G17d - Perimeter and area of composite shapes made up of polygons ... - Free Printable

Let's solve Alpha Exercise 1 step by step.

We are given two composite shapes, each made up of two rectangles. For each shape, we need to find:

- a) The perimeter
- b) The area

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🔷 Shape (1): Green L-shaped figure

Given dimensions:
- Top rectangle: width = 13 cm, height = ?
- Bottom rectangle: width = 7 cm, height = 5 cm
- Total height = 9 cm → So the top part must be $ 9 - 5 = 4 $ cm tall.

So, we can split the shape into two rectangles:

#### ✔ Rectangle A (top):
- Length = 13 cm
- Height = 4 cm

#### ✔ Rectangle B (bottom right):
- Length = 7 cm
- Height = 5 cm

Note: The bottom left is not a separate rectangle; the total shape is formed such that the bottom rectangle extends 7 cm to the right and 5 cm down, and the top rectangle is 13 cm wide and 4 cm high.

But wait — actually, let’s re-analyze the geometry.

Looking at the diagram:

- The total height is 9 cm.
- The bottom rectangle has a height of 5 cm → so the top rectangle must be $ 9 - 5 = 4 $ cm tall.
- The bottom rectangle has a base of 7 cm.
- The top rectangle is 13 cm wide.

So the top rectangle sticks out to the left by $ 13 - 7 = 6 $ cm.

Therefore, we can break it into:

- Top rectangle: 13 cm × 4 cm
- Bottom rectangle: 7 cm × 5 cm

They share a common vertical side of 7 cm, but only internally.

Now, let's compute:

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🔹 a) Perimeter of Shape (1)

To find the perimeter, we sum all the outer edges.

Let’s trace around the shape:

Starting from the top-left corner, go clockwise:

1. Top edge: 13 cm
2. Right edge: 9 cm (full height)
3. Bottom edge: 7 cm (right part)
4. Left edge of bottom rectangle: 5 cm upward
5. Then go left: 6 cm (since 13 − 7 = 6 cm overhang on the left)
6. Then up: 4 cm (height of top rectangle)

Wait — better to list all outer sides clearly.

Break down the outer boundary:

- Top: 13 cm
- Right: 9 cm
- Bottom: 7 cm (only the bottom part of the lower rectangle)
- Left side of lower rectangle: 5 cm (vertical)
- Horizontal gap: from the end of the lower rectangle to the start of the upper one? No — actually, the left side of the top rectangle is not fully exposed.

Actually, let's label all sides.

We'll use coordinates or just logic.

From the figure:

- The leftmost point is at 13 cm width on the top, but only 6 cm of the top rectangle extends beyond the bottom one.
- So the left side of the shape has:
- A vertical segment of 4 cm (from top of top rectangle down to where bottom starts)
- Then a horizontal segment of 6 cm (to the left), then down 5 cm?

Wait — better approach: add all external sides.

List all outer edges:

1. Top: 13 cm
2. Right: 9 cm
3. Bottom: 7 cm
4. Left side of bottom rectangle: 5 cm (going up)
5. Inner horizontal segment: from the end of the bottom rectangle to the start of the top one? Wait — no.

Wait — actually, the left side of the shape is composed of:
- A vertical line of 4 cm (the left side of the top rectangle)
- Then a horizontal line of 6 cm (the overhang on the left) — but this is not a side unless it's exposed.

Wait — actually, no. The left side of the entire shape is:

- From top-left to bottom-left: it goes down 4 cm, then right 6 cm, then down 5 cm?

No — that would make it non-rectangular.

Let me sketch mentally:

- The top rectangle is 13 cm wide and 4 cm tall.
- The bottom rectangle is 7 cm wide and 5 cm tall.
- The bottom rectangle is placed under the right 7 cm of the top rectangle.
- So the left 6 cm of the top rectangle is overhanging.

