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

Let’s solve each problem one by one. We’ll find the perimeter (add all outer sides) and area (break shape into rectangles, calculate each area, then add).

---

Problem 1:

Shape is L-shaped.

Given sides:
Left side = 10 in
Bottom = 25 in
Top right horizontal = 15 in
Vertical drop on right = 4 in

We need to find missing sides:

- Top left horizontal: total bottom is 25 in, right part is 15 in → so left top = 25 - 15 = 10 in
- Right vertical full height: left side is 10 in, but there’s a 4 in drop → so lower right vertical = 10 - 4 = 6 in

Now list all outer sides for perimeter:

Start from top left, go clockwise:

→ 10 in (top left)
→ 4 in (down)
→ 15 in (right)
→ 6 in (down)
→ 25 in (bottom)
→ 10 in (up left side)

Wait — that double counts? Let’s trace carefully.

Actually, better way: walk around the outside.

From bottom-left corner:

→ Up 10 in
→ Right 10 in (since 25 - 15 = 10)
→ Down 4 in
→ Right 15 in
→ Down 6 in (because 10 - 4 = 6)
→ Left 25 in (back to start)

But wait — when going down 6 in, then left 25 in, we’re back to start? That doesn’t close properly.

Better approach: use “rectangle completion” or just sum all visible + inferred outer edges.

Standard method for L-shape:

Perimeter = sum of all outer sides.

Label vertices:

Imagine coordinates:

Bottom-left: (0,0)
Bottom-right: (25,0)
Then up to (25,6) because right side has 6 in vertical (since total height is 10, minus 4 drop)
Then left to (10,6) — because 25 - 15 = 10
Then up to (10,10)
Then left to (0,10)
Then down to (0,0)

So sides:

(0,0) to (25,0): 25
(25,0) to (25,6): 6
(25,6) to (10,6): 15
(10,6) to (10,10): 4
(10,10) to (0,10): 10
(0,10) to (0,0): 10

Add them: 25 + 6 + 15 + 4 + 10 + 10 = 70 inches

Area: break into two rectangles.

Option 1: big rectangle minus small cutout? Or split vertically/horizontally.

Split horizontally:

Bottom rectangle: width 25, height 6 → area = 25 × 6 = 150
Top rectangle: width 10, height 4 → area = 10 × 4 = 40
Total area = 150 + 40 = 190 sq in

Check another way: split vertically.

Left rectangle: 10 wide × 10 high = 100
Right rectangle: 15 wide × 6 high = 90
Total = 190 → same.

✔ Perimeter: 70 in
✔ Area: 190 in²

---

Problem 2:

Given:

Top = 33 in
Right side = 23 in
Left side has 13 in horizontal, then 12 in down.

Find missing parts.

Total height should be consistent.

From left: goes right 13 in, then down 12 in. Then must go right more to reach total width 33 in → so next horizontal = 33 - 13 = 20 in

Then down to match right side 23 in → so far down 12 in, need to go down another 23 - 12 = 11 in

Then left along bottom: should be 33 in? Wait, let's map it.

Vertices:

Start bottom-left: (0,0)
Up to (0,23) — since right side is 23, assume symmetric height? Not necessarily.

Actually, given:

Top edge: 33 in
Right edge: 23 in
On left: from top, go right 13 in, then down 12 in.

So from top-left (0,23), go right 13 to (13,23), then down 12 to (13,11), then right to (33,11) — because total width 33, so 33 - 13 = 20 in right
Then down to (33,0) — distance = 11 in (since 23 - 12 = 11)
Then left to (0,0) — 33 in

Sides:

(0,0) to (0,23): 23
(0,23) to (13,23): 13
(13,23) to (13,11): 12
(13,11) to (33,11): 20
(33,11) to (33,0): 11
(33,0) to (0,0): 33

Sum: 23 + 13 + 12 + 20 + 11 + 33 = let’s compute:

23+13=36; 36+12=48; 48+20=68; 68+11=79; 79+33=112 in

Area: split into two rectangles.

Top rectangle: 13 wide × 12 high? No.

Better: entire bounding box minus nothing? Actually, it’s like a rectangle with a bite taken out? No, it’s stepped.

Split at y=11.

Bottom rectangle: full width 33, height 11 → 33×11 = 363
Top rectangle: only left part, width 13, height 12 (from y=11 to y=23) → 13×12 = 156
Total area = 363 + 156 = 519 sq in

Check: if we did full rectangle 33×23 = 759, minus the missing part? The missing part would be from x=13 to 33, y=11 to 23 → width 20, height 12 → 240. 759 - 240 = 519 → yes.

