Volume of a cylinder - Variation Theory - Free Printable

It looks like you've uploaded an image showing eight cylinders, each with labeled dimensions (radius or diameter and height). However, the task isn't explicitly stated in your message. Since no specific question is given (e.g., "find the volume", "compare surface areas", etc.), I'll assume a common type of problem associated with such images: calculating the volume or surface area of each cylinder.

Let’s go through each cylinder, determine its volume, and then summarize.

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🔷 Formula for Volume of a Cylinder:

$$
V = \pi r^2 h
$$
Where:
- $ r $ = radius of the base
- $ h $ = height
- $ \pi \approx 3.14 $

We’ll use $ \pi \approx 3.14 $ unless specified otherwise.

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Let’s label the cylinders from left to right, top to bottom:

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🔹 Cylinder 1:

- Radius $ r = 4 $ cm
- Height $ h = 10 $ cm
$$
V = \pi (4)^2 (10) = \pi \cdot 16 \cdot 10 = 160\pi \approx 502.4 \text{ cm}^3
$$

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🔹 Cylinder 2:

- Radius $ r = 4 $ cm
- Height $ h = 5 $ cm
$$
V = \pi (4)^2 (5) = \pi \cdot 16 \cdot 5 = 80\pi \approx 251.2 \text{ cm}^3
$$

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🔹 Cylinder 3:

- Radius $ r = 4 $ cm
- Height $ h = 10 $ cm
→ Same as Cylinder 1
$$
V = 160\pi \approx 502.4 \text{ cm}^3
$$

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🔹 Cylinder 4:

- Radius $ r = 40 $ cm
- Height $ h = 10 $ cm
$$
V = \pi (40)^2 (10) = \pi \cdot 1600 \cdot 10 = 16,000\pi \approx 50,240 \text{ cm}^3
$$

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🔹 Cylinder 5:

- Radius $ r = 2 $ cm
- Height $ h = 5 $ cm
$$
V = \pi (2)^2 (5) = \pi \cdot 4 \cdot 5 = 20\pi \approx 62.8 \text{ cm}^3
$$

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🔹 Cylinder 6:

- This one is drawn sideways.
- Diameter = 5 cm → Radius $ r = 2.5 $ cm
- Height = 4 m → But wait! Units differ: cm vs m.

⚠️ Unit inconsistency: The diameter is 5 cm, but height is 4 meters = 400 cm.

So convert height to cm: $ h = 400 $ cm

$$
V = \pi (2.5)^2 (400) = \pi \cdot 6.25 \cdot 400 = 2500\pi \approx 7,850 \text{ cm}^3
$$

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🔹 Cylinder 7:

- Radius $ r = 0.4 $ cm
- Height $ h = 10 $ cm
$$
V = \pi (0.4)^2 (10) = \pi \cdot 0.16 \cdot 10 = 1.6\pi \approx 5.024 \text{ cm}^3
$$

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🔹 Cylinder 8:

- This is also sideways.
- Diameter = 5 cm → Radius $ r = 2.5 $ cm
- Height = 2 m = 200 cm

$$
V = \pi (2.5)^2 (200) = \pi \cdot 6.25 \cdot 200 = 1250\pi \approx 3,925 \text{ cm}^3
$$

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## ✔ Summary Table: Volumes

| Cylinder | Radius | Height | Volume (cm³) |
|--------|--------|--------|--------------|
| 1 | 4 cm | 10 cm | $160\pi$ ≈ 502.4 |
| 2 | 4 cm | 5 cm | $80\pi$ ≈ 251.2 |
| 3 | 4 cm | 10 cm | $160\pi$ ≈ 502.4 |
| 4 | 40 cm | 10 cm | $16,000\pi$ ≈ 50,240 |
| 5 | 2 cm | 5 cm | $20\pi$ ≈ 62.8 |
| 6 | 2.5 cm | 400 cm | $2500\pi$ ≈ 7,850 |
| 7 | 0.4 cm | 10 cm | $1.6\pi$ ≈ 5.024 |
| 8 | 2.5 cm | 200 cm | $1250\pi$ ≈ 3,925 |

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## 📌 Observations:
- Cylinder 4 has the largest volume due to large radius (40 cm).
- Cylinder 7 has the smallest volume (very small radius).
- Cylinders 1 and 3 are identical.
- Cylinders 6 and 8 have same radius but different heights; 6 has double the height → twice the volume.

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❓ What might the task be?

Possible tasks based on this image:
1. Calculate the volume of each cylinder
2. Order them by volume
3. Find which two are equal
4. Compare volumes across different units
5. Identify the cylinder with maximum/minimum volume

If you can specify the actual task, I can tailor the answer better!

But assuming the goal was to calculate volumes, here’s the complete solution.

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Let me know if you want:
- Surface area instead?
- A comparison?
- Or perhaps identifying matching cylinders?

I'm happy to help further!