{ mm} \times 6 \text{ mm} = 24 \text{ mm}^2$.
* Total Area:
* Top + Left Leg + Right Leg
* $55 + 24 + 24 = 103 \text{ mm}^2$.
* Check for overlap: The "6 mm" height label is for the leg sticking down. The "2.5 mm" is the thickness of the top bar. They are distinct parts. The diagram shows the legs attached to the bottom of the top bar. So simple addition works.

Final Answer:
1. Area = 224 m²
2. Area = 230 cm²
3. Area = 50 yd²
4. Area = 36 mm²
5. Area = 48 yd²
6. Area = 20.06 ft²
7. Area = 180 km²
8. Area = 103 mm²