✔ Perimeter: 112 in
✔ Area: 519 in²

---

Problem 3:

Given:

Top = 18 in
Right side = 15 in
Left side: down 5 in, then right 6 in.

So from top-left (0,15), down 5 to (0,10), then right 6 to (6,10), then up? To meet top? No.

Actually, after going right 6 in at y=10, we need to go up to connect to top? But top is at y=15.

Wait — probably: from (6,10) go up to (6,15)? But then how to connect to top-right?

Assume:

Start bottom-left: (0,0)
Up to (0,10) — because left side has 5 in down from top, so if total height is 15, then from bottom to first step is 10? Let’s think.

Given: left side has segment labeled 5 in (vertical) and 6 in (horizontal). Probably meaning: from top-left, go down 5 in, then right 6 in.

So:

Top-left (0,15)
Down 5 to (0,10)
Right 6 to (6,10)
Then up? To where? Must go up to meet top? But top is already passed.

Perhaps the 15 in is the full right side.

So from (6,10), go up to (6,15)? Then right to (18,15)? Then down to (18,0)? Then left to (0,0)? But then left side from (0,0) to (0,10) is 10 in, not matching.

I think I have it:

The shape is like a rectangle with a notch on the left bottom.

Total width 18 in, total height 15 in.

On left side: from bottom, go up 6 in? Label says "5 in" and "6 in" on left.

Looking at diagram description: "5 in" vertical, "6 in" horizontal on left.

Probably: from top-left, go down 5 in, then right 6 in, then down to bottom? But then height wouldn't match.

Alternative interpretation: the 5 in is the height of the left protrusion, 6 in is its width.

So the main body is 18x15, but on the left bottom, there's a rectangle sticking out? No, usually it's indented.

Standard way: the shape has a "step" on the left.

Let me define points:

Start at bottom-left: (0,0)
Go right to (6,0) — this is the 6 in horizontal
Then up to (6,5) — this is the 5 in vertical
Then right to (18,5) — because total width 18, so 18-6=12? But no label.

Given: top is 18 in, right side is 15 in.

After (6,5), go up to (6,15)? Then right to (18,15), then down to (18,0), then left to (0,0).

But then from (18,0) to (0,0) is 18 in, but we have a point at (6,0), so actually from (18,0) to (6,0) is 12 in, then to (0,0) is 6 in? Messy.

Better: the path is:

From (0,0) to (18,0) — bottom, 18 in
To (18,15) — right, 15 in
To (6,15) — left, 12 in (since 18-6=12)
To (6,5) — down, 10 in? But given 5 in.

I think the "5 in" and "6 in" are the dimensions of the indentation.

Assume the shape is mostly 18x15, but on the left side, from y=0 to y=5, it's only 6 in wide instead of 18.

So:

- From (0,0) to (6,0): 6 in
- (6,0) to (6,5): 5 in
- (6,5) to (18,5): 12 in
- (18,5) to (18,15): 10 in
- (18,15) to (0,15): 18 in
- (0,15) to (0,0): 15 in? But that would include the left side which is not straight.

From (0,15) to (0,5) is 10 in, then to (0,0) is 5 in, but we have a horizontal at y=5 from x=0 to x=6? No.

Correct vertex order for perimeter:

Start at (0,0)
→ (6,0) : 6 in
→ (6,5) : 5 in
→ (18,5) : 12 in
→ (18,15) : 10 in
→ (0,15) : 18 in
→ (0,0) : 15 in? But from (0,15) to (0,0) is 15 in, but we already have the left side covered? No, in this path, from (0,15) directly to (0,0) skips the indentation.

Mistake.

After (0,15), we should go down to (0,5), then right to (6,5), but (6,5) is already visited.

Standard way for such shapes: the outer boundary is:

- Bottom: from x=0 to x=18 at y=0? But at y=0, only from x=0 to x=6 is present? No.

Let's think of the shape as having:

- A rectangle from x=6 to x=18, y=0 to y=15: size 12x15
- Plus a rectangle from x=0 to x=6, y=5 to y=15: size 6x10

Then the total shape has:

Width 18, height 15, but missing the bottom-left 6x5 rectangle.

Yes! That makes sense.

So area = total bounding box minus missing part.

Bounding box: 18 * 15 = 270
Missing part: 6 * 5 = 30
Area = 270 - 30 = 240 sq in

Perimeter: when you remove a rectangle from corner, perimeter increases by twice the depth.

Original rectangle perimeter: 2*(18+15)=66
Remove 6x5 from bottom-left: you remove two sides (6 and 5) but add two new sides (6 and 5) so net change zero? No.

When you cut out a rectangle from the corner, you replace the outer corner with an inner corner, so you add two sides equal to the cut.

