Perimeter of irregular shapes worksheet - Free Printable

Let’s solve each problem step by step.

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Problem 1:

We have an L-shaped figure. We need to find the missing side labeled X, then calculate the perimeter.

Look at the bottom side: it’s 15 cm long.
The top part has two horizontal segments: one is X cm, and the other is 9 cm.
Since the shape is made of straight lines and right angles, the total length across the top (X + 9) must equal the bottom (15 cm).

So:

> X + 9 = 15
> → X = 15 - 9
> → X = 6 cm

Now, let’s find the perimeter — that means adding up all the outer sides.

List all the sides going around the shape:

- Left vertical: 11 cm
- Top left horizontal: X = 6 cm
- Small vertical drop down: ? Let’s think… The full height on the left is 11 cm, and the right side is 9 cm tall. So the small vertical step down must be 11 - 9 = 2 cm
- Then the top right horizontal: 9 cm
- Right vertical: 9 cm
- Bottom horizontal: 15 cm

Wait — actually, we don’t need to guess the small vertical if we just walk around the entire outside.

Let’s trace the perimeter clockwise from bottom-left corner:

1. Bottom: 15 cm
2. Right side up: 9 cm
3. Top-right horizontal left: 9 cm
4. Vertical down (the “step”): this is the difference between left height (11 cm) and right height (9 cm) → 11 - 9 = 2 cm
5. Top-left horizontal left: X = 6 cm
6. Left side down: 11 cm

But wait — when you go down the left side, you’re already including the full 11 cm. Actually, let’s list only the outer edges without double-counting.

Better way: add all visible sides plus the ones we can deduce.

Actually, here’s a trick for these shapes: the perimeter of such rectilinear shapes is the same as the perimeter of the bounding rectangle IF there are no holes — but in this case, since it’s indented, we still add every outer edge.

Let me label all six sides clearly:

Starting from bottom-left, going clockwise:

- Side 1 (bottom): 15 cm
- Side 2 (right): 9 cm
- Side 3 (top-right, leftward): 9 cm
- Side 4 (downward step): this is 11 - 9 = 2 cm
- Side 5 (top-left, leftward): X = 6 cm
- Side 6 (left side downward): 11 cm

Now add them:

15 + 9 + 9 + 2 + 6 + 11 = ?

Let’s compute:

15 + 9 = 24
24 + 9 = 33
33 + 2 = 35
35 + 6 = 41
41 + 11 = 52 cm

Alternatively, notice that the total horizontal distances: top adds to 15 (same as bottom), and verticals: left is 11, right is 9, but the step adds extra 2 cm vertically? Wait — actually, in perimeter, every segment counts.

Another way: imagine unfolding the shape — the total horizontal contribution is 2 × 15 = 30 cm (top and bottom). The vertical contributions: left side 11, right side 9, but also the inner vertical step of 2 cm — so total vertical = 11 + 9 + 2? No, that would be wrong because the step is internal? Wait no — in perimeter, we walk along the outside.

Actually, correct breakdown:

Horizontal sides:
- Bottom: 15
- Top: X + 9 = 6 + 9 = 15 → so total horizontal = 15 + 15 = 30

Vertical sides:
- Left: 11
- Right: 9
- Plus the little vertical drop between the two top parts: which is 11 - 9 = 2 → this is also part of the perimeter!

So vertical total = 11 + 9 + 2 = 22? But wait — that’s not right because when you go up the right side (9 cm), then left 9 cm, then down 2 cm, then left 6 cm, then down 11 cm — yes, those are all separate.

Actually, listing again:

Sides:

1. Bottom: 15
2. Right: 9
3. Top-right horizontal: 9
4. Inner vertical down: 2
5. Top-left horizontal: 6
6. Left vertical: 11

Sum: 15+9=24; +9=33; +2=35; +6=41; +11=52 → Yes.

So perimeter = 52 cm

✔ Confirmed.

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Problem 2:

This is another L-shape, with measurements in mm.

We need to find X (the bottom side), then the perimeter.

Looking at the top: there are two horizontal segments: 6 mm and 14 mm.
So total width at the top = 6 + 14 = 20 mm.

Therefore, the bottom side X must also be 20 mm, because it spans the same total width.

→ X = 20 mm

Now, perimeter: add all outer sides.

Trace clockwise from bottom-left:

1. Bottom: X = 20 mm
2. Right side up: 6 mm (given on the far right)
3. Top-right horizontal left: 14 mm
4. Vertical up (the step): the total height on the left is 15 mm, and the right part is only 6 mm high, so the step up is 15 - 6 = 9 mm (which matches the given 9 mm label!)
5. Top-left horizontal left: 6 mm
6. Left side down: 15 mm

Now add them:

20 (bottom)
+ 6 (right)
+ 14 (top-right)
+ 9 (vertical step)
+ 6 (top-left)
+ 15 (left)

Calculate:

20 + 6 = 26
26 + 14 = 40
40 + 9 = 49
49 + 6 = 55
55 + 15 = 70 mm

Double-check:

Horizontal totals: top = 6 + 14 = 20, bottom = 20 → total horizontal = 40 mm
Vertical totals: left = 15, right = 6, plus the middle vertical step = 9 → 15 + 6 + 9 = 30 mm
Total perimeter = 40 + 30 = 70 mm → Correct.

✔ Confirmed.

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Final Answer:

1)
a) X cm = 6 cm
b) Perimeter = 52 cm

2)
a) X = 20 mm
b) Perimeter = 70 mm