Math worksheet for calculating area and perimeter of irregular shapes.

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

Remember:
- Perimeter = add up all the outside sides.
- Area = for rectangles, multiply length × width. For irregular shapes (like L-shapes), break them into rectangles, find each area, then add them.

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Problem 1: Rectangle

Sides: 25 yd and 23 yd

→ Perimeter = 2 × (length + width) = 2 × (25 + 23) = 2 × 48 = 96 yd
→ Area = length × width = 25 × 23
Let’s calculate:
25 × 20 = 500
25 × 3 = 75
Total = 500 + 75 = 575 sq yd

✔ Check: 25×23 = 575 → correct.

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Problem 2: Rectangle

Sides: 48 in and 24 in

→ Perimeter = 2 × (48 + 24) = 2 × 72 = 144 in
→ Area = 48 × 24
Break it down:
48 × 20 = 960
48 × 4 = 192
Total = 960 + 192 = 1152 sq in

✔ Check: 48×24 = 1152 → correct.

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Problem 3: L-shaped figure

Given sides:
Left side = 12 ft
Bottom = 24 ft
Top right horizontal = 14 ft
Vertical step = 5 ft

We need to find missing sides to get perimeter and area.

First, let’s sketch mentally:

The full bottom is 24 ft. The top part has a horizontal segment of 14 ft on the right. So the left horizontal part must be 24 - 14 = 10 ft.

The total height on the left is 12 ft. There’s a vertical drop of 5 ft on the right side of the top rectangle. So the lower rectangle’s height is 12 - 5 = 7 ft.

Now we can split into two rectangles:

Rectangle A (top left): 10 ft wide × 5 ft high → Area = 10 × 5 = 50 sq ft
Rectangle B (bottom): 24 ft wide × 7 ft high → Area = 24 × 7 = 168 sq ft
Total Area = 50 + 168 = 218 sq ft

Perimeter: Walk around the outside.

Start at bottom left corner:

- Right along bottom: 24 ft
- Up right side: 7 ft (height of bottom rect)
- Left along top of bottom rect? Wait — no, better to trace outer edges only.

Actually, let’s list all outer sides:

From bottom left:

1. Right → 24 ft
2. Up → 7 ft (to start of inner corner)
3. Left → 14 ft? No — wait, after going up 7 ft, you go left? Actually, from bottom right, you go up 7 ft, then left 14 ft? That would be inside.

Better approach: Use the “outer path”.

Imagine walking around:

Start at bottom-left corner:

→ Go right 24 ft (bottom)
→ Go up 7 ft (right side of bottom rectangle)
→ Go left 14 ft (this is the top edge of the bottom rectangle? But that’s not outer — actually, after going up 7 ft, you’re at the inner corner. Then you go up another 5 ft? Let me re-draw mentally.

Actually, the shape looks like this:

```
+-------+
| | 5 ft
| A +-------+
| | B |
+-------+-------+
10 14
<------24------>
```

Height of A is 5 ft, height of B is 7 ft (since 12 - 5 = 7).

So outer perimeter:

Start at bottom-left:

1. Right → 24 ft
2. Up → 7 ft (right side of B)
3. Left → 14 ft (top of B) ← but this is NOT outer! Wait, no — after going up 7 ft, you are at the bottom-right of rectangle A? I’m confusing myself.

Alternative method: For any polygon, perimeter is sum of all outer sides. Given labeled sides:

Labeled: 12 ft (left), 5 ft (inner vertical), 14 ft (inner horizontal), 24 ft (bottom)

Missing sides:

- Top horizontal: since bottom is 24, and the right part is 14, the left top must be 24 - 14 = 10 ft
- Right vertical: total height is 12 ft, and the top part is 5 ft, so the bottom part is 12 - 5 = 7 ft

Now, trace the outline:

Start at bottom-left:

→ Right 24 ft (bottom)
→ Up 7 ft (right side of lower part)
→ Left 14 ft (this is the top of the lower rectangle — but is this outer? Actually, no — because above it is the upper rectangle. Wait, in an L-shape, when you go up 7 ft on the right, then you turn left and go 14 ft — but that 14 ft is actually the *bottom* of the upper rectangle? I think I have the orientation wrong.

Let me define coordinates to avoid confusion.

Place bottom-left corner at (0,0)

Then:

- Bottom goes to (24, 0)
- From (24,0) go up to (24, 7) [since lower height is 7]
- Then left to (10, 7) [because 24 - 14 = 10]
- Then up to (10, 12) [since total height is 12]
- Then left to (0,12)
- Then down to (0,0)

Yes! That makes sense.

