Perimeter of Rectilinear Shapes Worksheet - Find missing sides and calculate perimeters.

Let’s solve each shape one by one. We’ll find the missing sides first, then add all the sides to get the perimeter.

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Shape 1:

We have:
- Left side = 11 cm
- Bottom = 15 cm
- Right side = 9 cm
- Top right horizontal = 9 cm

Missing top left horizontal and top vertical? Wait — actually, look at the shape: it’s like a rectangle with a bite taken out of the top right.

Actually, let’s think vertically and horizontally.

Vertical sides:
Left = 11 cm
Right = 9 cm → so the “step” down on the right must be 11 - 9 = 2 cm (this is the missing vertical segment on the top right)

Horizontal sides:
Bottom = 15 cm
Top right = 9 cm → so the top left must be 15 - 9 = 6 cm

Now list ALL sides going around:

Start from bottom left, go clockwise:

→ Bottom: 15 cm
→ Right: 9 cm
→ Up step: 2 cm (we just found)
→ Left along top right: 9 cm
→ Down step? No — wait, we’re going around the outside.

Better way: trace the outer edge.

From bottom-left corner:

1. Right along bottom: 15 cm
2. Up right side: 9 cm
3. Left along top-right part: 9 cm
4. Up the small vertical step: 2 cm (because total height is 11, and right side is only 9, so difference is 2)
5. Left along top-left part: 6 cm (because total width is 15, minus 9 already used)
6. Down left side: 11 cm

Wait — that double-counts? Let me draw mentally.

Actually, better method: for rectilinear shapes, you can pair opposite sides.

Total horizontal lengths: top and bottom should both equal 15 cm when added together in parts.

Similarly, total vertical lengths: left and right should both equal 11 cm when added in parts.

So for Shape 1:

Horizontal segments:
- Bottom: 15 cm
- Top: two parts: ? + 9 cm → must also sum to 15 → so missing top-left = 6 cm

Vertical segments:
- Left: 11 cm
- Right: two parts: 9 cm + ? → must sum to 11 → so missing top-right vertical = 2 cm

Now, perimeter = sum of all outer sides.

Going around:

Start at bottom-left:

→ Right: 15 cm
→ Up: 9 cm
→ Left: 9 cm
→ Up: 2 cm (the little step up)
→ Left: 6 cm
→ Down: 11 cm

Add them: 15 + 9 + 9 + 2 + 6 + 11

Calculate:

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

Perimeter = 52 cm

✔ Check: Another way — if it were full rectangle 15x11, perimeter would be 2*(15+11)=52. But since we cut out a rectangle from top-right, did we change perimeter? Actually no — because removing a corner adds two new edges but removes none — wait, in this case, we didn’t remove, we indented. Actually, in this shape, the perimeter is same as full rectangle? Let’s see:

Original rectangle 15x11: perimeter 52.

Our shape: we went up 9 instead of 11 on right, then left 9, then up 2, then left 6 — compared to original which would be up 11 then left 15.

But path length: original right+top = 11 + 15 = 26

Our path: 9 (up) + 9 (left) + 2 (up) + 6 (left) = 9+9+2+6=26 → same!

So yes, perimeter is still 52 cm. Good.

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Shape 2:

Given:
- Left: 22 mm
- Bottom: 18 mm
- Top: 25 mm
- Right-top: 6 mm

This is an L-shape or stepped shape.

Find missing sides.

First, vertical:

Left = 22 mm
Right has two parts: top part = 6 mm, so bottom part must be 22 - 6 = 16 mm

Horizontal:

Bottom = 18 mm
Top = 25 mm → so the "overhang" on top right must be 25 - 18 = 7 mm? Wait, no.

Actually, looking at shape: it's like a big rectangle with a notch on bottom right? Or top right?

Labeling:

Top side: 25 mm
Right side has a step: top part 6 mm, then goes left, then down.

Actually, standard way:

The total width at top is 25 mm. The bottom width is 18 mm. So the horizontal step on the right must be 25 - 18 = 7 mm (this is the missing horizontal segment on the right side, going left).

Vertically: left is 22 mm. On the right, we have top vertical 6 mm, then after going left 7 mm, we go down the rest. Since total height is 22, and we’ve gone down 6 mm already? Wait.

Actually, from top-right corner:

Go down 6 mm → then left 7 mm → then down ? to meet bottom.

Total height is 22 mm, so the remaining vertical drop is 22 - 6 = 16 mm

Yes.

