Math worksheet with six compound shapes requiring area and perimeter calculations.

Let’s solve each compound shape one by one. We’ll find both the area and the perimeter for each.

We’ll break each shape into simpler parts (like rectangles, triangles, semicircles), calculate their areas and perimeters, then combine them carefully — making sure not to double-count any shared sides when finding perimeter.

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Problem 1)

Shape: A rectangle with a quarter-circle on top left.

Dimensions:
- Rectangle: width = 22.2 in, height = 11.5 in
- Quarter-circle: radius = 12.1 in (since from D to F is 12.1 in, and it’s vertical; also E to F is horizontal 10.1 in? Wait — let’s check labels.)

Actually, looking at diagram:

Points: A-B-C-D-E-F-A

From A to B: 22.2 in (bottom)
A to F: 11.5 in (left side)
D to C: 10.1 in (top right part of rectangle)
F to D: 12.1 in (vertical up from D to F? But that would make total height 11.5 + 12.1? That doesn’t match.)

Wait — better interpretation:

The shape has:
- Bottom rectangle: AB = 22.2 in, AF = 11.5 in → so rectangle ABCD? But point D is connected to C which is 10.1 in from D? Hmm.

Actually, looking again:

It seems like:
- The main body is a rectangle from A to B to C to some point, but there's a cutout or addition.

Wait — perhaps it’s composed of:
- A rectangle: 22.2 in wide × 11.5 in high
- Plus a quarter-circle attached on the top-left corner, with radius 12.1 in? But 12.1 > 11.5, so that can't be.

Alternative: Maybe the quarter-circle is inside? No.

Let me re-read the diagram description.

Actually, standard interpretation for such diagrams:

Shape 1: It looks like a rectangle with a quarter-circle “bite” taken out of the top-left? Or added?

Looking at points:

Start at A (bottom left), go right to B (22.2 in), up to C, left to D (10.1 in), then up to F (12.1 in), then curve back to E (which is above A?), then down to A.

Wait — label says:

E is top-left of curve, F is bottom-right of curve? And EF is curved.

Actually, likely:

- From A to B: 22.2 in (base)
- B to C: up 11.5 in? But no, C is connected to D which is 10.1 in left, then D to F is 12.1 in up, then F to E is arc, then E to A is 11.5 in down? That doesn’t add up.

Perhaps:

The shape consists of:
- A rectangle: 22.2 in (width) × 11.5 in (height) — this is the bottom part.
- On top of the left part, there is a quarter-circle extending upward, with radius 12.1 in? But then the total height would be more than 11.5.

Wait — maybe the 11.5 in is the height of the rectangle, and the quarter-circle is sitting on top of it, but only over a portion.

Another way: Let’s assume the figure is made of:

- A rectangle: length 22.2 in, width 11.5 in
- Attached to its top-left corner is a quarter-circle of radius 12.1 in, but since 12.1 > 11.5, that suggests the circle extends beyond the rectangle.

But that might be okay.

However, looking at the dashed lines: from D to C is 10.1 in, and D to F is 12.1 in, and F to E is arc, E to A is 11.5 in.

Perhaps the shape is:

Start at A, go up 11.5 in to E, then arc to F (quarter-circle), then down 12.1 in to D, then right 10.1 in to C, then down to B? But B to A is 22.2 in.

This is confusing without seeing the actual image, but based on common problems, I think:

Shape 1 is a rectangle with dimensions 22.2 in by 11.5 in, and on the top-left, instead of a straight corner, there is a quarter-circle bulging outward with radius 12.1 in. But that would mean the quarter-circle replaces the corner.

Standard problem: often it's a rectangle with a semicircle or quarter-circle added or removed.

Given the labels, let's try this:

Assume the shape has:
- A rectangular part: from A to B to C to D to ... but D is not directly above A.

Perhaps:

Total base AB = 22.2 in

Left side AE = 11.5 in (vertical)

Then from E, there is a quarter-circle arc to F, with center at some point.

If EF is a quarter-circle, and DF = 12.1 in, DC = 10.1 in, then perhaps the quarter-circle has radius 12.1 in, and it's attached to the top of the rectangle.

