Area and Perimeter of Compound Shapes (A) - Free Printable

Let's solve each of the compound shapes in your worksheet step by step. We'll calculate both area and perimeter for each shape.

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1) Shape 1: Rectangle with a Quarter Circle on Top

Given:
- Rectangle: AB = 22.2 in, AD = 11.5 in
- Quarter circle (quarter of a circle) on top: radius = 12.1 in

#### Area:
- Area of rectangle = length × width = $ 22.2 \times 11.5 $
$$
= 255.3 \text{ in}^2
$$
- Area of quarter circle = $ \frac{1}{4} \pi r^2 = \frac{1}{4} \pi (12.1)^2 $
$$
= \frac{1}{4} \pi (146.41) \approx \frac{1}{4} \times 3.1416 \times 146.41 \approx 114.08 \text{ in}^2
$$
- Total area = $ 255.3 + 114.08 = 369.38 \text{ in}^2 $

#### Perimeter:
- Perimeter includes:
- Bottom: AB = 22.2 in
- Left side: AD = 11.5 in
- Right side: BC = 11.5 in
- Arc of quarter circle: $ \frac{1}{4} \times 2\pi r = \frac{1}{2} \pi r $
$$
= \frac{1}{2} \times 3.1416 \times 12.1 \approx 19.02 \text{ in}
$$
- The straight edge from D to F is already part of the rectangle (vertical), but we don't include it again.
- However, the top is replaced by the arc — so instead of DF and FC (which are not included), we have the arc.
- So total perimeter:
$$
AB + BC + \text{arc} + DA = 22.2 + 11.5 + 19.02 + 11.5 = 64.22 \text{ in}
$$

> ✔ Answer 1:
> - Area: ≈ 369.4 in²
> - Perimeter: ≈ 64.2 in

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2) Shape 2: Rectangle with a Semicircle on Top

Given:
- Rectangle: AB = 20.2 mm, AD = 3.1 mm
- Semicircle on top: diameter = 20.2 mm → radius = 10.1 mm

#### Area:
- Rectangle: $ 20.2 \times 3.1 = 62.62 \text{ mm}^2 $
- Semicircle: $ \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (10.1)^2 $
$$
= \frac{1}{2} \pi (102.01) \approx 160.2 \text{ mm}^2
$$
- Total area = $ 62.62 + 160.2 = 222.82 \text{ mm}^2 $

#### Perimeter:
- Rectangle sides: AB (bottom), BC (right), CD (top? no — covered), DA (left)
- But top is replaced by semicircular arc.
- So:
- Bottom: 20.2 mm
- Left: 3.1 mm
- Right: 3.1 mm
- Semicircular arc: $ \pi d / 2 = \pi \times 20.2 / 2 = \pi \times 10.1 \approx 31.7 \text{ mm} $
- Total perimeter = $ 20.2 + 3.1 + 3.1 + 31.7 = 58.1 \text{ mm} $

> ✔ Answer 2:
> - Area: ≈ 222.8 mm²
> - Perimeter: ≈ 58.1 mm

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3) Shape 3: Rectangle with a Triangle Cut Out

Given:
- Outer rectangle: AF = 11.1 cm, FB = 12.1 cm → AB = 11.1 + 12.1 = 23.2 cm?
Wait — let’s examine:

Actually:
- AB = ? From diagram:
- AF = 11.1 cm (vertical)
- FD = 11.1 cm (horizontal)
- DC = 12.1 cm (horizontal)
- CB = 12.1 cm (vertical)

Wait — this seems like a rectangle ABCD with a triangle removed?

But look at the dashed lines: triangle EFC is cut out.

From the figure:
- Rectangle ABCD: AB = 23.2 cm? Wait:
- AF = 11.1 cm
- FB = 12.1 cm → AB = 11.1 + 12.1 = 23.2 cm
- AD = 11.1 cm (vertical)
- DC = 12.1 cm (horizontal)
- Wait — inconsistent unless it's a rectangle with different dimensions.

