Math worksheet for calculating area and perimeter of rectilinear shapes, featuring four problems with labeled dimensions and instructions.

Let’s solve each shape one by one. We’ll find missing sides, split shapes into rectangles, calculate area and perimeter carefully — and double-check our work!

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Shape 1)

Given:
- Top horizontal = 5 cm
- Right vertical = 7 cm
- Left “step” vertical = 6 cm → so bottom part of left side is 7 - 6 = 1 cm
- Bottom horizontal on left = 8 cm → so total bottom width = 8 + 5 = 13 cm

We can split this into two rectangles:

→ Rectangle A (bottom): 13 cm wide × 1 cm high
→ Rectangle B (top right): 5 cm wide × 6 cm high

Area = (13 × 1) + (5 × 6) = 13 + 30 = 43 cm²

Perimeter: Add all outer edges.

Start at bottom-left corner, go clockwise:

- Bottom: 13 cm
- Right: 7 cm
- Top: 5 cm
- Down inner step: 6 cm
- Left inner step: 8 cm
- Up left side: 1 cm

Wait — that’s not correct for perimeter! Perimeter is only the *outer* boundary.

Better way: trace the outside.

From bottom-left:

→ Right along bottom: 13 cm
→ Up right side: 7 cm
→ Left along top: 5 cm
→ Down inner vertical: 6 cm? No — wait, we must follow the outline.

Actually, let's list all outer segments:

Bottom: 13 cm
Right side: 7 cm
Top: 5 cm
Left-down from top: 6 cm (but that’s internal?) — no.

Let me draw mentally:

The shape looks like an L rotated.

Outer path:

Start at bottom-left:

→ Right 13 cm (full bottom)
→ Up 7 cm (right side)
→ Left 5 cm (top)
→ Down 6 cm (inner drop) — but this is NOT outer! Wait — actually, after going left 5 cm, you go down 6 cm to meet the lower rectangle? But then you have to go left again?

No — better method: use the fact that for rectilinear shapes, perimeter equals the perimeter of the bounding box if there are no holes — but here it’s indented.

Alternative: add all visible outer sides.

List them:

- Bottom: 13 cm
- Right: 7 cm
- Top: 5 cm
- Inner vertical down: 6 cm — but this is inside? Actually, no — in the diagram, the 6cm is labeled on the left vertical of the upper block, which is exposed on the left side of the upper block, but since the lower block sticks out to the left, that 6cm is actually part of the outer perimeter? Let me think.

Actually, looking at standard approach:

For such L-shapes, perimeter = sum of all outer edges.

Label points:

Assume bottom-left corner is (0,0)

Then:

- Go right to (13,0) → 13 cm
- Up to (13,7) → 7 cm
- Left to (8,7) → 5 cm (since 13-8=5)
- Down to (8,1) → 6 cm (because 7-1=6)
- Left to (0,1) → 8 cm
- Down to (0,0) → 1 cm

Now add these:

13 + 7 + 5 + 6 + 8 + 1 = let’s compute:

13+7=20; 20+5=25; 25+6=31; 31+8=39; 39+1=40 cm

But the question asks for perimeter in meters? That seems odd — probably a typo in the worksheet. The units given are cm, so perimeter should be in cm. But it says “Perimeter = ____ m”. Hmm.

Wait — look back: Shape 1 has all dimensions in cm, but perimeter blank says “____ m”. That must be a mistake. Probably meant cm. But let’s check other shapes.

Shape 2 uses meters and mm — so maybe they want conversion.

But for Shape 1, all inputs are cm, output area in cm², perimeter in m? That would be strange.

Perhaps it’s a typo, and it should be cm. I’ll assume it’s cm unless told otherwise. But let’s see the instruction: “Perimeter = ____ m” — explicitly says m.

That means we need to convert cm to m.

40 cm = 0.4 m

But let’s confirm area first.

Area: as above, 43 cm² — correct.

Perimeter: 40 cm = 0.4 m

But let’s verify perimeter another way.

Another method: imagine extending lines to make a full rectangle.

The overall width is 13 cm, height is 7 cm.

If it were a full rectangle, perimeter = 2*(13+7)=40 cm.