But then the rectangle's top would be from D to C = 10.1 in, and from A to D horizontally would be 22.2 - 10.1 = 12.1 in, which matches the radius.

Ah! That makes sense.

So:

- The shape is composed of:
- A rectangle on the right: width 10.1 in, height 11.5 in (from D to C to B to some point)
- A square or rectangle on the left: width 12.1 in, height 11.5 in, but with the top-left corner replaced by a quarter-circle of radius 12.1 in.

More precisely:

Imagine a full rectangle 22.2 in wide × 11.5 in high.

On the left part, which is 12.1 in wide, instead of having a flat top, we have a quarter-circle curving upward from the top-left corner.

But if it's curving upward, then the area would include the quarter-circle extra.

In many such problems, the quarter-circle is added externally.

Let's calculate based on that.

So for Shape 1:

Area = area of rectangle + area of quarter-circle

Rectangle: 22.2 in × 11.5 in

Quarter-circle: radius r = 12.1 in, so area = (1/4) * π * r²

Perimeter: we need to trace the outer boundary.

Starting from A, go up to E (11.5 in), then along the arc to F (quarter-circle circumference = (1/4)*2*π*r = (1/2)*π*r), then down to D (12.1 in), then right to C (10.1 in), then down to B (11.5 in), then left to A (22.2 in).

But when we go from F to D, that's vertical down 12.1 in, but D is at the same level as C, and C is at height 11.5 in from bottom? This is inconsistent.

Perhaps the 11.5 in is the height from A to the line DC, and the quarter-circle is above that.

Let's define coordinates to clarify.

Set point A at (0,0)

Then B at (22.2, 0)

C at (22.2, h) but what is h? Not given directly.

From the diagram, likely:

- From A to F is not direct; rather, from A up to some point, but label says "11.5 in" next to A to F? In the text: "11.5 in" is written near A to F, and "12.1 in" near D to F, "10.1 in" near D to C, "22.2 in" near A to B.

Also, there is a right angle at D and at B.

So probably:

- Points: A(0,0), B(22.2,0), C(22.2, y), D(x,y), F(x, y+12.1), E(0, y+12.1) or something.

Assume that the rectangle part is from y=0 to y=11.5, so height 11.5 in.

Then D is at (a, 11.5), C at (22.2, 11.5), and DC = 10.1 in, so a = 22.2 - 10.1 = 12.1 in.

So D is at (12.1, 11.5)

Then F is at (12.1, 11.5 + 12.1) = (12.1, 23.6)? But then E is at (0, 23.6), and arc from E to F is quarter-circle with center at (0,11.5) or (12.1,11.5)?

If arc from E to F is quarter-circle, and E is at (0, b), F at (12.1, b), then it would be a semicircle, but it's labeled as quarter-circle.

Typically, if it's a quarter-circle bulging up, center at D(12.1,11.5), then F would be at (12.1,11.5+12.1)=(12.1,23.6), and E at (12.1-12.1,11.5)=(0,11.5), but then E is at (0,11.5), which is the top-left of the rectangle.

Then the arc from E(0,11.5) to F(12.1,23.6) is not a quarter-circle; it would be a quarter-circle only if it's 90 degrees.

From (0,11.5) to (12.1,23.6) with center at (0,23.6) or (12.1,11.5)? Let's calculate distance.

If center at D(12.1,11.5), then to E(0,11.5): distance = 12.1 in (horizontal)

To F(12.1,23.6): distance = 12.1 in (vertical)

And the angle at D between DE and DF is 90 degrees, so yes, arc from E to F is a quarter-circle with center D, radius 12.1 in.

Perfect.

So the shape is:

- From A(0,0) to B(22.2,0) to C(22.2,11.5) to D(12.1,11.5) to F(12.1,23.6) via arc? No, from D to F is straight up, then arc from F to E? Let's see the path.

The boundary is: start at A(0,0), go to B(22.2,0), to C(22.2,11.5), to D(12.1,11.5), then up to F(12.1,23.6), then arc to E(0,23.6)? But E should be at (0,11.5) if center is D.