So the outer perimeter consists of:

1. Top: 13 cm
2. Right side: 9 cm (from top-right down to bottom-right)
3. Bottom: 7 cm (from bottom-right to bottom-left of bottom rectangle)
4. Left side of bottom rectangle: 5 cm (upward)
5. Horizontal segment from the left end of the bottom rectangle to the left end of the top rectangle: this is 6 cm (because 13 - 7 = 6 cm), going left, then up?
- Actually, from the bottom-left of the bottom rectangle, we go left for 6 cm? No — that's not correct.

Wait — actually, the left side of the shape has:
- A vertical segment of 4 cm (from top-left to the bottom of the top rectangle)
- Then a horizontal segment of 6 cm (from there to the left end of the bottom rectangle)? No — that's internal.

Wait — no. The left side of the shape is:

- From top-left corner: go down 4 cm (left side of top rectangle)
- Then go right 6 cm? That doesn't make sense.

Better: Let's assign coordinates.

Assume bottom-left corner is at (0,0):

- Bottom rectangle: from (0,0) to (7,5)
- Top rectangle: from (6,5) to (13,9) → because it extends 6 cm to the left of the bottom rectangle? Wait — no.

Wait — if the top rectangle is 13 cm wide and sits above the bottom one, and the bottom rectangle is only 7 cm wide, then the overlap is 7 cm.

So the bottom rectangle is from x=0 to x=7, y=0 to y=5.

The top rectangle is from x=0 to x=13, y=5 to y=9.

Yes — that makes sense.

So:
- Bottom rectangle: 0 ≤ x ≤ 7, 0 ≤ y ≤ 5
- Top rectangle: 0 ≤ x ≤ 13, 5 ≤ y ≤ 9

Then the full shape is from x=0 to x=13, y=0 to y=9, but with a "step" at x=7, y=5.

Wait — no — the top rectangle is wider, so it extends from x=0 to x=13.

So the combined shape has:

- Bottom: x from 0 to 7, y=0 to 5
- Top: x from 0 to 13, y=5 to 9

So the overall shape is like an "L" but with the long arm on the top.

Now, the outer perimeter:

Start at (0,0):

1. Go right to (7,0): 7 cm
2. Up to (7,5): 5 cm
3. Right to (13,5): 6 cm
4. Up to (13,9): 4 cm
5. Left to (0,9): 13 cm
6. Down to (0,0): 9 cm

Wait — no! From (13,9) to (0,9) is 13 cm, but we already have the top.

Wait — let's trace carefully:

Start at bottom-left: (0,0)

1. → (7,0): 7 cm (bottom edge)
2. → (7,5): 5 cm (right side of bottom rectangle)
3. → (13,5): 6 cm (horizontal extension of top rectangle)
4. → (13,9): 4 cm (right side of top rectangle)
5. → (0,9): 13 cm (top edge)
6. → (0,0): 9 cm (left side)

But wait — from (0,9) to (0,0) is 9 cm, yes.

But now check: does this include all outer edges?

Yes.

So total perimeter:

- 7 + 5 + 6 + 4 + 13 + 9 = ?

Add:

7 + 5 = 12
12 + 6 = 18
18 + 4 = 22
22 + 13 = 35
35 + 9 = 44 cm

✔ Perimeter of Shape (1) = 44 cm

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🔹 b) Area of Shape (1)

Area = Area of top rectangle + Area of bottom rectangle

- Top rectangle: 13 cm × 4 cm = 52 cm²
- Bottom rectangle: 7 cm × 5 cm = 35 cm²

Total area = 52 + 35 = 87 cm²

✔ Area of Shape (1) = 87 cm²

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## 🟦 Shape (2): Blue L-shaped figure

Dimensions:

- Left rectangle: 5 cm wide, 4 cm high
- Right rectangle: 4 cm wide, 2 cm high
- But total width = 9 cm → so the left rectangle is 5 cm wide, and the right one is 4 cm wide → 5 + 4 = 9 cm ✔️
- Total height = 4 + 2 = 6 cm? But labeled as 4 cm on the left, and 2 cm on the right → so the left rectangle is 4 cm tall, and the right rectangle is 2 cm tall, but they are stacked vertically?

