How to find lengths of missing sides and finding perimeter of irregular shape - Free Printable

Let’s solve each problem step by step.

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Problem 1: What is the perimeter?

We are given a shape that looks like an “L” made of two rectangles. The outer sides are labeled, and some inner lines are drawn to help us see how the shape breaks down.

The key idea: Perimeter = total distance around the outside of the shape.

Even though there are internal lines (like the dashed ones), we only care about the outer edges.

Looking at the diagram:

- Bottom side: 20
- Right side: 4 (vertical) + 16 (horizontal top part of lower rectangle) → wait, no — let’s trace the full outer path.

Actually, it’s easier to notice that this L-shape can be thought of as having:

→ Total width = 20
→ Total height = 10

But because it’s indented, we need to add up all the outer segments.

Let’s walk around the shape clockwise starting from bottom-left corner:

1. Go right along bottom: 20
2. Go up on right side: 4
3. Go left along top of lower rectangle: 16 (since 20 - 4 = 16)
4. Go up on the vertical part of the upper rectangle: 6 (because total height is 10, and lower part is 4, so 10 - 4 = 6)
5. Go left along top of upper rectangle: 4
6. Go down on left side: 10

Now add them:

20 + 4 + 16 + 6 + 4 + 10 = ?

Let’s compute:

20 + 4 = 24
24 + 16 = 40
40 + 6 = 46
46 + 4 = 50
50 + 10 = 60

✔ So the perimeter is 60.

That matches option c).

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Problem 2: Find sides a and b, then find perimeter.

This is a more complex L-shaped figure. We’re told to find lengths a and b, then compute the full perimeter.

First, let’s understand what a and b are.

From the diagram:

- Side a is the bottom horizontal side (missing label).
- Side b is the rightmost vertical side (also missing).

We can find these using the fact that opposite sides in such shapes often match when you “complete” the rectangle.

Look at the vertical direction:

On the left side, we have two parts: 16 (top) and 36 (bottom) → total height = 16 + 36 = 52

On the right side, we have: b (top part) and 28 (bottom part). Since the total height must be the same on both sides:

→ b + 28 = 52
→ b = 52 - 28 = 24

Now for horizontal direction:

Top side is labeled 60.

Bottom side is a.

On the right side, there’s a horizontal segment of 20 sticking out.

So, if we look at the bottom, the total length should equal the top minus the overhang? Wait — actually, think of it as:

The entire shape fits inside a big rectangle of width 60 and height 52.

The bottom side a is shorter than 60 because there’s a notch on the right.

Specifically, on the right side, the horizontal part going inward is 20 units long.

So, the bottom side a = 60 - 20 = 40

Wait — let me double-check.

Actually, looking again:

The top is 60.

On the right, after going down 16, we go right 20? No — the 20 is labeled on the *inner* horizontal part on the right.

Actually, better approach: use the horizontal alignment.

The left side goes straight down 16 + 36 = 52.

The right side has a vertical drop of b, then a horizontal move left of 20, then down 28.

So the total height on right: b + 28 = 52 → b = 24 ✔️

For width:

The top is 60.

The bottom is a.

Between them, on the right, there’s a horizontal segment of 20 that sticks out to the right? Actually, no — the 20 is the length of the horizontal part on the right side, which means the bottom part is shorter by 20.

Wait — imagine completing the rectangle.

If you extend the bottom to the right, it would be 60. But currently, the bottom stops early — the missing part is exactly the 20-unit segment on the right.

So yes: a = 60 - 20 = 40

Alternatively, look at the left side: from left edge to the inner corner is... well, maybe not necessary.

Another way: the horizontal distance from left to right at the bottom level is a.

At the top level, it’s 60.

The difference is due to the 20-unit indentation on the right.

So yes, a = 40

Now, we have:

- a = 40
- b = 24

Now, let’s compute the perimeter.

We’ll walk around the entire outer edge.

Start at bottom-left corner, go clockwise:

1. Bottom side: a = 40
2. Up on right-bottom vertical: 28
3. Left on the small horizontal: 20
4. Up on the right-top vertical: b = 24
5. Left on top: 60
6. Down on left-top vertical: 16
7. Right on the middle horizontal? Wait — no, we already did top.

Wait — I think I missed something.

Actually, after step 5 (left on top: 60), we go down the left side: but the left side is split into 16 (top) and 36 (bottom). So:

After top (60), go down left side: first 16, then we hit the inner corner, then we go right? No — actually, the left side is continuous downward? Let me sketch mentally.

Better to list all outer segments:

Going clockwise:

- Start at bottom-left
- → Right: a = 40
- ↑ Up: 28
- ← Left: 20 (this is the inner horizontal on the right)
- ↑ Up: b = 24
- ← Left: 60 (top)
- ↓ Down: 16 (left-top vertical)
- → Right: ??? Wait — now we’re at the inner corner on the left.

Actually, after going down 16 on the left, we are at the point where the shape turns right — but that’s an internal corner? No — in the perimeter, we don’t go inside.

I think I made a mistake.

Let me define the vertices.

Label the corners:

Assume the shape has these outer points (going clockwise):

A: bottom-left
B: bottom-right (end of side a)
C: up 28 to point C
D: left 20 to point D
E: up b=24 to point E
F: left 60 to point F (top-left)
G: down 16 to point G
H: right ? to connect back to A?

Wait — from G, we need to go down to A? But there’s a gap.

Actually, from G (which is 16 down from top-left), we go down another 36 to reach the bottom-left? But that would mean the left side is 16 + 36 = 52, which matches.

But between G and A, is there a direct vertical line? Yes — the left side is straight down 52 units.

In my earlier walk, I went down only 16, but I need to go all the way down.

Let me restart the perimeter walk carefully.