So now, list all sides going clockwise from bottom-left:

1. Right along bottom: 18 mm
2. Up right-bottom vertical: 16 mm
3. Left along middle horizontal: 7 mm
4. Up right-top vertical: 6 mm
5. Left along top: 25 mm
6. Down left side: 22 mm

Wait — that doesn't close. From top-left, down 22 mm brings us to bottom-left, but we started at bottom-left and went right 18, etc.

Actually, let's trace properly:

Start at bottom-left corner:

→ Right: 18 mm (bottom)
→ Up: 16 mm (right-bottom vertical)
→ Left: 7 mm (middle horizontal)
→ Up: 6 mm (right-top vertical)
→ Left: 25 mm? No — from there, going left should be to the left end, but top is 25 mm total, and we've already moved left 7 mm from the right, so the remaining leftward distance is 25 - 7 = 18 mm? I'm confusing myself.

Better: use the pairing method.

Total horizontal movement: top and bottom must each sum to the same total when considering all horizontal segments.

Actually, for perimeter, we just need to add all outer edges.

List all unique outer sides:

- Bottom: 18 mm
- Right-bottom vertical: 16 mm (since 22 - 6)
- Middle horizontal (going left): 7 mm (since 25 - 18)
- Right-top vertical: 6 mm
- Top: 25 mm
- Left: 22 mm

But when we go from top-left down to bottom-left, that's 22 mm, but we already accounted for the right side in parts.

Actually, the six sides are:

1. Bottom: 18
2. Right-lower vertical: 16
3. Middle horizontal (leftward): 7
4. Right-upper vertical: 6
5. Top: 25
6. Left: 22

But adding these: 18+16+7+6+25+22

Calculate:

18+16=34
34+7=41
41+6=47
47+25=72
72+22=94

Is that correct? Let me verify with another approach.

Imagine the full rectangle if no step: width 25, height 22 → perimeter 2*(25+22)=94. And since we indented the bottom-right corner, but in doing so, we removed a rectangle of size 7x16, but when you indent, you remove two sides but add two new ones of same length — so perimeter unchanged.

In this case, we have a step inward on the bottom-right, so we lose the outer corner but gain inner edges — actually, perimeter remains the same as bounding box.

Bounding box is 25 mm wide and 22 mm high → perimeter 2*(25+22)=94 mm. Yes.

So Perimeter = 94 mm

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Shape 3:

Units in meters.

Given:
- Top: 2.5 m
- Left-top: 0.5 m
- Left-bottom horizontal: 0.5 m
- Bottom: 1 m
- Right-bottom vertical: 0.8 m
- Right-top: 1.2 m

This looks like a T-shape or something.

Find missing sides.

First, vertical totals.

Left side: has two parts: top 0.5 m, then after going right 0.5 m, we go down some amount, then right 1 m, then up 0.8 m, then right? Wait.

Actually, let's find the total height.

On the right, we have top part 1.2 m and bottom part 0.8 m, so total height = 1.2 + 0.8 = 2.0 m

On the left, we have top 0.5 m, then we go right 0.5 m, then down ? , then right 1 m, then up 0.8 m.

The vertical drop on the left-middle must be such that from top to bottom is 2.0 m.

From top-left, down 0.5 m, then later we go up 0.8 m on the right, but the bottom is flat.

Actually, the lowest point is the bottom of the stem.

Total height is 2.0 m (from right side: 1.2 + 0.8).

On the left, from top to the start of the bottom horizontal, we go down 0.5 m, then we need to go down further to reach the bottom level.

The bottom level is 2.0 m below top.

After going down 0.5 m on left, we are at height 0.5 m from top, so to reach bottom (2.0 m down), we need to go down another 1.5 m.

But on the diagram, after going right 0.5 m from the left, we go down to the bottom, then right 1 m, then up 0.8 m.

So the vertical segment on the left-middle (after the 0.5 m right) is the drop to the bottom.

Since total height is 2.0 m, and we've already descended 0.5 m on the far left, the remaining descent is 2.0 - 0.5 = 1.5 m

But wait, on the right, we ascend 0.8 m from bottom to the shoulder, so the bottom is 0.8 m below that shoulder.

Perhaps better to calculate missing horizontal and vertical.

Horizontal:

Top: 2.5 m

Bottom: consists of left part 0.5 m (given), middle 1 m (given), and right part?

The right part should be such that total bottom width equals top width? Not necessarily, but in this shape, the top is wider.

Actually, the top is 2.5 m.

The bottom has three parts: left horizontal 0.5 m, middle vertical? No.

Let's list the path.