Wait — look at the diagram:

- The left part is 5 cm wide, 4 cm high
- The right part is 4 cm wide, 2 cm high
- The total width is 9 cm → 5 + 4 = 9 cm ✔️
- The total height is not uniform: left is 4 cm, right is 2 cm, but the bottom is continuous?

Wait — the right rectangle is attached to the bottom of the left one?

But the left rectangle is 4 cm high, and the right one is 2 cm high — so the right one is shorter.

So the shape is:

- Left rectangle: 5 cm × 4 cm
- Right rectangle: 4 cm × 2 cm
- They are attached at the bottom — so the right rectangle sits below the left one?

But the total height on the left is 4 cm, and on the right is 2 cm → so the right rectangle is lower.

But the base is 9 cm long, so the bottom is continuous.

Wait — looking at the diagram:

- The left rectangle is 5 cm wide and 4 cm high
- The right rectangle is 4 cm wide and 2 cm high
- The right rectangle is attached to the bottom of the left one, but only extending 2 cm in height

So the left rectangle goes from y=0 to y=4
- The right rectangle goes from y=0 to y=2

So they share the bottom part.

Thus, the total shape is:

- From x=0 to 5, y=0 to 4 (left rectangle)
- From x=5 to 9, y=0 to 2 (right rectangle)

So the bottom is 9 cm long, from (0,0) to (9,0)

The top is not flat — it has a step at x=5, y=2 to y=4.

Now, compute:

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🔹 a) Perimeter of Shape (2)

Trace the outer boundary:

Start at (0,0):

1. → (5,0): 5 cm (bottom of left rectangle)
2. → (5,4): 4 cm (up)
3. → (9,4): 4 cm (right edge of top?)
- Wait — no. At x=5, y=4, the right rectangle only goes up to y=2 → so from (5,4), we cannot go right.

Wait — actually, the right rectangle is only 2 cm high → so the top of the shape is:

- From x=0 to 5: y=4
- From x=5 to 9: y=2

So the top edge is not straight.

So the outer boundary:

Start at (0,0):

1. → (5,0): 5 cm (bottom)
2. → (5,4): 4 cm (left side of left rectangle)
3. → (9,4): 4 cm (but wait — no, the right rectangle only goes up to y=2)

Wait — no! From (5,4), we cannot go to (9,4) because there's no shape there.

Instead, from (5,4), we go right along the top of the left rectangle until we reach the end, but the right rectangle only goes up to y=2.

So the top edge of the shape is:

- From (0,4) to (5,4): 5 cm
- Then from (5,4) to (5,2): down 2 cm (inner edge? No — that's internal)

Wait — no. The top of the shape ends at x=5, y=4.

Then the right side of the shape is:

- From (5,4) down to (5,2): 2 cm (this is internal — not part of perimeter)

Wait — no. The right rectangle is from x=5 to 9, y=0 to 2.

So the right side of the shape is:

- From (9,2) to (9,0): 2 cm
- From (9,0) to (5,0): 4 cm (bottom)
- From (5,0) to (5,4): 4 cm
- From (5,4) to (0,4): 5 cm
- From (0,4) to (0,0): 4 cm

Wait — but (0,4) to (0,0) is 4 cm, yes.

But is that correct?

Let’s list all outer edges:

Start at (0,0):

1. → (5,0): 5 cm (bottom of left rectangle)
2. → (5,4): 4 cm (left side of left rectangle)
3. → (0,4): 5 cm (top of left rectangle) ← but wait — this is from (5,4) to (0,4)? That’s leftward.

Wait — no. We should go around the shape.

Better:

Start at (0,0):

1. → (5,0): 5 cm (bottom)
2. → (5,4): 4 cm (up)
3. → (0,4): 5 cm (left)
4. → (0,0): 4 cm (down)

But this ignores the right rectangle!

Wait — the right rectangle is from x=5 to 9, y=0 to 2.

So the shape includes the region from x=5 to 9, y=0 to 2.

