Finding Area and Perimeter of Irregular shapes - Math Worksheets ... - Free Printable

Let’s solve each problem one by one. We’ll find both the area and perimeter for each irregular shape.

We’ll break each shape into rectangles, calculate area of each part, then add them up. For perimeter, we’ll add all the outer side lengths — being careful not to double-count or miss any sides.

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Problem 1:

Shape looks like an “L” turned sideways.

Given sides:
Top = 18 in
Right = 21 in
Bottom left vertical = 8 in
Left middle horizontal = 5 in

We can split this into two rectangles:

- Rectangle A (top): width = 18 in, height = ?
The total right side is 21 in, and bottom part is 8 in → so top rectangle height = 21 - 8 = 13 in
Area A = 18 × 13 = 234 in²

- Rectangle B (bottom left): width = 5 in, height = 8 in
Area B = 5 × 8 = 40 in²

Total Area = 234 + 40 = 274 in²

Perimeter: Walk around the outside.

Start at top-left corner:

→ Right 18 in
↓ Down 13 in (to where the step starts)
← Left 5 in (the inner step)
↓ Down 8 in
← Left? Wait — actually, let’s list all outer edges:

Actually, better way: trace the full outer boundary.

From top-left:

1. Right 18 in
2. Down 21 in (full right side)
3. Left ? — but wait, bottom has a step. Let’s reconstruct:

Actually, from diagram:

The full outer path:

Start at top-left:

- Right 18 in
- Down 21 in
- Left (18 - 5) = 13 in? No — look again.

Wait — better to label missing sides.

We know:

Total width on top = 18 in
On bottom, there’s a 5 in segment going left, then down 8 in, then right to meet the big rectangle.

So the bottom horizontal must be 18 - 5 = 13 in? But that doesn’t match.

Actually, let’s think differently.

The shape is made of:

- A big rectangle 18 in wide × 21 in tall, BUT with a bite taken out of the bottom-left corner: 5 in wide × 8 in high.

But no — looking at labels:

It says:

Top: 18 in
Right: 21 in
Then on the left side, it shows 5 in (horizontal inward), then 8 in (down).

So the full left side is not straight — it goes down 13 in (since 21 - 8 = 13), then right 5 in, then down 8 in.

So outer perimeter:

Start at top-left:

1. Right 18 in
2. Down 21 in
3. Left (18 - 5) = 13 in? Wait — no.

Actually, after going down 21 in on the right, you go left along the bottom. How long is the bottom?

The bottom consists of two parts: the part under the 5-in indent, and the rest.

Actually, the total width is 18 in. The indent is 5 in from the left, so the bottom horizontal segment on the right is 18 - 5 = 13 in.

Then you go up 8 in (the vertical of the indent), then left 5 in (the horizontal of the indent), then up 13 in to close.

Wait — let's list all outer sides in order:

Imagine walking clockwise:

Start at top-left corner:

1. Move right → 18 in
2. Move down → 21 in
3. Move left → 13 in (because total width 18, minus the 5-in indent on left)
4. Move up → 8 in (this is the vertical part of the indent)
5. Move left → 5 in (this is the horizontal part of the indent)
6. Move up → 13 in (to get back to start; since 21 - 8 = 13)

Now add them:

18 + 21 + 13 + 8 + 5 + 13 = let’s compute:

18+21=39
39+13=52
52+8=60
60+5=65
65+13=78 in

Perimeter = 78 in

Area we already did: 274 in²

Wait — let me verify area another way.

Alternative: Full rectangle 18x21 = 378 in²
Minus the missing rectangle: 5x8 = 40 in²
So 378 - 40 = 338? That contradicts earlier.

Oh! I see my mistake.

In my first method, I said top rectangle is 18x13 — but if the total height is 21, and the bottom part is 8, then the top part should be 21 - 8 = 13, yes.

But then the bottom rectangle is only 5 in wide? But in reality, the bottom part is not just 5x8 — because the 5 in is the indent, meaning the bottom rectangle is actually the part that sticks out.

I think I confused myself.

Let me redraw mentally.

The shape has:

- Top edge: 18 in
- Right edge: 21 in
- On the left, instead of going straight down, it goes down some amount, then right 5 in, then down 8 in.