But because it’s indented, does that change perimeter? In this case, the indentation adds two sides: when you cut out a rectangle, you remove two sides but add two new ones of same length — so perimeter remains same as bounding box.

Is that true?

In this shape: from full 13x7 rectangle, we removed a rectangle of size 8cm x 6cm from the top-left? No.

Actually, the shape is made by adding a 5x6 rectangle on top of a 13x1 rectangle? Earlier calculation was correct.

Visualize:

Full bounding box: 13 cm wide, 7 cm tall.

The missing part is a rectangle of 8 cm wide and 6 cm tall? No — actually, the shape occupies:

- From x=0 to 13, y=0 to 1 (bottom strip)
- Plus from x=8 to 13, y=1 to 7 (upper right block)

So the "missing" part is from x=0 to 8, y=1 to 7 — which is 8x6.

When you remove a rectangle from the corner, the perimeter increases by twice the depth of the cut? Let's calculate properly.

Original full rectangle perimeter: 2*(13+7)=40 cm.

When you remove a rectangle of size a x b from the corner, you remove two segments of length a and b, but add two new segments of length a and b — so net change zero? Only if you remove from corner.

In this case, we’re removing from the top-left corner: a rectangle 8 cm wide and 6 cm high.

Removing it:

- You lose the top edge of the removed part: 8 cm (which was part of the original top)
- You lose the left edge of the removed part: 6 cm (part of original left side)
- But you gain two new edges: the bottom of the removed part (8 cm) and the right of the removed part (6 cm) — but these are now internal? No, in the resulting shape, those become part of the perimeter.

Actually, for a rectangular notch taken from a corner, the perimeter remains unchanged.

Example: take a 10x10 square, perimeter 40. Cut out a 2x2 square from corner. New shape: still has perimeter 40, because you remove 2+2=4 cm of outer edge, but add 2+2=4 cm of new inner edge — but since it's a corner, the new edges are exposed, so yes, perimeter same.

In this case, our shape is equivalent to a 13x7 rectangle with a 8x6 rectangle removed from top-left corner.

So perimeter should be same as 13x7 rectangle: 2*(13+7)=40 cm.

Yes.

So perimeter = 40 cm = 0.4 m

But the blank says "____ m", so we'll put 0.4

Area = 43 cm²

Okay.

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Shape 2)

Dimensions in meters:

- Left side: 0.9 m
- Bottom: 1 m
- Right side: 0.5 m
- Indentation: 0.4 m (horizontal)

So, the shape is like a rectangle with a bite taken out of the bottom-right.

Find missing sides.

Total width: 1 m

The indent is 0.4 m, so the left part of the bottom is 1 - 0.4 = 0.6 m? Not necessarily.

Actually, the right side is 0.5 m, left side is 0.9 m, so the difference is 0.4 m — which matches the indent depth.

Split into two rectangles:

Option 1: large rectangle minus small rectangle.

Or: left rectangle and top rectangle.

Better: divide vertically or horizontally.

Notice: the full height on left is 0.9 m, on right is 0.5 m, so the drop is 0.4 m.

The indent is 0.4 m wide.

So, we can think of:

- A main rectangle: width 1 m, height 0.5 m (the bottom part)
- Plus a smaller rectangle on top-left: width ? , height 0.4 m

What is the width of the top rectangle?

Since the indent is 0.4 m on the right, the top rectangle extends only to where the indent starts.

Total width is 1 m, indent is 0.4 m, so the top rectangle width is 1 - 0.4 = 0.6 m

Height of top rectangle: 0.9 - 0.5 = 0.4 m

So:

Rectangle A (bottom): 1 m × 0.5 m = 0.5 m²

Rectangle B (top-left): 0.6 m × 0.4 m = 0.24 m²

Total area = 0.5 + 0.24 = 0.74 m²

Perimeter: trace outer edges.