So the outer boundary must include:

- From (5,0) → (9,0): 4 cm
- From (9,0) → (9,2): 2 cm
- From (9,2) → (5,2): 4 cm (but this is internal? No — it's the top of the right rectangle)

Wait — but (5,2) to (5,4) is the side of the left rectangle.

So the entire shape has:

- Bottom: from (0,0) to (9,0): 9 cm
- Left: from (0,0) to (0,4): 4 cm
- Top: from (0,4) to (5,4): 5 cm
- Then from (5,4) to (5,2): 2 cm (this is internal — not outer)
- Then from (5,2) to (9,2): 4 cm (top of right rectangle)
- Then from (9,2) to (9,0): 2 cm (right side)
- Then from (9,0) to (0,0): 9 cm — but we already did that.

Wait — better to trace:

Start at (0,0):

1. → (9,0): 9 cm (bottom)
2. → (9,2): 2 cm (right side)
3. → (5,2): 4 cm (top of right rectangle)
4. → (5,4): 2 cm (up along left side of right rectangle? No — from (5,2) to (5,4) is up, but that's inside the left rectangle? No — the left rectangle is from x=0 to 5, y=0 to 4 → so (5,2) to (5,4) is on the boundary.

Wait — actually, the line x=5 from y=2 to y=4 is shared between the two rectangles — but it's internal if both rectangles are adjacent.

But in terms of perimeter, only the outer edges count.

So the outer boundary:

- Start at (0,0)
- → (9,0): 9 cm (bottom)
- → (9,2): 2 cm (right side)
- → (5,2): 4 cm (top of right rectangle)
- → (5,4): 2 cm (up along the shared edge? But this is internal — no!)

Wait — no. The top of the shape is:

- From (0,4) to (5,4): 5 cm
- Then from (5,4) to (5,2): 2 cm — but this is vertical, and since the right rectangle is only 2 cm high, this is not part of the outer edge.

Wait — actually, the outer edge from (5,4) is not connected directly to (5,2) — instead, the top of the shape ends at (5,4), and the right side starts at (9,2).

But what about the gap?

Wait — the shape is continuous:

- From x=0 to 5, y=0 to 4
- From x=5 to 9, y=0 to 2

So the boundary:

- Bottom: (0,0) to (9,0): 9 cm
- Right: (9,0) to (9,2): 2 cm
- Top of right: (9,2) to (5,2): 4 cm
- Then from (5,2) to (5,4): 2 cm — this is vertical, and since the left rectangle extends up to y=4, this is part of the outer edge — because it's the right side of the left rectangle, and the right rectangle is shorter.

So yes, (5,2) to (5,4) is external — it's the right edge of the left rectangle, and it's exposed because the right rectangle doesn’t cover it.

Then from (5,4) to (0,4): 5 cm (top of left rectangle)
Then from (0,4) to (0,0): 4 cm (left side)

Now, list all segments:

1. (0,0) → (9,0): 9 cm
2. (9,0) → (9,2): 2 cm
3. (9,2) → (5,2): 4 cm
4. (5,2) → (5,4): 2 cm
5. (5,4) → (0,4): 5 cm
6. (0,4) → (0,0): 4 cm

Now sum:

9 + 2 = 11
11 + 4 = 15
15 + 2 = 17
17 + 5 = 22
22 + 4 = 26 cm

✔ Perimeter of Shape (2) = 26 cm

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🔹 b) Area of Shape (2)

Area = Area of left rectangle + Area of right rectangle

- Left: 5 cm × 4 cm = 20 cm²
- Right: 4 cm × 2 cm = 8 cm²

Total area = 20 + 8 = 28 cm²

✔ Area of Shape (2) = 28 cm²

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## ✔ Final Answers:

Shape (1):

- a) Perimeter = 44 cm
- b) Area = 87 cm²

Shape (2):

- a) Perimeter = 26 cm
- b) Area = 28 cm²

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Let me know if you'd like diagrams or explanations in another format!