Start at bottom-left corner (call it P1).

Move right along bottom: length a = 40 → to P2.

Move up: 28 → to P3.

Move left: 20 → to P4. (this is the horizontal part on the right)

Move up: b = 24 → to P5. (now at top-right corner)

Move left: 60 → to P6. (top-left corner)

Move down: 16 → to P7. (this is the top part of the left side)

Now, from P7, we need to get back to P1. But P7 is not directly above P1 — there’s a horizontal offset?

No — actually, from P7, if we go down, we should reach the bottom-left? But the total left side is 52, and we’ve only gone down 16 so far.

Ah! Here’s the issue: after moving down 16 from top-left, we are at a point that is still 36 units above the bottom. And since the left side is straight, we continue down 36 more to reach bottom-left.

But in the diagram, is there a horizontal segment between the end of the 16-down and the start of the 36-down? No — it’s a single vertical line on the left.

Looking back at the original diagram description:

Left side: labeled 16 (top part) and 36 (bottom part) — meaning the entire left side is 16 + 36 = 52, and it’s one straight line? Or is there a jog?

Actually, in standard L-shapes like this, the left side is usually straight. The 16 and 36 are just labels for the two segments, but they are colinear.

Similarly, on the right, b and 28 are vertical segments, but with a horizontal jog in between.

So, for perimeter, when we go down the left side, we go the full 52 units? But that can’t be, because we already accounted for parts.

I think I confused myself.

Let me try a different method: add all the outer sides without walking.

List all the outer edges:

Horizontal sides:

- Top: 60
- Bottom: a = 40
- Also, on the right, there’s a horizontal segment of 20 (the one connecting the two vertical parts on the right) — but is this outer or inner?

In the diagram, the 20 is labeled on the horizontal part that is on the "inside" of the L, but for perimeter, we only count the outer boundary.

Actually, in an L-shape, the perimeter includes the outer frame plus the inner "notch" if it's concave.

Standard way: for such polygons, perimeter is sum of all outer edges, including the indentations.

So, let's list every segment that forms the boundary.

From the diagram:

Vertical segments:

- Left side: total 52 (16 + 36) — but is it one segment or two? For perimeter, if it's straight, it's one, but here it might be considered as two parts, but length is still 52.

Actually, no — in terms of counting, we need to see how many distinct sides.

Perhaps it's better to use the "bounding box" method.

Notice that this shape can be seen as a large rectangle with a smaller rectangle cut out.

Large rectangle: width 60, height 52 (since 16+36=52)

Cut out a rectangle from the bottom-right corner: width 20, height 28? Let's see.

If we cut out a rectangle of size 20x28 from the bottom-right, then:

- The bottom side becomes 60 - 20 = 40 = a
- The right side becomes 52 - 28 = 24 = b

Yes, that makes sense.

When you cut out a rectangle from the corner, the perimeter changes.

Original perimeter of large rectangle: 2*(60 + 52) = 2*112 = 224

When you cut out a rectangle of size w x h from a corner, you remove two sides (w and h) but add two new sides (w and h) — so perimeter remains the same!

Is that right?

Imagine a rectangle. Cut out a small rectangle from the corner. You remove the outer corner, but you add two new edges: one horizontal and one vertical, each equal to the cut-out dimensions.

So net change: you lose the two outer segments (length w and h), but gain two inner segments (also w and h), so perimeter unchanged.

Therefore, perimeter of this L-shape is the same as the bounding rectangle: 2*(60 + 52) = 224

Oh! So we don't even need to find a and b for the perimeter — it's automatically 224.

But the problem asks to find a and b first, then perimeter.

So let's confirm a and b.

As before:

Total height = 16 + 36 = 52

On right: b + 28 = 52 → b = 24

Total width = 60

On bottom: a + 20 = 60? No.

If the cut-out is 20 wide and 28 high, then:

- The bottom side of the L is the full width minus the cut-out width: a = 60 - 20 = 40

- The right side of the L is full height minus cut-out height: b = 52 - 28 = 24

Yes.

Now, to verify perimeter by adding all sides:

List all outer edges in order:

Start at bottom-left:

1. Right: a = 40

2. Up: 28 (this is the right-bottom vertical)

3. Left: 20 (this is the horizontal part of the cut-out — but is this outer? In the L-shape, this is actually an inner edge? No.

I think I have a confusion about what is outer.

In the standard L-shape like this, when you have the main rectangle and you remove a corner, the perimeter includes the three sides of the cut-out that are now exposed.

Let me draw it mentally:

- The shape has:

- Bottom: from left to right, length a=40

- Then up 28

- Then left 20 (this is the top of the cut-out rectangle)

- Then up b=24 (to reach the top)

- Then left 60 (top)

- Then down 52 (left side) — but wait, that would overlap.

No.

Correct path:

Start at bottom-left corner.

- Move right along bottom: 40 (to the point where the cut-out starts)

- Move up: 28 (along the right side of the cut-out)

- Move left: 20 (along the top of the cut-out — this is now part of the outer perimeter)

- Move up: 24 (b) to reach the top-right corner

- Move left: 60 (top)

- Move down: 52 (left side) back to start

But when we move down 52, that includes the entire left side, which is correct.

Now, sum these:

40 (bottom) + 28 (up right) + 20 (left) + 24 (up) + 60 (left top) + 52 (down left) = ?

Calculate:

40 + 28 = 68

68 + 20 = 88

88 + 24 = 112

112 + 60 = 172

172 + 52 = 224

Yes! Perimeter = 224

And we have a=40, b=24

Now, looking at the options for problem 2:

a) 168
b) 200
c) 224
d) 300

So answer is c) 224

Perfect.

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Final Answer:
For problem 1: c) 60
For problem 2: c) 224