Actually, from labeling:
- A to F: 11.1 cm (vertical)
- F to D: 11.1 cm (horizontal)
- D to C: 12.1 cm (horizontal)
- C to B: 12.1 cm (vertical)
- B to A: bottom base

So likely:
- Rectangle ABCD has:
- AB = 11.1 + 12.1 = 23.2 cm
- Height = 11.1 cm? But then from C down to B is 12.1 cm? That doesn’t match.

Wait — maybe it's a trapezoid or compound?

Looking closely: there's a triangle EFC cut out from rectangle ABCD.

Let me re-analyze:

- Rectangle ABCD:
- AB = ? Let's see: from A to B via F and C?
- Actually, from diagram:
- AF = 11.1 cm (vertical)
- FD = 11.1 cm (horizontal)
- DC = 12.1 cm (horizontal)
- CB = 12.1 cm (vertical)
- So AD = AF + FD? No — AD is vertical.

Wait — better interpretation:

- Points:
- A (bottom-left), B (bottom-right), C (top-right), D (top-left)
- F is on top edge? Or inside?

Actually:
- F is a point such that:
- AF = 11.1 cm (vertical)
- FB = 12.1 cm (horizontal)
- But F is connected to D and C?

Wait — labels:
- F is at top-left corner?
- D is top-right?
- Then FD = 11.1 cm (top edge)
- DC = 12.1 cm (right edge)? But then height is 11.1 cm?

Wait — actually, the shape is a rectangle ABCD, where:
- AB = 23.2 cm (bottom)
- AD = 11.1 cm (left)
- DC = 11.1 cm (top)
- CB = 11.1 cm (right)? But labeled as 12.1?

Wait — inconsistency.

Wait — look carefully:
- From D to C: 12.1 cm (horizontal)
- From C to B: 12.1 cm (vertical)
- From B to A: ?
- From A to F: 11.1 cm (vertical)
- From F to D: 11.1 cm (horizontal)

Ah! So:
- Rectangle ABCD has:
- AB = 23.2 cm (bottom)
- AD = 11.1 cm (left)
- DC = 11.1 cm (top)
- But wait, DC is labeled 12.1 cm? Contradiction.

Wait — perhaps F is not a corner.

Let me reinterpret based on standard layout:

This looks like a rectangle ABCD with a right triangle EFC removed from the top right.

But the label says:
- AF = 11.1 cm (vertical)
- FD = 11.1 cm (horizontal)
- DC = 12.1 cm (horizontal)
- CB = 12.1 cm (vertical)
- EB = 11.1 cm (dashed line)
- EF = 11.1 cm (dashed line)
- EC = 12.1 cm (dashed line)

Wait — actually, the triangle is EFC, with:
- EF = 11.1 cm
- FC = 11.1 cm
- EC = 12.1 cm?

No — dashed lines show:
- From F to E: 11.1 cm
- From E to C: 11.1 cm
- And angle at E is right angle?

Wait — the dashed triangle is EFC, with:
- EF = 11.1 cm
- EC = 11.1 cm
- Angle at E is 90°

But then FC would be hypotenuse.

But FC is labeled as 12.1 cm?

Wait — yes: "FC" is labeled as 12.1 cm.

So triangle EFC has:
- EF = 11.1 cm
- EC = 11.1 cm
- FC = 12.1 cm

But that can't be a right triangle unless Pythagoras holds:
$ 11.1^2 + 11.1^2 = 2 \times 123.21 = 246.42 $
$ 12.1^2 = 146.41 $ → Not equal.

So not right triangle.

Wait — the dashed line is from E to F and E to C, and angle at E is marked as right angle.

So EF ⊥ EC, both legs = 11.1 cm.

Then hypotenuse FC = $ \sqrt{11.1^2 + 11.1^2} = 11.1\sqrt{2} \approx 15.7 $ cm — but labeled 12.1 cm? Contradiction.

Wait — perhaps the label 12.1 cm is not FC?

Look: “FC” is labeled as 12.1 cm, but also “DC” is labeled 12.1 cm.

Wait — now I see: D to C is 12.1 cm, and C to B is 12.1 cm, so it’s a square-like top.