I think I have it backward.

If center is at D(12.1,11.5), and radius 12.1, then:

- Point E: if it's west of D, at (12.1 - 12.1, 11.5) = (0,11.5)

- Point F: north of D, at (12.1, 11.5 + 12.1) = (12.1,23.6)

Then the arc from E to F is a quarter-circle in the northwest direction.

But in the shape, from D, we go to F (up), then arc to E, then down to A.

From E(0,11.5) to A(0,0) is down 11.5 in.

Yes.

So the shape has vertices: A(0,0), B(22.2,0), C(22.2,11.5), D(12.1,11.5), F(12.1,23.6), then arc to E(0,11.5), then down to A(0,0).

But E is at (0,11.5), which is the same y as D and C.

Now, for area:

The shape consists of:
- Rectangle from x=0 to 22.2, y=0 to 11.5, but wait, from x=0 to 12.1, y=0 to 11.5 is included, but above that, from y=11.5 to 23.6, only the quarter-circle is added.

Actually, the area is:
- The rectangle from (0,0) to (22.2,11.5) MINUS the rectangle from (0,11.5) to (12.1,11.5) but that's a line.

Better: the shape includes:
- The large rectangle: 22.2 in wide × 11.5 in high (from y=0 to y=11.5)
- Plus the quarter-circle above it, centered at D(12.1,11.5), radius 12.1 in, which is in the region x<12.1, y>11.5.

Since the quarter-circle is outside the rectangle, we add its area.

So area = area_rectangle + area_quarter_circle

Area_rectangle = 22.2 * 11.5

Calculate that: 22.2 * 11.5 = 22.2 * 10 + 22.2 * 1.5 = 222 + 33.3 = 255.3 in²

Area_quarter_circle = (1/4) * π * r² = (1/4) * π * (12.1)^2

First, 12.1^2 = 146.41

So (1/4)*π*146.41 = (146.41/4)*π = 36.6025 * π

Use π ≈ 3.1416, so 36.6025 * 3.1416 ≈ let's compute:

36.6025 * 3 = 109.8075

36.6025 * 0.1416 ≈ 36.6025 * 0.14 = 5.12435, and 36.6025 * 0.0016 = 0.058564, so total approx 5.182914

So total area_quarter_circle ≈ 109.8075 + 5.1829 = 114.9904 in²

So total area ≈ 255.3 + 114.99 = 370.29 in²

But let's keep more precision or use exact.

Since the problem likely expects numerical answer, we'll use π=3.14 for simplicity, as common in such worksheets.

Check the other problems; they have decimals, so probably expect decimal answers.

In problem 2, diameter 20.2 mm, so likely use π=3.14.

So for consistency, use π = 3.14

So area_quarter_circle = (1/4) * 3.14 * (12.1)^2 = (0.25) * 3.14 * 146.41

First, 3.14 * 146.41 = 3.14 * 146 = 3.14*100=314, 3.14*46=144.44, total 458.44, then 3.14*0.41=1.2874, so total 459.7274

Then *0.25 = 114.93185 in²

Area_rectangle = 22.2 * 11.5

22.2 * 11.5 = 22.2 * (10 + 1.5) = 222 + 33.3 = 255.3 in²

Total area = 255.3 + 114.93185 = 370.23185 ≈ 370.23 in²

But let's confirm if the quarter-circle is added or if it's replacing something.

In this case, since the arc is bulging out, it should be added.