Actually, from the diagram description:

It says:

Left side: 5 in (probably horizontal), then 8 in (vertical down)

And right side is 21 in.

So the total height on the right is 21 in.

The left side has a "step": from top, go down X inches, then right 5 in, then down 8 in.

Since the total height is 21, and the last down is 8, then the first down must be 21 - 8 = 13 in.

So the shape is:

- A rectangle on top: 18 in wide × 13 in high
- Plus a rectangle on the bottom-left: 5 in wide × 8 in high

Yes, that makes sense.

Area = (18×13) + (5×8) = 234 + 40 = 274 in²

For perimeter, as I walked:

1. Right 18
2. Down 21
3. Left 13 (because 18 - 5 = 13 — the bottom-right part)
4. Up 8 (the vertical of the step)
5. Left 5 (the horizontal of the step)
6. Up 13 (back to start)

Sum: 18+21+13+8+5+13 = 78 in

Yes.

But let's confirm with another approach.

Total outer sides:

Top: 18
Right: 21
Bottom: consists of two segments: the part under the main body and the part under the step.

Actually, the bottom-most edge is only the 5 in part? No.

When you go down the right side 21 in, then you go left along the bottom. How far? Until you hit the step. Since the step is 5 in from the left, and total width 18, you go left 13 in to reach the step's right edge.

Then you go up 8 in (along the step's vertical), then left 5 in (along the step's top), then up 13 in to close.

Yes, same as before.

Perimeter = 78 in

Area = 274 in²

Okay, moving on.

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Problem 2:

Shape: L-shape, wider on top.

Given:

Top: 41 in
Right: 17 in
Middle horizontal: 24 in (going left from right side)
Vertical drop: 9 in

So, we can split into two rectangles:

- Top rectangle: width 41 in, height ?
Total right side is 17 in, and there's a drop of 9 in, so the top part height = 17 - 9 = 8 in? Not necessarily.

Actually, the 9 in is the vertical part of the step down.

So, the shape has:

- A large rectangle on top: 41 in wide, and height H1
- Then a smaller rectangle hanging down on the right: width W2, height 9 in

From the diagram:

After the top 41 in, on the right side, it goes down 17 in total.

But there's a horizontal segment labeled 24 in going left from the right side, at some point.

Probably, the 24 in is the length of the top part of the lower section.

Standard way: the shape is like a rectangle with a bite taken out of the bottom-left, but here it's extended on the right.

Actually, it's an L-shape rotated.

Let me define:

The full height on the right is 17 in.

There is a horizontal line inside, 24 in long, which is probably the top of the lower rectangle.

And a vertical drop of 9 in.

So, likely:

- The upper rectangle is 41 in wide, and its height is 17 - 9 = 8 in? But that might not be right.

Perhaps the 9 in is the height of the lower part.

Assume:

The shape consists of:

- Rectangle A (top): width 41 in, height h1
- Rectangle B (bottom-right): width w2, height 9 in

From the diagram, the horizontal segment labeled 24 in is likely the width of the bottom rectangle, or the overlap.

Notice that the top is 41 in, and there's a 24 in segment going left from the right side — so probably, the bottom rectangle extends 24 in to the left from the right edge.

So, the width of the bottom rectangle is 24 in.

Then, the remaining part on the left is 41 - 24 = 17 in, which is the width of the top-only part.

Height of top part: since total right height is 17 in, and bottom part is 9 in, then top part height = 17 - 9 = 8 in.

Is that correct? Let's see.

If bottom rectangle is 24 in wide and 9 in high, sitting at the bottom-right.

Then above it, the top rectangle spans the full 41 in wide, but only for the height above the bottom rectangle.

The total height on the right is 17 in, which includes the bottom 9 in and the top part, so top part height = 17 - 9 = 8 in.

Yes.

So:

Area A (top) = 41 × 8 = 328 in²
Area B (bottom) = 24 × 9 = 216 in²
Total Area = 328 + 216 = 544 in²

Perimeter: walk around.