But F to D is 11.1 cm, so F is not at D.

Wait — perhaps the shape is:

- Rectangle ABCD
- With a triangle cut out from the top-right corner: triangle EFC
- Where:
- F is on AD, at height 11.1 cm from A
- E is on AB, at distance 11.1 cm from B
- Then triangle EFC is removed

But the labels:
- AF = 11.1 cm → F is on AD
- FD = 11.1 cm → so AD = 22.2 cm?
- But then DC = 12.1 cm → so width = 12.1 cm?
- But AB = ? From A to B: if E is on AB, BE = 11.1 cm, then AE = ?

Wait — let's define points clearly:

Assume:
- A (bottom-left)
- B (bottom-right)
- C (top-right)
- D (top-left)
- F is on AD, AF = 11.1 cm, FD = 11.1 cm → so AD = 22.2 cm
- But DC = 12.1 cm → so rectangle is 22.2 cm high, 12.1 cm wide?

But then AB should be 12.1 cm.

But from diagram, AB has a point E such that BE = 11.1 cm → so AE = 12.1 - 11.1 = 1.0 cm

And triangle EFC is cut out, with:
- EF = 11.1 cm
- EC = 11.1 cm
- Angle at E is 90°

So E is at bottom, F is up on left, C is top-right.

So:
- Coordinates:
- A = (0, 0)
- B = (12.1, 0)
- C = (12.1, 22.2)
- D = (0, 22.2)
- F = (0, 11.1) [since AF = 11.1]
- E = (12.1 - 11.1, 0) = (1.0, 0)? But BE = 11.1 → so E is at x = 12.1 - 11.1 = 1.0, y=0 → (1.0, 0)

Now, triangle EFC has:
- E = (1.0, 0)
- F = (0, 11.1)
- C = (12.1, 22.2)

But then EF = distance between (1.0,0) and (0,11.1) = √[(1)^2 + (11.1)^2] ≈ √(1 + 123.21) = √124.21 ≈ 11.15 → close to 11.1

Similarly, EC = from (1.0,0) to (12.1,22.2): dx=11.1, dy=22.2 → √(11.1² + 22.2²) = √(123.21 + 492.84) = √616.05 ≈ 24.8 — not 11.1

So something is wrong.

Alternatively, perhaps EF = 11.1 cm, EC = 11.1 cm, and angle at E is 90°, so triangle EFC is a right triangle with legs 11.1 cm.

Then FC = √(11.1² + 11.1²) = 11.1√2 ≈ 15.7 cm

But labeled FC = 12.1 cm — contradiction.

Wait — maybe the label 12.1 cm is not FC?

Look at the diagram: the label “12.1 cm” is next to DC, not FC.

Yes! In the image:
- “DC” is labeled 12.1 cm
- “CB” is labeled 12.1 cm
- “AF” = 11.1 cm
- “FD” = 11.1 cm
- “BE” = 11.1 cm
- “EF” = 11.1 cm
- “EC” = 11.1 cm

So:
- Rectangle ABCD:
- AD = AF + FD = 11.1 + 11.1 = 22.2 cm
- DC = 12.1 cm
- So width = 12.1 cm, height = 22.2 cm

- Point E is on AB, BE = 11.1 cm → so AE = AB - BE = 12.1 - 11.1 = 1.0 cm
- Point F is on AD, AF = 11.1 cm
- Triangle EFC is cut out, with EF = 11.1 cm, EC = 11.1 cm, and right angle at E

So E is at (1.0, 0), F is at (0, 11.1), C is at (12.1, 22.2)

But then EF = √[(1.0)^2 + (11.1)^2] = √(1 + 123.21) = √124.21 ≈ 11.15 → close to 11.1, maybe rounding.

EC = √[(12.1 - 1.0)^2 + (22.2 - 0)^2] = √[11.1² + 22.2²] = √[123.21 + 492.84] = √616.05 ≈ 24.8 — not 11.1

So contradiction.

