Similar Polygons Worksheets - Math Monks - Free Printable

Let's solve each problem on the "Similar Polygons and Scale Factor Worksheet" step by step.

We are asked to find the scale factor from the smaller figure to the larger figure. The scale factor is the ratio of corresponding sides of the larger figure to the smaller one.

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🔷 Problem 1

- Smaller triangle: sides = 9, 18, 21
- Larger triangle: sides = 15, 30, 35

Compare corresponding sides:
- 15 / 9 = 5/3 ≈ 1.67
- 30 / 18 = 5/3 ≈ 1.67
- 35 / 21 = 5/3 ≈ 1.67

✔ All ratios are equal → Scale factor = 5/3

> ✔ Answer: 5/3

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🔷 Problem 2

- Smaller trapezoid: sides = 19, 25, 40
- Larger trapezoid: sides = 22.8, 30, 48

Compare:
- 22.8 / 19 = 1.2
- 30 / 25 = 1.2
- 48 / 40 = 1.2

✔ All ratios = 1.2 → Scale factor = 1.2 or 6/5

> ✔ Answer: 6/5 or 1.2

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🔷 Problem 3

- Smaller rectangle: 54 × 36
- Larger rectangle: 63 × 42

Compare:
- 63 / 54 = 7/6 ≈ 1.166...
- 42 / 36 = 7/6

✔ Same ratio → Scale factor = 7/6

> ✔ Answer: 7/6

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🔷 Problem 4

- Smaller square: side = 414 (top), 388.7 (side)
- Larger square: side = 405.6 (top), 432 (side)

Wait — this looks like a rectangle, not a square.

Smaller: 414 × 388.7
Larger: 405.6 × 432

But wait: let’s check if they're similar.

Compare:
- 405.6 / 414 = ? → 405.6 ÷ 414 = 0.979... → less than 1
- But 432 / 388.7 ≈ 1.111...

Wait — that doesn't match. Is the larger one actually bigger?

Wait: 405.6 vs 414 → smaller? But the problem says "scale factor from smaller to larger".

So maybe I have it backward?

Let’s double-check:

- Left figure: 414 (top), 388.7 (side)
- Right figure: 405.6 (top), 432 (side)

So:
- Top: 405.6 < 414 → smaller
- Side: 432 > 388.7 → larger

Not consistent? Wait — unless the figures are oriented differently.

But both are rectangles.

Let’s assume the larger figure is the one with greater dimensions, so we need to compare corresponding sides.

But look at the labels:
- Left: 414 and 388.7
- Right: 405.6 and 432

So:
- 414 vs 405.6 → left is larger in one dimension
- 388.7 vs 432 → right is larger in other

But for similarity, ratios must be proportional.

Let’s compute ratios:

Try:
- 405.6 / 414 = 0.9797
- 432 / 388.7 ≈ 1.111

Not equal.

Wait — maybe the left is smaller? But no.

Wait — perhaps the right is the larger one?

But 405.6 < 414 → top side is shorter

Unless the corresponding sides are paired correctly.

Let’s suppose:
- 414 corresponds to 405.6 → ratio = 405.6 / 414 = 0.9797
- 388.7 corresponds to 432 → 432 / 388.7 ≈ 1.111

Still not matching.

Wait — maybe the left is the larger one?

Then scale factor from smaller to larger would be >1.

But 414 > 405.6, 388.7 < 432 → inconsistent.

Wait — could there be a typo?

Wait — look again: the right rectangle has 405.6 and 432, while the left has 414 and 388.7

Let’s try comparing longest sides:

- Left: max = 414
- Right: max = 432 → so right is larger

Now, check if they are similar:

Left: 414 × 388.7
Right: 405.6 × 432

Wait — but 405.6 < 414 → so if 414 corresponds to 405.6, then scaling down.

But 432 > 388.7 → scaling up.

So unless sides are swapped.

Maybe the sides are labeled differently.

Let’s suppose the heights are corresponding:

- Left height = 388.7
- Right height = 432

Ratio: 432 / 388.7 ≈ 1.111

Widths:
- Left width = 414
- Right width = 405.6

405.6 / 414 ≈ 0.9797 ≠ 1.111

No match.

Wait — unless the labels are switched?

Wait — perhaps the left figure is not the smaller one?

But the problem says: “Find the scale factor from the smaller figure to the larger figure.”

Let’s suppose the right figure is the larger one.

But its width is 405.6 < 414 → so shorter.

Unless the 405.6 is the height?

Wait — the label "405.6" is on the top, and "432" on the side.

Similarly, left: "414" on top, "388.7" on side.

So:
- Left: width = 414, height = 388.7
- Right: width = 405.6, height = 432

Now check ratios:
- Width: 405.6 / 414 ≈ 0.9797
- Height: 432 / 388.7 ≈ 1.111

Not equal → not similar?

But the worksheet says they are similar.

Wait — perhaps I misread.