Start at top-left:

→ Right: 2.5 m (top)
→ Down: 1.2 m (right-top)
→ Left: ? (this is the missing horizontal on the right-middle)
→ Down: 0.8 m (right-bottom)
→ Left: 1 m (bottom-middle)
→ Up: ? (left-bottom vertical)
→ Left: 0.5 m (left-bottom horizontal)
→ Up: 0.5 m (left-top) back to start.

We need to find the missing horizontal on the right-middle and the missing vertical on the left-bottom.

First, horizontal balance.

The total width at the top is 2.5 m.

At the bottom, we have segments: left 0.5 m, middle 1 m, and the right part must make up the difference.

When we go from the right-top down 1.2 m, then left some amount, then down 0.8 m, then left 1 m, then up, then left 0.5 m.

The key is that the horizontal distance from left to right at any level should be consistent, but it's stepped.

Note that the bottom-most horizontal is from left to right: 0.5 m (left) + 1 m (middle) + ? (right) = total bottom width.

But the top is 2.5 m, and the shape is symmetric? Not necessarily.

Another way: the overhang on the left and right.

From the left: at the top, we start at left end. After going right 2.5 m, we are at right end.

Then down 1.2 m, then left x m, then down 0.8 m, then left 1 m, then up y m, then left 0.5 m, then up 0.5 m to start.

The net horizontal displacement should be zero.

But perhaps easier: the total horizontal span.

The leftmost point is the left end.

The rightmost point is the right end of the top.

The bottom extends from left to some point.

Specifically, the bottom has a segment of 0.5 m on left, then 1 m in middle, so from left end to the start of the right-bottom vertical is 0.5 + 1 = 1.5 m.

Then from there, we go up 0.8 m, then left? No, after going up 0.8 m, we are at the shoulder, then we go left to connect to the top part.

Actually, after going up 0.8 m on the right, we are at a point that is 0.8 m above bottom, and we need to go left to meet the vertical from the top.

The top is at height 2.0 m (since right side 1.2 + 0.8 = 2.0), so from the shoulder (0.8 m above bottom), we go left to the line where the top starts descending.

The missing horizontal on the right-middle is the distance from the right end to the point above the bottom-right corner.

Since the bottom has width from left to the right-bottom corner: let's call it W_bottom.

From the diagram, the bottom has: left horizontal 0.5 m, then a vertical down? No, after left 0.5 m, we go down to bottom, then right 1 m, so the bottom horizontal is 0.5 m (left) + 1 m (middle) = 1.5 m, but this is not the full bottom; there is also the right part.

I think I have it.

Let me define coordinates.

Set top-left corner as (0,0).

Then:

- Go right 2.5 m to (2.5, 0)
- Go down 1.2 m to (2.5, -1.2)
- Go left x m to (2.5 - x, -1.2) [this is the missing horizontal]
- Go down 0.8 m to (2.5 - x, -2.0) [since 1.2 + 0.8 = 2.0]
- Go left 1 m to (2.5 - x - 1, -2.0) = (1.5 - x, -2.0)
- Go up y m to (1.5 - x, -2.0 + y)
- Go left 0.5 m to (1.5 - x - 0.5, -2.0 + y) = (1.0 - x, -2.0 + y)
- Go up 0.5 m to (1.0 - x, -2.0 + y + 0.5) = (1.0 - x, -1.5 + y)

This must equal the start point (0,0), so:

x-coordinate: 1.0 - x = 0 ⇒ x = 1.0

y-coordinate: -1.5 + y = 0 ⇒ y = 1.5

Perfect.

So missing horizontal on right-middle: x = 1.0 m

Missing vertical on left-bottom: y = 1.5 m

Now, list all sides for perimeter:

1. Top: 2.5 m
2. Right-top vertical: 1.2 m
3. Right-middle horizontal (left): 1.0 m
4. Right-bottom vertical: 0.8 m
5. Bottom-middle horizontal (left): 1.0 m
6. Left-bottom vertical (up): 1.5 m
7. Left-bottom horizontal (left): 0.5 m
8. Left-top vertical (up): 0.5 m

Add them:

2.5 + 1.2 = 3.7
3.7 + 1.0 = 4.7
4.7 + 0.8 = 5.5
5.5 + 1.0 = 6.5
6.5 + 1.5 = 8.0
8.0 + 0.5 = 8.5
8.5 + 0.5 = 9.0

Perimeter = 9.0 m

Check: bounding box width 2.5 m, height 2.0 m, perimeter 2*(2.5+2.0)=9.0 m. And since it's a rectilinear shape with no holes, and we're going around, it should be the same as bounding box if it's convex, but here it's concave? In this case, because we have indentations, but in our calculation, we got 9.0, and bounding box is 9.0, so it matches. Actually, for any rectilinear polygon that is orthogonally convex or not, the perimeter can be calculated by summing all turns, but in this case, since we traced all outer edges and got 9.0, and it makes sense.