Unless EC = 11.1 cm, but then C cannot be at (12.1, 22.2)

Wait — perhaps C is not the top-right corner?

Wait — the triangle is EFC, and E is on AB, F is on AD, and C is the corner.

But EC = 11.1 cm, and C is at (12.1, 22.2), E at (x,0), so EC = √[(12.1 - x)^2 + 22.2^2] = 11.1

Then:
$$
(12.1 - x)^2 + 492.84 = 123.21 \\
(12.1 - x)^2 = 123.21 - 492.84 = negative → impossible
$$

So impossible.

Thus, my interpretation must be wrong.

Alternative idea: The triangle is not EFC, but EFC is a right triangle with legs 11.1 cm, and hypotenuse 12.1 cm?

Check: 11.1² + 11.1² = 2×123.21 = 246.42
12.1² = 146.41 → not equal.

Or 11.1² + x² = 12.1² → x² = 146.41 - 123.21 = 23.2 → x ≈ 4.82 — not matching.

Perhaps the 12.1 cm is not FC.

Wait — looking back at the diagram:

- “DC” = 12.1 cm
- “CB” = 12.1 cm
- “AF” = 11.1 cm
- “FD” = 11.1 cm
- “BE” = 11.1 cm
- “EF” = 11.1 cm
- “EC” = 11.1 cm
- “FC” is not labeled

But in the triangle, it's EFC, with EF and EC both 11.1 cm, and angle at E is 90°, so FC = √(11.1² + 11.1²) = 11.1√2 ≈ 15.7 cm

But the label “12.1 cm” is near DC and CB, so not FC.

So FC is not labeled.

So we can assume:

- Rectangle ABCD:
- AB = DC = 12.1 cm
- AD = BC = 22.2 cm (since AF + FD = 11.1 + 11.1 = 22.2)
- But then AB = 12.1 cm, so BE = 11.1 cm → AE = 1.0 cm
- E is on AB, 11.1 cm from B
- F is on AD, 11.1 cm from A
- Triangle EFC is removed, with EF = 11.1 cm, EC = 11.1 cm, and right angle at E

But then EC = 11.1 cm, and C is at (12.1, 22.2), E at (1.0, 0), so distance is √(11.1² + 22.2²) = √(123.21 + 492.84) = √616.05 ≈ 24.8 — not 11.1

So impossible.

Unless the rectangle is smaller.

Wait — perhaps AD = 11.1 cm, and FD = 11.1 cm means F is beyond D?

That doesn't make sense.

Another possibility: F is not on AD, but is a separate point.

Wait — the diagram shows:
- A to F: 11.1 cm (vertical)
- F to D: 11.1 cm (horizontal)
- D to C: 12.1 cm (horizontal)
- C to B: 12.1 cm (vertical)
- B to A: bottom
- E on AB, BE = 11.1 cm
- EF = 11.1 cm
- EC = 11.1 cm
- Angle at E is 90°

So perhaps:
- A = (0,0)
- F = (0,11.1)
- D = (11.1,11.1)
- C = (11.1 + 12.1, 11.1) = (23.2, 11.1)
- B = (23.2, 0)
- E = (23.2 - 11.1, 0) = (12.1, 0)

Then:
- EF = from E(12.1,0) to F(0,11.1): dx=12.1, dy=11.1 → dist = √(12.1² + 11.1²) = √(146.41 + 123.21) = √269.62 ≈ 16.42 — not 11.1

Not matching.

I think there's a misinterpretation.

Perhaps the shape is a rectangle with a triangle cut out, and the triangle has legs of 11.1 cm, and the rectangle has dimensions:

- Width: 11.1 + 12.1 = 23.2 cm
- Height: 11.1 cm

But then the triangle is cut from the top.

Given the complexity and time, and since this is a common type, let's assume:

Shape 3: Rectangle of width 23.2 cm, height 11.1 cm, with a right triangle cut out from the top-right corner, with legs 11.1 cm each.

But then the area would be:
- Rectangle: 23.2 × 11.1 = 256.32 cm²
- Triangle: (1/2)(11.1)(11.1) = 61.605 cm²
- Area = 256.32 - 61.605 = 194.715 cm²

Perimeter: original rectangle minus two legs of triangle, plus hypotenuse.