Wait — maybe the left rectangle is smaller, but its sides are 414 and 388.7, and the right is 405.6 and 432.

But 405.6 < 414 → so width is smaller, but 432 > 388.7 → height is larger.

So unless the orientation is different.

Alternatively, maybe the corresponding sides are:
- 414 ↔ 405.6
- 388.7 ↔ 432

But ratios don’t match.

Wait — what if we reverse?

Try: 414 / 405.6 ≈ 1.0207
388.7 / 432 ≈ 0.900 → not same.

Wait — maybe the right figure is scaled up from left?

But 405.6 < 414 → so width decreased.

This suggests a mistake in labeling or assumption.

Wait — perhaps the right rectangle is the larger one, but its width is 405.6, and height is 432.

And left: width 414, height 388.7

But 405.6 < 414 → width smaller

So only if height increases, but width decreases → not possible for similar figures unless aspect ratio is preserved.

Check aspect ratios:

Left: 414 / 388.7 ≈ 1.064
Right: 405.6 / 432 ≈ 0.939 → not equal

Wait — that can’t be.

Wait — maybe I have the numbers wrong.

Look again:

Problem 4:
- Left rectangle: top = 414, side = 388.7
- Right rectangle: top = 405.6, side = 432

But 405.6 < 414 → so top is smaller

432 > 388.7 → side is larger

Aspect ratio:
- Left: 414 / 388.7 ≈ 1.064
- Right: 405.6 / 432 ≈ 0.939 → not same

But the worksheet says they are similar → contradiction?

Wait — unless the sides are not aligned.

Wait — maybe the side of the right rectangle is not 432?

Wait — the number 432 is on the right side, and 405.6 on the top.

But the left rectangle has 414 on top, 388.7 on side.

So if we assume:
- Top sides correspond: 414 ↔ 405.6 → ratio = 405.6 / 414 = 0.9797
- Side sides: 388.7 ↔ 432 → 432 / 388.7 ≈ 1.111

No.

Wait — unless the right rectangle is the larger one, and the top side is 405.6, but maybe the length is 405.6, and width is 432?

But then it’s wider than tall.

Left: 414 (long), 388.7 (short) → longer

Right: 405.6 (short), 432 (long) → taller

So orientation changed.

But for similarity, shape must be same.

So check:

Left: length / width = 414 / 388.7 ≈ 1.064
Right: 432 / 405.6 ≈ 1.065 → very close!

Ah! So the right rectangle has height = 432, width = 405.6

So aspect ratio: 432 / 405.6 ≈ 1.065
Left: 414 / 388.7 ≈ 1.064 → almost equal

But not exactly.

Wait — let’s compute more precisely.

414 / 388.7 = ?

388.7 × 1.064 = 388.7 × 1 + 388.7 × 0.064 = 388.7 + 24.8848 = 413.5848 ≈ 414 → yes

432 / 405.6 = ?

405.6 × 1.065 = 405.6 × 1 + 405.6 × 0.065 = 405.6 + 26.404 = 432.004 → yes!

So both have aspect ratio ≈ 1.064–1.065

So they are similar.

Now, which is larger?

Left: area = 414 × 388.7 ≈ 160,500
Right: 405.6 × 432 ≈ 175,300 → larger

So right is larger.

Now, find scale factor from smaller to larger.

So we need to find ratio of corresponding sides.

But how to pair?

Since both are rectangles, and assuming orientation is preserved, we should pair:

- Left width = 414 ↔ Right width = 405.6? No — 405.6 < 414

But earlier we saw that the right rectangle has height = 432, width = 405.6

But left has width = 414, height = 388.7

So likely:
- Left width = 414 ↔ Right width = 405.6 → ratio = 405.6 / 414 = 0.9797
- Left height = 388.7 ↔ Right height = 432 → ratio = 432 / 388.7 ≈ 1.111

Still not same.

Wait — unless the corresponding sides are:
- 414 ↔ 432
- 388.7 ↔ 405.6

Try:
- 432 / 414 = 1.0435
- 405.6 / 388.7 ≈ 1.0435 → YES!

Oh! So:
- 414 ↔ 432 (ratio = 432 / 414 = 1.0435)
- 388.7 ↔ 405.6 → 405.6 / 388.7 ≈ 1.0435

Yes! So the scale factor from smaller to larger is:

432 / 414 = 432 ÷ 414 = simplify:

Divide numerator and denominator by 6:
- 432 ÷ 6 = 72
- 414 ÷ 6 = 69
→ 72/69 = 24/23

Or 432 / 414 = 24/23

✔ Scale factor = 24/23

> ✔ Answer: 24/23

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🔷 Problem 5

Two rectangles:
- Large rectangle: AB = 45 (height), OP = 40 (width)
- Small rectangle inside: MN = 30 (height), OP = 40 (width)? Wait — no.

Wait: Figure shows:
- Outer rectangle: A-B (top), O-P (bottom), width = 40, height = 45
- Inner rectangle: M-N, height = 30, width = ?

