Finding Area and Perimeter of Irregular shapes - Math Worksheets ... - Free Printable

Let’s solve problem #8 step by step.

We are given an irregular L-shaped figure with these measurements:

- Left vertical side: 26 yd
- Bottom horizontal side: 48 yd
- Right vertical “notch” down: 11 yd
- Top right horizontal part: 24 yd

We need to find:
→ Perimeter (P) = total distance around the shape
→ Area (A) = total space inside the shape

---

Step 1: Find missing sides for perimeter

This is an L-shape made of rectangles. We can think of it as a big rectangle with a smaller rectangle cut out — but for perimeter, we just add all outer edges.

Label the unknown sides:

The top-left horizontal side? Let’s call it x.
The inner vertical side (between the two horizontal parts)? Call it y.

From the bottom: total width = 48 yd
Top right part = 24 yd → so top left part must be:
→ x = 48 - 24 = 24 yd

From the left: total height = 26 yd
Right notch goes down 11 yd → so the inner vertical drop (y) must be:
→ y = 26 - 11 = 15 yd

Now list ALL outer sides going clockwise from top-left corner:

1. Top-left horizontal: 24 yd
2. Inner vertical down: 15 yd
3. Top-right horizontal: 24 yd
4. Right vertical down: 11 yd
5. Bottom horizontal: 48 yd
6. Left vertical up: 26 yd

Wait — that’s not quite right. Actually, when you go around the outside, you don’t include the inner vertical unless it’s on the boundary. Let me redraw mentally:

Actually, better way: trace the entire outer edge.

Start at bottom-left corner:

→ Go right along bottom: 48 yd
→ Go up right side: 11 yd (this is the short right leg)
→ Go left along top of that little rectangle: 24 yd
→ Go up the inner vertical: this is the difference in heights: 26 - 11 = 15 yd
→ Go left along top: 24 yd (since 48 - 24 = 24)
→ Go down left side: 26 yd back to start.

So sides are:

- Bottom: 48
- Right-up: 11
- Top-right-horizontal: 24
- Inner-vertical-up: 15
- Top-left-horizontal: 24
- Left-down: 26

Add them:

48 + 11 = 59
59 + 24 = 83
83 + 15 = 98
98 + 24 = 122
122 + 26 = 148 yd

✔ Perimeter = 148 yd — matches what’s written!

---

Step 2: Find area

Break the L-shape into two rectangles.

Option A: Big rectangle minus small rectangle? Not necessary here.

Better: Split vertically or horizontally.

Let’s split it into:

→ Rectangle 1: left part, full height 26 yd, width = 24 yd (because 48 - 24 = 24)
→ Rectangle 2: right part, only the bottom part, height = 11 yd, width = 24 yd

Wait — actually, if we look:

Total width = 48 yd
The right protrusion is 24 yd wide and 11 yd tall.
The left part is 24 yd wide (48 - 24) and 26 yd tall.

But wait — the left part includes the area under the right part? No.

Actually, correct split:

Imagine the whole shape as:

- A large rectangle on the left: 24 yd wide × 26 yd high
- Plus a smaller rectangle on the bottom right: 24 yd wide × 11 yd high

BUT — they overlap? No, because the left rectangle goes full height, and the right one is attached to its bottom right.

Actually, no — if the left part is 24 yd wide and 26 yd tall, and the right part is 24 yd wide and 11 yd tall, then together they make the L-shape without overlapping.

Check total width: 24 + 24 = 48 ✔️
Height on left: 26, on right: 11 — which matches the diagram.

So area = area of left rectangle + area of right rectangle

Left: 24 × 26
Right: 24 × 11

Calculate:

24 × 26:
20×26 = 520
4×26 = 104
Total = 520 + 104 = 624

24 × 11 = 264

Total area = 624 + 264 = 888 yd²

Wait — but the worksheet says A = 984 yd². That doesn’t match.

Hmm… did I split wrong?

Alternative split:

Think of the whole bounding box: 48 yd wide × 26 yd high = 1248 yd²

Then subtract the missing rectangle on top right.

What’s missing? The top-right corner that’s cut out.

Width of missing part: 24 yd (same as the top-right horizontal)
Height of missing part: 26 - 11 = 15 yd

So missing area = 24 × 15 = 360 yd²

Then actual area = 1248 - 360 = 888 yd²

Still 888.

But worksheet says 984.

That suggests maybe my interpretation of the diagram is wrong.

Wait — let’s re-read the labels.

In problem #8:

Left side: 26 yd
Bottom: 48 yd
Top-right horizontal: 24 yd
Right-side vertical drop: 11 yd

Perhaps the 11 yd is NOT the height of the right leg, but the depth of the notch?

Wait — looking again: the label “11 yd” is placed on the vertical segment that drops down from the top-right horizontal. So yes, that should be the height of the right leg.

But then why does the worksheet say 984?

Let me try another split.

Maybe the shape is composed of:

- Bottom rectangle: 48 yd wide × 11 yd high → area = 528
- Top-left rectangle: (48 - 24) = 24 yd wide × (26 - 11) = 15 yd high → area = 24 × 15 = 360
Total = 528 + 360 = 888 again.