Start at top-left:

1. Right 41 in
2. Down 17 in (full right side)
3. Left 24 in (along the bottom of the bottom rectangle)
4. Up 9 in (left side of bottom rectangle)
5. Left (41 - 24) = 17 in (along the bottom of the top-only part)
6. Up 8 in (left side of top part, to close)

Add them:

41 + 17 + 24 + 9 + 17 + 8

Calculate:

41+17=58
58+24=82
82+9=91
91+17=108
108+8=116 in

Perimeter = 116 in

Verify area another way: full rectangle 41x17 = 697 in², minus the missing part on bottom-left.

Missing part: width = 41 - 24 = 17 in, height = 9 in? But the bottom-left is empty, size 17 in wide × 9 in high.

So area = 41*17 - 17*9 = 17*(41-9) = 17*32 = 544 in² — same as before. Good.

Perimeter: when you remove a rectangle from the corner, the perimeter changes.

Original rectangle perimeter: 2*(41+17)=116 in

When you cut out a rectangle from the bottom-left corner, you remove two sides but add two new sides of the same length, so perimeter remains the same? In this case, we're not cutting out; we have an L-shape that is additive.

In our calculation, we got 116 in, which matches the outer rectangle perimeter, but is that correct?

In our path, we have:

- Top: 41
- Right: 17
- Bottom: 24 (of bottom rect) + 17 (of top-only part) = 41? No, in the path, we went left 24, then up 9, then left 17, so the bottom is not continuous.

The actual bottom edge is only the 24 in part? No, the very bottom is the 24 in segment, and above it on the left is the 17 in segment at a higher level.

In terms of outer boundary, the lowest points are the bottom of the bottom rectangle, which is 24 in long.

Then on the left, the bottom of the top part is at a higher elevation, so it's not part of the bottom perimeter.

In our earlier path, we have:

After going down 17 on right, we go left 24 (this is the bottom edge), then up 9 (this is the left side of the bottom rectangle), then left 17 (this is the bottom edge of the top-only part, which is at y=9 from bottom), then up 8 to top.

So the perimeter includes both horizontal segments at different heights.

Total perimeter 116 in seems correct.

Another way: sum all outer sides.

List:

- Top: 41
- Right: 17
- Bottom-right horizontal: 24
- Left side of bottom rectangle: 9 (up)
- Bottom-left horizontal (of top part): 17 (left)
- Left side of top part: 8 (up)

Same as before.

Yes.

So Problem 2: Area = 544 in², Perimeter = 116 in

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Problem 3:

Units in yards.

Shape: similar to problem 1.

Given:

Top: 18 yd
Left middle horizontal: 11 yd
Right: 12 yd
Bottom left vertical: 7 yd

So, total height on right is 12 yd.

The step on left: 11 yd horizontal, then 7 yd down.

So, the first down part on left is 12 - 7 = 5 yd.

Split into two rectangles:

- Top: 18 yd wide × 5 yd high (since 12 - 7 = 5)
Area A = 18 × 5 = 90 yd²

- Bottom-left: 11 yd wide × 7 yd high
Area B = 11 × 7 = 77 yd²

Total Area = 90 + 77 = 167 yd²

Perimeter:

Walk clockwise:

Start top-left:

1. Right 18 yd
2. Down 12 yd
3. Left (18 - 11) = 7 yd (bottom-right part)
4. Up 7 yd (vertical of step)
5. Left 11 yd (horizontal of step)
6. Up 5 yd (to close; 12 - 7 = 5)

Sum: 18 + 12 + 7 + 7 + 11 + 5

Calculate:

18+12=30
30+7=37
37+7=44
44+11=55
55+5=60 yd

Perimeter = 60 yd

Verify area: full rectangle 18x12=216, minus missing part: width 18-11=7, height 7, so 7x7=49, 216-49=167 — good.

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Problem 4:

Similar to problem 2.

Given:

Top: 46 in
Right: 25 in
Middle horizontal: 28 in
Vertical drop: 10 in

So, bottom rectangle width = 28 in (assumed, as in prob 2)

Height of bottom part = 10 in

Total right height = 25 in, so top part height = 25 - 10 = 15 in

Width of top-only part = 46 - 28 = 18 in

Area:

Top rectangle: 46 × 15 = 690 in²
Bottom rectangle: 28 × 10 = 280 in²
Total Area = 690 + 280 = 970 in²

Perimeter:

Path:

1. Right 46
2. Down 25
3. Left 28
4. Up 10
5. Left (46 - 28) = 18
6. Up 15

Sum: 46 + 25 + 28 + 10 + 18 + 15

Calculate:

46+25=71
71+28=99
99+10=109
109+18=127
127+15=142 in

Perimeter = 142 in

Verify area: full 46x25=1150, minus missing 18x10=180, 1150-180=970 — good.