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Shape 4:

Units in cm.

Given:
- Bottom: 5 cm
- Left-bottom: 1 cm
- Left-middle horizontal: 1.5 cm
- Left-top vertical: 2.5 cm
- Top: 2.5 cm
- Right: 2 cm

Find missing sides.

This looks like a staircase or stepped shape.

Total width: bottom is 5 cm.

Top is 2.5 cm.

So the overhangs must account for the difference.

Similarly, height: left has 1 cm (bottom) + ? + 2.5 cm (top) — wait.

List the path.

Start at bottom-left:

→ Right: 5 cm (bottom)
→ Up: 2 cm (right)
→ Left: ? (missing horizontal on top-right)
→ Up: ? (missing vertical on top-right) — wait, no.

From the diagram:

After going up 2 cm on right, we go left some amount, then up to the top.

The top is 2.5 cm wide.

Also, on the left, we have: from bottom-left, up 1 cm, then right 1.5 cm, then up 2.5 cm to top-left.

So total height on left: 1 + 2.5 = 3.5 cm? But on right, we have only 2 cm given, so there must be more.

Actually, the right side has two parts: bottom 2 cm, and then after going left, we go up to the top.

Total height should be the same on left and right.

On left: from bottom to top: 1 cm (first vertical) + 2.5 cm (second vertical) = 3.5 cm

On right: we have 2 cm (given), so the remaining vertical must be 3.5 - 2 = 1.5 cm

Now horizontal.

Bottom: 5 cm

Top: 2.5 cm

The difference is 5 - 2.5 = 2.5 cm, which must be distributed in the steps.

On the left, after going up 1 cm, we go right 1.5 cm.

On the right, after going up 2 cm, we go left some amount, say z cm, then up 1.5 cm.

The horizontal segments:

From left: at height 1 cm, we go right 1.5 cm.

At the top, we have 2.5 cm.

The total width at the base is 5 cm.

The shape has steps.

Using coordinates again.

Set bottom-left as (0,0).

→ Right 5 cm to (5,0)
→ Up 2 cm to (5,2)
→ Left z cm to (5-z,2) [missing horizontal]
→ Up w cm to (5-z,2+w) [missing vertical]
→ Left 2.5 cm to (5-z-2.5,2+w) = (2.5-z,2+w) [top]
→ Down 2.5 cm to (2.5-z,2+w-2.5) = (2.5-z,w-0.5) [left-top vertical]
→ Left 1.5 cm to (2.5-z-1.5,w-0.5) = (1.0-z,w-0.5) [left-middle horizontal]
→ Down 1 cm to (1.0-z,w-0.5-1) = (1.0-z,w-1.5) [left-bottom vertical]

This must equal (0,0), so:

x: 1.0 - z = 0 ⇒ z = 1.0

y: w - 1.5 = 0 ⇒ w = 1.5

Perfect.

So missing horizontal on right-middle: z = 1.0 cm

Missing vertical on right-top: w = 1.5 cm

Now, list all sides:

1. Bottom: 5 cm
2. Right-bottom vertical: 2 cm
3. Right-middle horizontal (left): 1.0 cm
4. Right-top vertical (up): 1.5 cm
5. Top: 2.5 cm
6. Left-top vertical (down): 2.5 cm
7. Left-middle horizontal (left): 1.5 cm
8. Left-bottom vertical (down): 1 cm

Add them:

5 + 2 = 7
7 + 1.0 = 8
8 + 1.5 = 9.5
9.5 + 2.5 = 12
12 + 2.5 = 14.5
14.5 + 1.5 = 16
16 + 1 = 17

Perimeter = 17 cm

Check: bounding box width 5 cm, height: from bottom to top, on left is 1 + 2.5 = 3.5 cm, on right is 2 + 1.5 = 3.5 cm, so height 3.5 cm. Perimeter of bounding box 2*(5+3.5)=17 cm. Same as our calculation. Good.

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Final Answers:

1) 52 cm
2) 94 mm
3) 9.0 m
4) 17 cm

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Final Answer:
1) 52
2) 94
3) 9.0
4) 17