Same result.

Wait — perhaps the 26 yd is not the full left height? But it’s labeled on the left side.

Or maybe the 11 yd is the length of the horizontal part? No, it’s drawn vertically.

Another idea: maybe the 24 yd is not the top-right horizontal, but something else?

Looking at the diagram description: “11 yd” is on the vertical drop, “24 yd” is on the horizontal part after the drop.

Perhaps the total height is not 26? But it’s labeled on the left.

Wait — let’s calculate what would give 984.

Suppose area = 984.

If we do 48 × 26 = 1248
1248 - 984 = 264 → so missing area would be 264.

264 ÷ 24 = 11 → so if missing rectangle is 24 × 11, then height of missing part is 11, meaning the right leg height is 26 - 11 = 15.

But in the diagram, the 11 yd is labeled on the vertical drop — which would be the height of the right leg, not the missing part.

Unless... the 11 yd is the height of the missing part?

Let me reinterpret the diagram based on common conventions.

In many such problems, when you see an L-shape with:

- Left side: 26
- Bottom: 48
- Then a horizontal segment labeled 24 on the top right
- And a vertical segment labeled 11 dropping down from the end of that 24-yd segment

Then the 11 yd is the height of the "arm" sticking down — so the right leg is 11 yd tall.

Then the top-left horizontal must be 48 - 24 = 24 yd.

And the inner vertical (from the end of the top-left to the start of the right leg) is 26 - 11 = 15 yd.

Area calculation still gives 888.

But the worksheet says 984.

Perhaps there's a mistake in the worksheet? Or in my reading?

Wait — let's check the other problems to see if their answers are correct.

For example, problem #1:

Shape: left 10 in, bottom 25 in, top-right horizontal 15 in, vertical drop 4 in.

Perimeter: they say 70 in.

Let's verify:

Sides: bottom 25, right-up 4, top-right 15, inner-vertical down? Wait.

Actually, similar logic:

Missing top-left horizontal: 25 - 15 = 10 in
Inner vertical: 10 - 4 = 6 in? No.

Trace perimeter:

Start bottom-left:

→ right 25
→ up 4 (right leg)
→ left 15 (top of right leg)
→ up ? This is the inner vertical: since left side is 10, and we've gone up 4, so remaining up is 6
→ left 10 (top-left)
→ down 10 (left side)

Sides: 25, 4, 15, 6, 10, 10

Sum: 25+4=29; +15=44; +6=50; +10=60; +10=70 ✔️

Area: split into two rectangles.

Left: 10 in wide × 10 in high = 100
Right: 15 in wide × 4 in high = 60
Total = 160? But worksheet says 190.

Wait, that doesn't match.

Alternative split:

Big rectangle 25×10 = 250
Minus missing top-right: width 15, height 6 (since 10-4=6) → 15×6=90
250 - 90 = 160 — still not 190.

But worksheet says A=190.

How to get 190?

25×10 = 250
If missing area is 60, then 250-60=190 → missing rectangle 15×4=60 — but that would mean the drop is 4, so the missing height is 4, implying the right leg height is 6.

But in the diagram, the 4 in is labeled on the vertical drop — which should be the height of the right leg.

I'm confused.

Perhaps for problem #1, the 4 in is the height of the missing part, not the leg.

Let me assume that.

In problem #1:

Left side: 10 in
Bottom: 25 in
Top-right horizontal: 15 in
Vertical drop: 4 in — but if this 4 in is the amount that the top is lower than the left, then the right leg height is 10 - 4 = 6 in.

Then area:

Left rectangle: 10 in wide × 10 in high = 100
Right rectangle: 15 in wide × 6 in high = 90
Total = 190 ✔️

Perimeter:

Sides: bottom 25, right-up 6, top-right 15, inner-vertical down 4, top-left 10, left-down 10

Sum: 25+6=31; +15=46; +4=50; +10=60; +10=70 ✔️

Ah! So the key is: the labeled vertical dimension (like 4 in in #1, 11 yd in #8) is the *difference* in height, i.e., how much the top-right is lower than the left side. It is NOT the height of the right leg.

In other words, the number labeled on the vertical drop is the amount that the upper surface is recessed, so the actual height of the right leg is total height minus that number.

Similarly, in problem #8:

Left side: 26 yd (total height)
Vertical drop labeled: 11 yd → this means the right leg height is 26 - 11 = 15 yd
Top-right horizontal: 24 yd
Bottom: 48 yd → so top-left horizontal = 48 - 24 = 24 yd

Now area:

Split into two rectangles:

1. Left rectangle: width = 24 yd, height = 26 yd → area = 24 × 26 = 624
2. Right rectangle: width = 24 yd, height = 15 yd (since right leg is 15 yd tall) → area = 24 × 15 = 360
Total area = 624 + 360 = 984 yd² ✔️

Perfect! Matches the worksheet.

Perimeter we already calculated as 148 yd, which also matches.

So for problem #8:

Perimeter = 148 yd
Area = 984 yd²

Final Answer:
P = 148 yd, A = 984 yd²