Now perimeter:

Trace the boundary:

Start at A(0,0) -> B(22.2,0): length 22.2 in

B(22.2,0) -> C(22.2,11.5): length 11.5 in

C(22.2,11.5) -> D(12.1,11.5): length |22.2 - 12.1| = 10.1 in

D(12.1,11.5) -> F(12.1,23.6): length 12.1 in (up)

F(12.1,23.6) -> E(0,11.5) via arc: this is a quarter-circle of radius 12.1 in, so arc length = (1/4) * 2 * π * r = (1/2) * π * r = 0.5 * 3.14 * 12.1

Compute: 0.5 * 3.14 = 1.57, *12.1 = 1.57*12 = 18.84, 1.57*0.1=0.157, total 18.997 in

Then E(0,11.5) -> A(0,0): length 11.5 in (down)

So total perimeter = 22.2 + 11.5 + 10.1 + 12.1 + 18.997 + 11.5

Add step by step:

22.2 + 11.5 = 33.7

33.7 + 10.1 = 43.8

43.8 + 12.1 = 55.9

55.9 + 18.997 = 74.897

74.897 + 11.5 = 86.397 in ≈ 86.40 in

But is E to A included? Yes, and it's straight down.

In the shape, from E to A is vertical, yes.

So for Shape 1:

Area ≈ 370.23 in²

Perimeter ≈ 86.40 in

But let's write with two decimals as inputs have one or two.

Inputs have one decimal mostly, so perhaps round to one decimal.

22.2, 11.5, 12.1, 10.1 — all one decimal, so answers should be to one decimal place.

So area = 370.2 in²? 370.23 is closer to 370.2, but let's calculate exactly.

With π=3.14:

Area_quarter_circle = 0.25 * 3.14 * 146.41 = first 3.14 * 146.41

3.14 * 146 = 3.14*100=314, 3.14*46=144.44, sum 458.44

3.14*0.41=1.2874, so 458.44 + 1.2874 = 459.7274

Then /4 = 114.93185

Area_rect = 22.2 * 11.5 = let's calculate: 222/10 * 115/10 = (222*115)/100

222*100=22200, 222*15=3330, total 25530, /100 = 255.3

Sum 255.3 + 114.93185 = 370.23185 ≈ 370.2 in² (to one decimal)

Perimeter: arc length = 0.5 * 3.14 * 12.1 = 1.57 * 12.1

1.57*12 = 18.84, 1.57*0.1=0.157, sum 18.997 ≈ 19.0 in (to one decimal)

Then sides: 22.2 + 11.5 + 10.1 + 12.1 + 19.0 + 11.5

Add: 22.2+11.5=33.7

33.7+10.1=43.8

43.8+12.1=55.9

55.9+19.0=74.9

74.9+11.5=86.4 in

So Shape 1: Area = 370.2 in², Perimeter = 86.4 in

But is the quarter-circle really added? In some interpretations, it might be that the shape is the rectangle minus the quarter-circle, but in this case, since the arc is convex outward, it should be added.

Moreover, in the diagram, it's likely shown as bulging out.

So I'll go with that.

Now Problem 2)

Shape: a rectangle with a semicircle on top.

Dimensions: rectangle width 20.2 mm, height 3.1 mm, and semicircle on top with diameter 20.2 mm.

Labels: A-B-C-D, with AB= ? , BC=3.1 mm, CD=20.2 mm, DA= ? , and semicircle on CD.

From diagram: points A,B,C,D, with D to C is 20.2 mm (dashed, so diameter), A to B is bottom, B to C is 3.1 mm up, A to D is up, and semicircle on top from D to C.

So it's a rectangle ABCD with AB parallel to DC, AD and BC perpendicular.

AD = BC = 3.1 mm, DC = 20.2 mm, so AB = 20.2 mm.

Semicircle on top of DC, so bulging upward.

Area = area_rectangle + area_semicircle

Area_rectangle = 20.2 * 3.1

20.2 * 3 = 60.6, 20.2 * 0.1 = 2.02, total 62.62 mm²

Area_semicircle = (1/2) * π * r², r = diameter/2 = 20.2 / 2 = 10.1 mm

So (1/2) * 3.14 * (10.1)^2 = 0.5 * 3.14 * 102.01

First, 3.14 * 102.01 = 3.14*100=314, 3.14*2.01=3.14*2=6.28, 3.14*0.01=0.0314, so 6.3114, total 314+6.3114=320.3114

Then *0.5 = 160.1557 mm²

Total area = 62.62 + 160.1557 = 222.7757 ≈ 222.8 mm² (to one decimal)

Perimeter: the outer boundary.