Wait — the small rectangle has:
- Height = 30 (from M to N)
- Width = ? But it's drawn inside, and seems to have same width as outer? But not labeled.

But the small rectangle has:
- Vertical side = 30 (labeled)
- Horizontal side: from M to N — but not labeled.

Wait — the horizontal side of small rectangle is not labeled.

But notice: the small rectangle has same width as large one? Because it goes from left to right?

No — it's centered.

But the width of small rectangle is not given.

But the height is 30, and large is 45.

So vertical sides: 30 vs 45 → ratio = 30/45 = 2/3

But since the small rectangle is inside and likely similar, and shares same width?

Wait — but the horizontal side is not labeled.

But the width of the large rectangle is 40.

The small rectangle appears to have same width? But not labeled.

Wait — the arrow from O to P is 40, and the small rectangle starts at some point, ends at another.

But no length is given.

Wait — perhaps the small rectangle has height = 30, and width = ?

But the problem says the polygons are similar.

But we only have one side of small rectangle: height = 30

Large rectangle: height = 45, width = 40

If they are similar, then:

Scale factor = 30 / 45 = 2/3

So width of small rectangle = 40 × (2/3) = 80/3 ≈ 26.67

But we don’t need to find that.

The question is: scale factor from smaller to larger

So smaller → larger = 45 / 30 = 3/2

✔ Answer: 3/2

> ✔ Answer: 3/2

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🔷 Problem 6

Two right triangles:
- Larger: legs = 12 and 16, hypotenuse = 20
- Smaller: legs = 6 and 8, hypotenuse = 10

Check:
- 12 / 6 = 2
- 16 / 8 = 2
- 20 / 10 = 2

So scale factor from smaller to larger = 2

✔ Answer: 2

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🔷 Problem 7

Two rectangles:
- Larger: 15 × 10
- Smaller: 6 × 4

Check ratios:
- 15 / 6 = 2.5
- 10 / 4 = 2.5

Same → scale factor = 2.5 = 5/2

✔ Answer: 5/2

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🔷 Problem 8

Two trapezoids:
- Smaller: sides = 4, 6, 8, 6
- Larger: sides = 12, 9, 6, 9

Wait — label:
- Smaller: top = 4, legs = 6, bottom = 8
- Larger: top = 12, legs = 9, bottom = 6

Wait — bottom of larger is 6? But smaller bottom is 8 → so smaller?

Wait — but larger has top = 12 > 4

So probably:
- Smaller: top = 4, bottom = 8, legs = 6
- Larger: top = 12, bottom = 6, legs = 9

But bottom of larger is 6 < 8 → so smaller?

Wait — inconsistency.

Wait — the larger trapezoid has:
- Top = 12
- Bottom = 6
- Legs = 9

But bottom is shorter than top → upside-down?

But smaller has:
- Top = 4
- Bottom = 8
- Legs = 6

So:
- Top: 12 / 4 = 3
- Bottom: 6 / 8 = 0.75 → not same

Wait — unless bottom of larger is 12?

No — label says bottom = 6

Wait — look carefully:

In the larger trapezoid:
- Top: 12
- Bottom: 6
- Legs: 9 and 9

But bottom = 6, top = 12 → so top is longer

In smaller:
- Top: 4
- Bottom: 8 → bottom is longer

So orientation is different.

But for similarity, shapes must be same.

Check ratios:

Try pairing:
- Smaller top = 4 ↔ Larger top = 12 → 12/4 = 3
- Smaller bottom = 8 ↔ Larger bottom = 6 → 6/8 = 0.75 → not 3

No.

Try:
- Smaller top = 4 ↔ Larger bottom = 6 → 6/4 = 1.5
- Smaller bottom = 8 ↔ Larger top = 12 → 12/8 = 1.5
- Legs: 6 ↔ 9 → 9/6 = 1.5

Yes! All ratios = 1.5

So the smaller trapezoid has:
- Top = 4
- Bottom = 8
- Legs = 6

The larger trapezoid has:
- Top = 12 (which corresponds to smaller's bottom = 8)
- Bottom = 6 (which corresponds to smaller's top = 4)
- Legs = 9

So it's rotated or flipped.

But since the shapes are similar, and ratios are all 1.5, scale factor from smaller to larger is:

1.5 = 3/2

✔ Answer: 3/2

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✔ Final Answers:

| Problem | Scale Factor |
|--------|--------------|
| 1 | 5/3 |
| 2 | 6/5 or 1.2 |
| 3 | 7/6 |
| 4 | 24/23 |
| 5 | 3/2 |
| 6 | 2 |
| 7 | 5/2 |
| 8 | 3/2 |

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📌 Summary of Solutions:

1. 5/3
2. 6/5
3. 7/6
4. 24/23
5. 3/2
6. 2
7. 5/2
8. 3/2

All scale factors are from the smaller figure to the larger figure.

Let me know if you'd like this formatted as a printable answer sheet!