Start at bottom-left:

→ Right along bottom: 1 m
→ Up right side: 0.5 m
→ Left along the indent top: 0.4 m (this is the horizontal part of the indent)
→ Up the vertical part of the indent: 0.4 m (since 0.9 - 0.5 = 0.4)
→ Left along top: 0.6 m (because total width 1 m, minus 0.4 m indent)
→ Down left side: 0.9 m

Add them:

1 + 0.5 + 0.4 + 0.4 + 0.6 + 0.9

Calculate:

1 + 0.5 = 1.5
1.5 + 0.4 = 1.9
1.9 + 0.4 = 2.3
2.3 + 0.6 = 2.9
2.9 + 0.9 = 3.8 m

But the question asks for perimeter in mm.

Convert 3.8 m to mm: 3.8 × 1000 = 3800 mm

Area is in m², which is fine.

So Area = 0.74 m², Perimeter = 3800 mm

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Shape 3)

Mixed units: cm and mm.

Given:

- Left side: 10 cm
- Top: 2 cm
- Right vertical of upper part: 55 mm
- Horizontal indent: 60 mm

First, convert everything to cm for consistency.

55 mm = 5.5 cm
60 mm = 6.0 cm

So:

- Left: 10 cm
- Top: 2 cm
- Upper right vertical: 5.5 cm
- Indent horizontal: 6.0 cm

Now, find missing sides.

The total height on left is 10 cm.

The upper part has height 5.5 cm, so the lower part height is 10 - 5.5 = 4.5 cm

The top width is 2 cm.

The indent is 6.0 cm to the right, so the total width at the bottom is 2 + 6.0 = 8.0 cm? Not necessarily.

Actually, the shape is like a backwards L.

From top-left:

Go right 2 cm, down 5.5 cm, right 6.0 cm, down ? , left ? , up 10 cm.

The last vertical down should be the remaining height: since total height is 10 cm, and we've gone down 5.5 cm already, the next down is 10 - 5.5 = 4.5 cm? But that might not be accurate.

Better to define coordinates.

Set top-left as (0,0)

→ Right to (2,0) — 2 cm
→ Down to (2, -5.5) — 5.5 cm (using negative for down)
→ Right to (8, -5.5) — 6 cm (since 2+6=8)
→ Down to (8, -10) — how much? Total height is 10 cm, so from y=0 to y=-10, so down 4.5 cm more? From y=-5.5 to y=-10 is 4.5 cm, yes.
→ Left to (0, -10) — 8 cm
→ Up to (0,0) — 10 cm

Now, perimeter: sum of all these segments:

2 + 5.5 + 6 + 4.5 + 8 + 10

Calculate:

2 + 5.5 = 7.5
7.5 + 6 = 13.5
13.5 + 4.5 = 18
18 + 8 = 26
26 + 10 = 36 cm

Area: split into two rectangles.

Rectangle A (left tall): width 2 cm, height 10 cm → area = 20 cm²

But wait, that includes the part that is overlapped? No.

Actually, the shape consists of:

- A left rectangle: 2 cm wide × 10 cm high = 20 cm²
- Plus a right rectangle attached to the bottom: width 6 cm, height 4.5 cm (since from y=-5.5 to y=-10 is 4.5 cm)

But is that correct? When we go from (2,-5.5) to (8,-5.5) to (8,-10) to (0,-10), the part from x=2 to 8, y=-5.5 to -10 is added, but the left part is already included.

Actually, better to split as:

- Top rectangle: 2 cm × 5.5 cm = 11 cm²
- Bottom rectangle: 8 cm × 4.5 cm = 36 cm²? But that would overlap or something.

No: the bottom rectangle should be from x=0 to 8, y=-10 to -5.5? Height 4.5 cm, width 8 cm.

But the top part is only from x=0 to 2, y=0 to -5.5.

So total area = area of top part + area of bottom part.

Top part: 2 cm × 5.5 cm = 11 cm²

Bottom part: 8 cm × 4.5 cm = 36 cm²

But do they overlap? At x=0 to 2, y=-5.5 to -10, it's included in both? No, because top part is y=0 to -5.5, bottom is y=-5.5 to -10, so no overlap.

However, the bottom part includes x=0 to 8, but the top part is only x=0 to 2, so together they cover the whole shape.

Total area = 11 + 36 = 47 cm²

But is that correct? Let me verify with another method.

Use the coordinate method: the shape is a polygon with vertices:

(0,0), (2,0), (2,-5.5), (8,-5.5), (8,-10), (0,-10), back to (0,0)

We can use shoelace formula, but perhaps overkill.