So the vertices are:

(0,0) → (24,0) → (24,7) → (10,7) → (10,12) → (0,12) → back to (0,0)

Now, compute distances between consecutive points:

1. (0,0) to (24,0): 24 ft
2. (24,0) to (24,7): 7 ft
3. (24,7) to (10,7): 14 ft (since 24-10=14)
4. (10,7) to (10,12): 5 ft
5. (10,12) to (0,12): 10 ft
6. (0,12) to (0,0): 12 ft

Add them: 24 + 7 + 14 + 5 + 10 + 12 = let's add:

24+7=31; 31+14=45; 45+5=50; 50+10=60; 60+12=72 ft

So perimeter = 72 ft

Area: as before, two rectangles:

- Lower: 24 ft × 7 ft = 168 sq ft
- Upper: 10 ft × 5 ft = 50 sq ft
Total = 168 + 50 = 218 sq ft

✔ Matches earlier calculation.

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Problem 4: Another L-shape

Given:
Left side = 11 ft
Bottom = 29 ft
Right horizontal = 17 ft
Inner vertical = 7 ft

Find missing sides.

Total bottom = 29 ft. Right part is 17 ft, so left top horizontal = 29 - 17 = 12 ft

Total height = 11 ft. Inner vertical is 7 ft, so lower rectangle height = 11 - 7 = 4 ft

Split into two rectangles:

Upper left: 12 ft × 7 ft → Area = 12 × 7 = 84 sq ft
Lower: 29 ft × 4 ft → Area = 29 × 4 = 116 sq ft
Total Area = 84 + 116 = 200 sq ft

Perimeter: use coordinate method again.

Vertices:

Start at (0,0)

→ (29,0) → (29,4) → (12,4) → (12,11) → (0,11) → (0,0)

Distances:

1. (0,0)-(29,0): 29
2. (29,0)-(29,4): 4
3. (29,4)-(12,4): 17 (29-12=17)
4. (12,4)-(12,11): 7
5. (12,11)-(0,11): 12
6. (0,11)-(0,0): 11

Sum: 29+4=33; +17=50; +7=57; +12=69; +11=80 ft

Perimeter = 80 ft

✔ Check: 29+4+17+7+12+11 = 80 → correct.

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Problem 5: Rectangle

Sides: 32 ft and 11 ft

Perimeter = 2 × (32 + 11) = 2 × 43 = 86 ft
Area = 32 × 11 = 32 × 10 + 32 × 1 = 320 + 32 = 352 sq ft

✔ Correct.

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Problem 6: Rectangle

Sides: 27 ft and 13 ft

Perimeter = 2 × (27 + 13) = 2 × 40 = 80 ft
Area = 27 × 13
27 × 10 = 270
27 × 3 = 81
Total = 270 + 81 = 351 sq ft

✔ Correct.

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Problem 7: Rectangle

Sides: 23 yd and 21 yd

Perimeter = 2 × (23 + 21) = 2 × 44 = 88 yd
Area = 23 × 21
23 × 20 = 460
23 × 1 = 23
Total = 460 + 23 = 483 sq yd

✔ Correct.

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Problem 8: L-shape

Given:
Left side = 18 ft
Bottom = 28 ft
Right horizontal = 14 ft
Inner vertical = 11 ft

Find missing sides.

Total bottom = 28 ft. Right part is 14 ft, so left top horizontal = 28 - 14 = 14 ft

Total height = 18 ft. Inner vertical is 11 ft, so lower rectangle height = 18 - 11 = 7 ft

Split into two rectangles:

Upper left: 14 ft × 11 ft → Area = 14 × 11 = 154 sq ft
Lower: 28 ft × 7 ft → Area = 28 × 7 = 196 sq ft
Total Area = 154 + 196 = 350 sq ft

Perimeter: coordinates.

Vertices:

(0,0) → (28,0) → (28,7) → (14,7) → (14,18) → (0,18) → (0,0)

Distances:

1. (0,0)-(28,0): 28
2. (28,0)-(28,7): 7
3. (28,7)-(14,7): 14 (28-14=14)
4. (14,7)-(14,18): 11
5. (14,18)-(0,18): 14
6. (0,18)-(0,0): 18

Sum: 28+7=35; +14=49; +11=60; +14=74; +18=92 ft

Perimeter = 92 ft

✔ Check: 28+7+14+11+14+18 = 92 → correct.

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Final Answer:

1. Perimeter: 96 yd, Area: 575 sq yd
2. Perimeter: 144 in, Area: 1152 sq in
3. Perimeter: 72 ft, Area: 218 sq ft
4. Perimeter: 80 ft, Area: 200 sq ft
5. Perimeter: 86 ft, Area: 352 sq ft
6. Perimeter: 80 ft, Area: 351 sq ft
7. Perimeter: 88 yd, Area: 483 sq yd
8. Perimeter: 92 ft, Area: 350 sq ft