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Problem 5:

Units in feet.

Shape: L-shape, but oriented differently.

Given:

Left: 11 ft
Top-right vertical: 5 ft
Bottom: 29 ft
Right-middle horizontal: 15 ft

So, total width on bottom is 29 ft.

The step on top-right: 5 ft down, then 15 ft left? Labels say:

"5 ft" is vertical, "15 ft" is horizontal.

Probably, from the top-right, it goes down 5 ft, then left 15 ft.

Total height on left is 11 ft.

So, the bottom part height is 11 - 5 = 6 ft? Let's see.

Split into two rectangles:

- Left rectangle: width w1, height 11 ft
- Bottom-right rectangle: width 15 ft, height h2

From diagram, the bottom is 29 ft total.

The left part width plus the right part width should be 29 ft.

The right part is 15 ft wide (given).

So left part width = 29 - 15 = 14 ft

Height of left part is 11 ft.

The right part: it has a vertical drop of 5 ft from the top, but since the left is 11 ft tall, and the right part is shorter, its height is 11 - 5 = 6 ft?

The label "5 ft" is the difference in height.

So, bottom-right rectangle: 15 ft wide × 6 ft high (since 11 - 5 = 6)

Area:

Left: 14 × 11 = 154 ft²
Right: 15 × 6 = 90 ft²
Total Area = 154 + 90 = 244 ft²

Perimeter:

Walk clockwise, start at top-left:

1. Right 14 ft (width of left part)
2. Down 5 ft (the step down on right)
3. Right 15 ft (to end of bottom-right)
4. Down 6 ft (height of bottom-right)
5. Left 29 ft (full bottom)
6. Up 11 ft (left side)

But after step 3, we are at bottom-right corner, then down 6 ft? No, if we go right 15 ft from the step, we are at the far right, then we need to go down to the bottom.

Actually, from top-left:

- Right 14 ft (to the step)
- Down 5 ft (this is the vertical drop)
- Right 15 ft (along the top of the bottom-right part)
- Down 6 ft (to bottom)
- Left 29 ft (along bottom)
- Up 11 ft (left side)

But when we go left 29 ft, that covers the entire bottom, including under the left part.

Then up 11 ft closes it.

Now, is there any overlap? Let's list the segments:

1. Top-left to step: right 14 ft
2. Step down: down 5 ft
3. To far right: right 15 ft
4. Down to bottom: down 6 ft
5. Left along bottom: left 29 ft
6. Up left side: up 11 ft

Sum: 14 + 5 + 15 + 6 + 29 + 11

Calculate:

14+5=19
19+15=34
34+6=40
40+29=69
69+11=80 ft

Perimeter = 80 ft

Verify area: full rectangle if no step: width 29, height 11 = 319 ft², minus the missing top-right rectangle: width 15 ft, height 5 ft = 75 ft², so 319 - 75 = 244 ft² — good.

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Problem 6:

Units in yards.

Shape: L-shape, standing vertically.

Given:

Left: 10 yd
Top-right vertical: 6 yd
Bottom: 10 yd
Middle horizontal: 4 yd

So, total width on bottom is 10 yd.

The step on top-right: 6 yd down, then 4 yd left? Labels:

"6 yd" is vertical, "4 yd" is horizontal.

Probably, from the top-right, it goes down 6 yd, then left 4 yd.

Total height on left is 10 yd.

So, the bottom part height is 10 - 6 = 4 yd? Let's see.

Split:

- Left rectangle: width w1, height 10 yd
- Bottom-right rectangle: width 4 yd, height h2

Bottom total width 10 yd.

The right part is 4 yd wide (given).

So left part width = 10 - 4 = 6 yd

Height of left part is 10 yd.

The right part: height is 10 - 6 = 4 yd? Since the drop is 6 yd from top, so the right part height is 4 yd.