Specifically, for a rectangle removed from corner, perimeter increases by 2*depth, but here it's not removed, it's that the shape doesn't include that part.

In this case, the shape has:

- Left side: from y=5 to y=15: 10 in
- Bottom: from x=0 to x=6: 6 in
- Then up 5 in to (6,5)
- Then right 12 in to (18,5)
- Then up 10 in to (18,15)
- Then left 18 in to (0,15)
- Then down 10 in to (0,5) — but (0,5) to (0,15) is already counted? Let's list unique segments.

Vertices in order:

Start at (0,5)
→ (0,15) : 10 in
→ (18,15) : 18 in
→ (18,5) : 10 in
→ (6,5) : 12 in
→ (6,0) : 5 in
→ (0,0) : 6 in
→ (0,5) : 5 in

Sum: 10 + 18 + 10 + 12 + 5 + 6 + 5 = let's calculate:

10+18=28; 28+10=38; 38+12=50; 50+5=55; 55+6=61; 61+5=66 in

But earlier I thought bounding box is 66, and we added some, but got 66 again? That can't be right because we have an indentation.

From (0,5) to (0,15): 10
(0,15) to (18,15): 18
(18,15) to (18,5): 10
(18,5) to (6,5): 12
(6,5) to (6,0): 5
(6,0) to (0,0): 6
(0,0) to (0,5): 5

Yes, 10+18+10+12+5+6+5 = 66 in

But intuitively, if you have a rectangle 18x15, perimeter 66, and you cut out a 6x5 rectangle from the bottom-left corner, you remove the two sides of length 6 and 5, but add two new sides of length 6 and 5, so perimeter remains the same. Yes! So 66 in is correct.

Area: as above, 18*15 - 6*5 = 270 - 30 = 240 sq in

✔ Perimeter: 66 in
✔ Area: 240 in²

---

Problem 4:

Given:

Left side = 26 ft
Bottom = 38 ft
Top right horizontal = 19 ft
Vertical drop on right = 11 ft

Similar to problem 1.

Find missing sides.

Total width 38 ft, right part is 19 ft, so left top horizontal = 38 - 19 = 19 ft

Total height 26 ft, right drop is 11 ft, so lower right vertical = 26 - 11 = 15 ft

Perimeter: walk around.

Start bottom-left (0,0)
→ (38,0) : 38 ft
→ (38,15) : 15 ft (since 26-11=15)
→ (19,15) : 19 ft left (38-19=19)
→ (19,26) : 11 ft up
→ (0,26) : 19 ft left
→ (0,0) : 26 ft down

Sides: 38, 15, 19, 11, 19, 26

Sum: 38+15=53; 53+19=72; 72+11=83; 83+19=102; 102+26=128 ft

Area: split into two rectangles.

Bottom: 38 ft wide × 15 ft high = 570
Top: 19 ft wide × 11 ft high = 209
Total = 570 + 209 = 779 sq ft

Or left rectangle: 19x26 = 494, right rectangle: 19x15 = 285, total 779.

✔ Perimeter: 128 ft
✔ Area: 779 ft²

---

Problem 5:

Given:

Top = 27 yd
Right side = 10 yd
Bottom has 16 yd and 3 yd segments.

Probably: from left, bottom has 16 yd, then a step up 3 yd, then to right.

So total width 27 yd, so after 16 yd, remaining is 27-16=11 yd, but there's a 3 yd vertical.

Assume shape is like a rectangle with a notch on bottom right.

Total height 10 yd.

From bottom-left (0,0)
→ (16,0) : 16 yd
→ (16,3) : 3 yd up
→ (27,3) : 11 yd right (27-16=11)
→ (27,10) : 7 yd up (10-3=7)
→ (0,10) : 27 yd left
→ (0,0) : 10 yd down

Sides: 16, 3, 11, 7, 27, 10

Sum: 16+3=19; 19+11=30; 30+7=37; 37+27=64; 64+10=74 yd

Area: can be seen as full rectangle minus missing part, or split.

Full bounding box: 27x10 = 270
Missing part: from x=16 to 27, y=0 to 3: width 11, height 3 → area 33
So area = 270 - 33 = 237 sq yd

Split: left part 16x10 = 160, right part 11x7 = 77, total 237.

✔ Perimeter: 74 yd
✔ Area: 237 yd²

---

Problem 6:

This looks like a simple rectangle!

Given: bottom = 34 yd, right side = 9 yd

So it's a rectangle 34 by 9.

Perimeter = 2*(34+9) = 2*43 = 86 yd

Area = 34*9 = 306 sq yd

No irregularity? But the worksheet says "irregular shapes", but this one is regular. Maybe typo, or perhaps it's intended to be straightforward.