Note that the full bounding box is 8 cm wide × 10 cm high = 80 cm²

Minus the missing rectangle: from x=2 to 8, y=0 to -5.5 — which is 6 cm × 5.5 cm = 33 cm²

So area = 80 - 33 = 47 cm² — same as before.

Good.

So Area = 47 cm²

Perimeter = 36 cm (as calculated)

The question asks for perimeter in cm, which matches.

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Shape 4)

All in cm.

Given:

- Top: 1.2 cm
- Right side: 0.8 cm
- Bottom: 1 cm
- Indent: 0.2 cm (vertical)

So, similar to shape 2 but mirrored.

Find missing sides.

Total width top is 1.2 cm, bottom is 1 cm, so the indent on the right is 1.2 - 1 = 0.2 cm wide? But it's labeled as 0.2 cm vertical.

Look: the indent is a step down on the right.

Specifically, from bottom-right, it goes up 0.2 cm, then left, then up to top.

So, the right side has two parts: bottom part 0.2 cm, then a horizontal left, then up to top.

Total height on right is 0.8 cm.

The indent vertical is 0.2 cm, so the upper part of the right side is 0.8 - 0.2 = 0.6 cm

The bottom width is 1 cm, top width is 1.2 cm, so the horizontal part of the indent is 1.2 - 1 = 0.2 cm

Now, split into rectangles.

Option:

- Main rectangle: width 1 cm, height 0.8 cm? But no, because the top is wider.

Better:

- Left rectangle: width 1 cm, height 0.8 cm? But then the extra on top-right.

Actually, the shape can be seen as:

- A large rectangle 1.2 cm wide × 0.6 cm high (top part)
- Plus a smaller rectangle below it on the left: width 1 cm, height 0.2 cm? Let's see.

Total height is 0.8 cm.

The indent is 0.2 cm up from bottom, so the lower part height is 0.2 cm, upper part height is 0.6 cm.

The upper part width is 1.2 cm.

The lower part width is 1 cm (given bottom).

So:

Rectangle A (top): 1.2 cm × 0.6 cm = 0.72 cm²

Rectangle B (bottom-left): 1 cm × 0.2 cm = 0.2 cm²

Total area = 0.72 + 0.2 = 0.92 cm²

Perimeter: trace outer edges.

Start at bottom-left:

→ Right along bottom: 1 cm
→ Up the indent: 0.2 cm
→ Left along the indent top: 0.2 cm (since 1.2 - 1 = 0.2)
→ Up the right side: 0.6 cm (0.8 - 0.2)
→ Left along top: 1.2 cm
→ Down left side: 0.8 cm

Add them:

1 + 0.2 + 0.2 + 0.6 + 1.2 + 0.8

Calculate:

1 + 0.2 = 1.2
1.2 + 0.2 = 1.4
1.4 + 0.6 = 2.0
2.0 + 1.2 = 3.2
3.2 + 0.8 = 4.0 cm

Check with bounding box: full rectangle 1.2 cm × 0.8 cm, perimeter 2*(1.2+0.8)=4.0 cm — same, since indent is from corner, perimeter unchanged.

Good.

So Area = 0.92 cm², Perimeter = 4.0 cm

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Now, summarize answers:

Shape 1:

Area = 43 cm²

Perimeter = 40 cm = 0.4 m (since asked in m)

Shape 2:

Area = 0.74 m²

Perimeter = 3.8 m = 3800 mm

Shape 3:

Area = 47 cm²

Perimeter = 36 cm

Shape 4:

Area = 0.92 cm²

Perimeter = 4.0 cm

Double-check Shape 1 perimeter unit: the blank says "____ m", and we have 40 cm = 0.4 m, so yes.

Similarly, Shape 2 perimeter in mm: 3.8 m = 3800 mm.

All good.

Final Answer:

1) Area = 43 square cm (cm²), Perimeter = 0.4 m
2) Area = 0.74 square m (m²), Perimeter = 3800 mm
3) Area = 47 square cm (cm²), Perimeter = 36 cm
4) Area = 0.92 square cm (cm²), Perimeter = 4.0 cm