Start at A, to B: 20.2 mm

B to C: 3.1 mm

C to D via semicircle: half circumference = (1/2)*2*π*r = π*r = 3.14 * 10.1 = 31.714 mm

D to A: 3.1 mm

Note: the diameter DC is not part of the perimeter because it's internal; the semicircle replaces it.

So perimeter = AB + BC + arc_CD + DA = 20.2 + 3.1 + 31.714 + 3.1

Add: 20.2 + 3.1 = 23.3

23.3 + 31.714 = 55.014

55.014 + 3.1 = 58.114 ≈ 58.1 mm

So Shape 2: Area = 222.8 mm², Perimeter = 58.1 mm

Problem 3)

Shape: a rectangle with a triangle attached on the right.

Dimensions: rectangle AFDE? Points A,F,D,E,C,B

From diagram: A to F: 11.1 cm (left side), F to D: 11.1 cm (top), D to C: 12.1 cm? Label says "12.1 cm" near D to C, but also "11.6 cm" and "12.1 cm" for the triangle.

Specifically: rectangle part: A to F: 11.1 cm, F to D: 11.1 cm, so square? Then D to C is diagonal? No.

Labels: "11.1 cm" for AF and FD, so rectangle AFDB or something.

Points: A, F, D, then C, B.

From A to B: bottom, B to R? Label has R, but probably typo, should be B.

In diagram: A to B is bottom, length not given directly.

From F to D: 11.1 cm (top of rectangle)

A to F: 11.1 cm (left side)

Then from D to C: 12.1 cm, but also there is a point E, and "11.6 cm" from E to C, and "12.1 cm" from D to E?

Label says: "12.1 cm" near D to C, "11.6 cm" near E to C, "12.1 cm" near D to E? And right angle at E.

So likely, triangle DEC is attached to the rectangle.

Rectangle is AFDB, with AF=11.1 cm, FD=11.1 cm, so it's a square 11.1 cm x 11.1 cm.

Then from D, there is a triangle DEC, with DE=12.1 cm, EC=11.6 cm, DC=12.1 cm? But 12.1, 11.6, 12.1 — isosceles?

DC is labeled 12.1 cm, DE=12.1 cm, EC=11.6 cm, and right angle at E.

So triangle DEC is right-angled at E, with legs DE and EC? But DE=12.1, EC=11.6, then hypotenuse DC should be sqrt(12.1^2 + 11.6^2) = sqrt(146.41 + 134.56) = sqrt(280.97) ≈ 16.76, but labeled as 12.1, contradiction.

Perhaps DC is not the hypotenuse.

Label says: "12.1 cm" for D to C, and "12.1 cm" for D to E, and "11.6 cm" for E to C, and right angle at E.

So in triangle DEC, DE=12.1, EC=11.6, angle at E is 90 degrees, so DC should be hypotenuse = sqrt(12.1^2 + 11.6^2) = as above ~16.76, but it's labeled 12.1, which is impossible.

Unless the 12.1 cm for D to C is a mistake, or perhaps it's not the side.

Looking back: in the text: "12.1 cm" is written twice: once for D to C, once for D to E? In the user input: "12.1 cm" for D to C, and "12.1 cm" for the other, but in diagram description, it might be different.

User said: "3) F 11.1 cm D 12.1 cm C 11.1 cm E 11.6 cm B 12.1 cm A" and "right angle at E and at B"

Perhaps the rectangle is A-F-D-B, with AF=11.1, FD=11.1, so AB=11.1, FB=11.1? No.

Standard: likely rectangle is A-B-R-D-F or something.

Assume points: A bottom left, B bottom right, R is probably B, D top right of rectangle, F top left.

So rectangle A-B-D-F, with AB = ? , BD = 11.1 cm? Label says "11.1 cm" for A to F and F to D, so if A to F is left side, F to D is top, then A to B should be bottom, B to D right side.

But length of AB not given, but from F to D is 11.1 cm, so if it's rectangle, AB = FD = 11.1 cm, and AF = BD = 11.1 cm, so it's a square.

Then from D, there is a triangle D-C-E, with C being the new vertex.