Looking back at image description: "6. [rectangle] 34 yd bottom, 9 yd right"

Yes, so it's a rectangle.

✔ Perimeter: 86 yd
✔ Area: 306 yd²

---

Problem 7:

Given:

Top = 38 in
Right side = 9 in
Left side: 15 in horizontal, then 4 in down.

So similar to others.

Total width 38 in, left part 15 in, so right top horizontal = 38 - 15 = 23 in

Total height 9 in, left drop is 4 in, so lower left vertical = 9 - 4 = 5 in

Perimeter:

Start bottom-left (0,0)
→ (38,0) : 38 in
→ (38,9) : 9 in
→ (15,9) : 23 in left (38-15=23)
→ (15,4) : 5 in down (9-4=5)
→ (0,4) : 15 in left
→ (0,0) : 4 in down

Sides: 38, 9, 23, 5, 15, 4

Sum: 38+9=47; 47+23=70; 70+5=75; 75+15=90; 90+4=94 in

Area: split.

Bottom: 38x5 = 190? Height from y=0 to y=5 is 5 in, but the left part is only up to y=4? Let's see.

Better: the shape has a rectangle on top left and bottom full.

From y=4 to y=9: width 15 in (left part) → area 15*5 = 75 (since 9-4=5)

From y=0 to y=4: full width 38 in → area 38*4 = 152

Total area = 75 + 152 = 227 sq in

Check: full rectangle 38x9=342, minus the missing part on bottom right: from x=15 to 38, y=0 to 4: width 23, height 4 → 92, 342-92=250? Not matching.

Mistake.

If from (0,0) to (38,0) to (38,9) to (15,9) to (15,4) to (0,4) to (0,0)

Then the area is:

- Rectangle A: x=0 to 15, y=4 to 9: 15*5=75
- Rectangle B: x=0 to 38, y=0 to 4: 38*4=152
Total 227

But is there overlap? No, they share the line y=4, but areas don't overlap.

The region from x=15 to 38, y=4 to 9 is not included? In this description, from (15,9) to (15,4) to (0,4), so yes, the top-right part is missing.

In the path, after (15,9) we go down to (15,4), then left to (0,4), so the area between x=15 to 38 and y=4 to 9 is not part of the shape. So it's like a rectangle with a bite taken out of the top-right.

So area should be total minus that bite.

Total bounding box: 38*9=342
Bite: width 23 (38-15), height 5 (9-4) → 23*5=115
Area = 342 - 115 = 227 sq in — matches.

✔ Perimeter: 94 in
✔ Area: 227 in²

---

Problem 8:

Given:

Left side = 26 yd
Bottom = 48 yd
Top right horizontal = 24 yd
Vertical drop on right = 11 yd

Similar to problem 4.

Total width 48 yd, right part 24 yd, so left top horizontal = 48 - 24 = 24 yd

Total height 26 yd, right drop 11 yd, so lower right vertical = 26 - 11 = 15 yd

Perimeter:

Start (0,0)
→ (48,0) : 48
→ (48,15) : 15
→ (24,15) : 24 left (48-24=24)
→ (24,26) : 11 up
→ (0,26) : 24 left
→ (0,0) : 26 down

Sides: 48, 15, 24, 11, 24, 26

Sum: 48+15=63; 63+24=87; 87+11=98; 98+24=122; 122+26=148 yd

Area: bottom rectangle 48x15 = 720
Top rectangle 24x11 = 264
Total = 720 + 264 = 984 sq yd

Or left: 24x26=624, right: 24x15=360, total 984.

✔ Perimeter: 148 yd
✔ Area: 984 yd²

---

Now, compiling all answers:

1. Perimeter: 70 in, Area: 190 in²
2. Perimeter: 112 in, Area: 519 in²
3. Perimeter: 66 in, Area: 240 in²
4. Perimeter: 128 ft, Area: 779 ft²
5. Perimeter: 74 yd, Area: 237 yd²
6. Perimeter: 86 yd, Area: 306 yd²
7. Perimeter: 94 in, Area: 227 in²
8. Perimeter: 148 yd, Area: 984 yd²

Final Answer:
1. Perimeter: 70 in, Area: 190 in²
2. Perimeter: 112 in, Area: 519 in²
3. Perimeter: 66 in, Area: 240 in²
4. Perimeter: 128 ft, Area: 779 ft²
5. Perimeter: 74 yd, Area: 237 yd²
6. Perimeter: 86 yd, Area: 306 yd²
7. Perimeter: 94 in, Area: 227 in²
8. Perimeter: 148 yd, Area: 984 yd²