Area:

Left: 6 × 10 = 60 yd²
Right: 4 × 4 = 16 yd²
Total Area = 60 + 16 = 76 yd²

Perimeter:

Start top-left:

1. Right 6 ft (width of left part)
2. Down 6 ft (step down)
3. Right 4 ft (to far right)
4. Down 4 ft (height of right part)
5. Left 10 ft (full bottom)
6. Up 10 ft (left side)

Sum: 6 + 6 + 4 + 4 + 10 + 10 = 40 yd

Calculate: 6+6=12, +4=16, +4=20, +10=30, +10=40 yd

Perimeter = 40 yd

Verify area: full 10x10=100, minus missing top-right: width 4, height 6, area 24, 100-24=76 — good.

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Problem 7:

Units in inches.

Shape: L-shape, long on bottom.

Given:

Left: 10 in
Top-right vertical: 5 in
Bottom: 34 in
Right-middle horizontal: 20 in

So, total width on bottom is 34 in.

The step on top-right: 5 in down, then 20 in left? Labels:

"5 in" vertical, "20 in" horizontal.

Probably, from the top-right, it goes down 5 in, then left 20 in.

Total height on left is 10 in.

So, the bottom part height is 10 - 5 = 5 in.

Split:

- Left rectangle: width w1, height 10 in
- Bottom-right rectangle: width 20 in, height 5 in

Bottom total width 34 in.

The right part is 20 in wide.

So left part width = 34 - 20 = 14 in

Area:

Left: 14 × 10 = 140 in²
Right: 20 × 5 = 100 in²
Total Area = 140 + 100 = 240 in²

Perimeter:

Start top-left:

1. Right 14 in
2. Down 5 in
3. Right 20 in
4. Down 5 in (since height of right part is 5 in)
5. Left 34 in
6. Up 10 in

Sum: 14 + 5 + 20 + 5 + 34 + 10

Calculate: 14+5=19, +20=39, +5=44, +34=78, +10=88 in

Perimeter = 88 in

Verify area: full 34x10=340, minus missing top-right: width 20, height 5, area 100, 340-100=240 — good.

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Problem 8:

Units in yards.

Similar to problem 7.

Given:

Left: 10 yd
Top-right vertical: 6 yd
Bottom: 37 yd
Right-middle horizontal: 22 yd

So, total width on bottom is 37 yd.

Step: down 6 yd, then left 22 yd.

Total height left: 10 yd.

So bottom part height = 10 - 6 = 4 yd.

Split:

- Left rectangle: width w1, height 10 yd
- Bottom-right rectangle: width 22 yd, height 4 yd

Left part width = 37 - 22 = 15 yd

Area:

Left: 15 × 10 = 150 yd²
Right: 22 × 4 = 88 yd²
Total Area = 150 + 88 = 238 yd²

Perimeter:

Start top-left:

1. Right 15 yd
2. Down 6 yd
3. Right 22 yd
4. Down 4 yd
5. Left 37 yd
6. Up 10 yd

Sum: 15 + 6 + 22 + 4 + 37 + 10

Calculate: 15+6=21, +22=43, +4=47, +37=84, +10=94 yd

Perimeter = 94 yd

Verify area: full 37x10=370, minus missing top-right: 22x6=132, 370-132=238 — good.

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Now, compiling all answers:

Problem 1: Area = 274 in², Perimeter = 78 in
Problem 2: Area = 544 in², Perimeter = 116 in
Problem 3: Area = 167 yd², Perimeter = 60 yd
Problem 4: Area = 970 in², Perimeter = 142 in
Problem 5: Area = 244 ft², Perimeter = 80 ft
Problem 6: Area = 76 yd², Perimeter = 40 yd
Problem 7: Area = 240 in², Perimeter = 88 in
Problem 8: Area = 238 yd², Perimeter = 94 yd

Final Answer:
1. Area: 274 in², Perimeter: 78 in
2. Area: 544 in², Perimeter: 116 in
3. Area: 167 yd², Perimeter: 60 yd
4. Area: 970 in², Perimeter: 142 in
5. Area: 244 ft², Perimeter: 80 ft
6. Area: 76 yd², Perimeter: 40 yd
7. Area: 240 in², Perimeter: 88 in
8. Area: 238 yd², Perimeter: 94 yd