Label: "12.1 cm" for D to C, "11.6 cm" for E to C, "12.1 cm" for D to E, and right angle at E.

So in triangle DEC, DE = 12.1 cm, EC = 11.6 cm, DC = 12.1 cm, and angle at E is 90 degrees.

But if angle at E is 90 degrees, then by Pythagoras, DC^2 = DE^2 + EC^2 = 12.1^2 + 11.6^2 = 146.41 + 134.56 = 280.97, so DC = sqrt(280.97) ≈ 16.76 cm, but it's labeled 12.1 cm, which is inconsistent.

Unless the 12.1 cm for D to C is not the side, but perhaps it's the length from D to C along the shape, but that doesn't make sense.

Perhaps "12.1 cm" for D to C is a typo, and it's the hypotenuse, but labeled as 12.1, while calculation shows 16.76.

Another possibility: the right angle is at E, but DE and EC are not the legs; perhaps DC is a leg.

Let's read the user input: "3) F 11.1 cm D 12.1 cm C 11.1 cm E 11.6 cm B 12.1 cm A" and "right angle at E and at B"

Perhaps the 12.1 cm for D to C is correct, and the triangle is not right-angled at E for those sides.

But it says "right angle at E", so likely at E, the angle is 90 degrees for the triangle.

Perhaps the sides are: from D to E is 12.1 cm, E to C is 11.6 cm, and D to C is the hypotenuse, but labeled as 12.1, which is wrong.

Or perhaps the 12.1 cm for D to C is for something else.

Another idea: perhaps "12.1 cm" is the length from D to C, but in the triangle, it's not the side; but that doesn't make sense.

Let's look at the numbers: 12.1, 11.6, 12.1 — perhaps it's isosceles with DE = DC = 12.1 cm, and EC = 11.6 cm, and right angle at E, but then it can't be right-angled at E if DE = DC.

If right-angled at E, then the two legs are DE and EC, hypotenuse DC.

So DC must be larger than both, but 12.1 < 12.1? 12.1 is not greater than 12.1.

Unless DE is not 12.1.

In the user input: "D 12.1 cm C" and "E 11.6 cm" and "B 12.1 cm A" — perhaps "B 12.1 cm A" means BA = 12.1 cm, but earlier F to D is 11.1 cm, so if rectangle, BA should be 11.1 cm.

I think there might be a mislabeling in my assumption.

Perhaps the rectangle is not square.

Let me assume that the rectangle has width AB = x, height AF = 11.1 cm, and FD = 11.1 cm, but FD is top, so if FD = 11.1 cm, and it's rectangle, then AB = 11.1 cm, so still square.

Unless FD is not the top side.

Another interpretation: points A, B, C, D, E, F.

From A to B: bottom

B to C: right side, but label has "B 12.1 cm A" — perhaps BA = 12.1 cm.

In user input: "3) F 11.1 cm D 12.1 cm C 11.1 cm E 11.6 cm B 12.1 cm A" and "right angle at E and at B"

Perhaps the lengths are: AF = 11.1 cm, FD = 12.1 cm? But it says "F 11.1 cm D" , so F to D is 11.1 cm.

Let's list the segments mentioned:

- F to D: 11.1 cm

- D to C: 12.1 cm

- C to E: 11.6 cm? "C 11.1 cm E" — perhaps C to E is 11.1 cm, but then "E 11.6 cm B" — E to B is 11.6 cm, and "B 12.1 cm A" — B to A is 12.1 cm.

And right angle at E and at B.

Also, likely rectangle involved.

Perhaps the shape is: rectangle A-B-C-D or something.

Assume that A-B is bottom, length 12.1 cm (from "B 12.1 cm A")

Then B to C: but C is connected to D and E.

From the right angle at B, and at E, likely B is corner of rectangle, E is corner of triangle.

Common configuration: rectangle on left, triangle on right.

So let's say rectangle A-B-G-F, with A bottom left, B bottom right, G top right, F top left.

Then from G, triangle G-C-E or something.

But in labels, we have D and C.

Perhaps D is G.

So