But without clear coordinates, it's hard.

Given the time and complexity, and since you asked for help, let's skip to simpler ones or clarify.

But for now, due to ambiguity in labeling, I recommend double-checking the diagram.

However, based on standard problems, here's a likely interpretation:

Revised Interpretation of Shape 3:

- Rectangle ABCD:
- AB = 23.2 cm (bottom)
- AD = 11.1 cm (left)
- DC = 11.1 cm (top)
- CB = 11.1 cm (right)
- But then DC = 11.1 cm, but labeled 12.1 cm — still conflict.

I think there might be a typo in my understanding.

Given the time, let's move to Shape 4, which is clearer.

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4) Shape 4: Rectangle with Two Triangles Cut Out

Given:
- Outer rectangle: AB = 31.1 ft, AD = 14.1 ft
- Two right triangles cut out:
- Left: AG = 4.1 ft, GC = 3.0 ft
- Right: BF = 4.1 ft, FC = 3.0 ft
- Also, top has a small rectangle?

Wait — from diagram:
- A to G = 4.1 ft
- G to C = 3.0 ft
- Similarly on right: B to F = 4.1 ft, F to C = 3.0 ft
- But C is on top edge?

Wait — the top is from D to F, and D to C is 3.1 ft, C to F is 3.1 ft?

Labels:
- DE = 3.1 ft
- EC = 3.1 ft
- CF = 3.1 ft
- FB = 3.1 ft
- But then top is divided into three parts?

Wait — actually:
- D to E = 3.1 ft
- E to C = 3.1 ft
- C to F = 3.1 ft
- F to B = 3.1 ft
- But then total top = 4 × 3.1 = 12.4 ft, but AB = 31.1 ft — not matching.

Wait — no, AB = 31.1 ft is bottom.

Top: D to F = 31.1 ft

Then:
- D to E = 3.1 ft
- E to C = 3.1 ft
- C to F = 3.1 ft
- But that's only 9.3 ft — not enough.

Wait — perhaps:
- DE = 3.1 ft
- EC = 3.1 ft
- CF = 3.1 ft
- FB = 3.1 ft
- But F is not on top.

Wait — the diagram shows:
- On top: D to E = 3.1 ft
- E to C = 3.1 ft
- C to F = 3.1 ft
- F to B = 3.1 ft
- But B is bottom-right, F is top-right?

No.

Better: the shape is a rectangle with two right triangles cut out from the bottom corners.

- Bottom: AB = 31.1 ft
- Height: AD = 14.1 ft
- At A: triangle AGC with AG = 4.1 ft, GC = 3.0 ft
- At B: triangle BFC with BF = 4.1 ft, FC = 3.0 ft
- But then the top is shortened.

But the top is from D to F, with D to E = 3.1 ft, etc.

This is very messy.

Given the complexity and potential for error in reading labels, and since you're asking for help, I recommend focusing on the first two, which are clear.

For the rest, please provide more clarity or confirm the dimensions.

But for now, here are the answers for 1 and 2:

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✔ Final Answers:

#### 1)
- Area: $ 22.2 \times 11.5 + \frac{1}{4}\pi(12.1)^2 = 255.3 + 114.08 = \boxed{369.4} \text{ in}^2 $
- Perimeter: $ 22.2 + 11.5 + 11.5 + \frac{1}{2}\pi(12.1) = 45.2 + 19.02 = \boxed{64.2} \text{ in} $

#### 2)
- Area: $ 20.2 \times 3.1 + \frac{1}{2}\pi(10.1)^2 = 62.62 + 160.2 = \boxed{222.8} \text{ mm}^2 $
- Perimeter: $ 20.2 + 3.1 + 3.1 + \pi(10.1) = 26.4 + 31.7 = \boxed{58.1} \text{ mm} $

For the remaining, please verify the diagrams or provide additional details.

Would you like me to continue with #3–6 with assumed interpretations?