Surface Area and Volume of Prism and Cylinder worksheet | Live ... - Free Printable

Let's solve for the surface area of each of the four 3D figures shown in the image. We will go one by one, using the appropriate formulas.

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🔹 Figure 1: Rectangular Prism (Cuboid)

Dimensions:
- Length = 10 m
- Width = 4 m
- Height = 6 m

Formula for Surface Area of a Rectangular Prism:
$$
SA = 2(lw + lh + wh)
$$

Plug in values:
$$
SA = 2(10 \times 4 + 10 \times 6 + 4 \times 6) = 2(40 + 60 + 24) = 2(124) = 248\ \text{m}^2
$$

✔ Surface Area = 248 m²

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🔹 Figure 2: Triangular Prism

Given dimensions:
- Base triangle sides: 3 ft, 4 ft, and 5 ft → This is a right triangle (since $3^2 + 4^2 = 5^2$)
- Length (height of prism) = 12 ft

We need to find the surface area, which includes:
- Two triangular bases
- Three rectangular lateral faces

#### Step 1: Area of one triangular base
$$
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 4 = 6\ \text{ft}^2
$$
So two bases: $2 \times 6 = 12\ \text{ft}^2$

#### Step 2: Lateral surface area (rectangles)
Each rectangle has height = 12 ft and width = side of triangle:

- Rectangle 1: $3 \times 12 = 36\ \text{ft}^2$
- Rectangle 2: $4 \times 12 = 48\ \text{ft}^2$
- Rectangle 3: $5 \times 12 = 60\ \text{ft}^2$

Total lateral area = $36 + 48 + 60 = 144\ \text{ft}^2$

#### Total Surface Area:
$$
SA = \text{Bases} + \text{Lateral} = 12 + 144 = 156\ \text{ft}^2
$$

✔ Surface Area = 156 ft²

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🔹 Figure 3: Cylinder

Given:
- Radius $r = 3$ ft
- Height $h = 4$ ft

Formula for Surface Area of a Cylinder:
$$
SA = 2\pi r^2 + 2\pi r h
$$

$$
= 2\pi (3)^2 + 2\pi (3)(4) = 2\pi (9) + 2\pi (12) = 18\pi + 24\pi = 42\pi\ \text{ft}^2
$$

Using $\pi \approx 3.14$:
$$
42 \times 3.14 = 131.88\ \text{ft}^2
$$

But we can leave it as exact:
✔ Surface Area = $42\pi\ \text{ft}^2$ or approximately 131.88 ft²

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🔹 Figure 4: Cuboid (Rectangular Prism)

Dimensions:
- Length = 8 m
- Width = 6 m
- Height = $x = 5$ cm? Wait — this is inconsistent!

⚠️ Problem: The height is labeled as $x = 5$ cm, but other dimensions are in meters. This is likely a typo.

Let’s check:
- The figure shows: 8 m, 6 m, and $x = 5$ cm → But 5 cm = 0.05 m, which seems too small.

Alternatively, maybe $x = 5$ cm is just labeling the variable, but the actual dimension is 5 m?

Wait — look at the diagram: It says "X = 5 cm" below the figure. But the other sides are 8 m and 6 m.

That would make no sense dimensionally.

➡️ Most likely, this is a typo, and $x = 5$ m, not cm. Otherwise, the unit mismatch makes the problem invalid.

Assuming x = 5 m, then:

Dimensions:
- Length = 8 m
- Width = 6 m
- Height = 5 m

Now compute surface area:

$$
SA = 2(lw + lh + wh) = 2(8 \times 6 + 8 \times 5 + 6 \times 5) = 2(48 + 40 + 30) = 2(118) = 236\ \text{m}^2
$$

✔ Surface Area = 236 m²

> ⚠️ Note: If $x = 5$ cm is correct, convert to meters: $0.05$ m, then:
>
> $$
> SA = 2(8 \times 6 + 8 \times 0.05 + 6 \times 0.05) = 2(48 + 0.4 + 0.3) = 2(48.7) = 97.4\ \text{m}^2
> $$
>
> But that would be odd — the height is tiny compared to others. So likely a typo, and it should be 5 m.

We'll proceed with 5 m.

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✔ Final Answers:

| Figure | Shape | Surface Area |
|-------|--------|--------------|
| 1 | Rectangular Prism | 248 m² |
| 2 | Triangular Prism | 156 ft² |
| 3 | Cylinder | $42\pi$ ft² ≈ 131.88 ft² |
| 4 | Rectangular Prism | 236 m² |

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📌 Summary of Solutions:

1. Rectangular Prism (10×4×6 m):
$$
2(10×4 + 10×6 + 4×6) = 2(40+60+24) = 2×124 = \boxed{248\ \text{m}^2}
$$

2. Triangular Prism (3–4–5 right triangle, length 12 ft):
- Bases: $2 × \frac{1}{2} × 3 × 4 = 12$
- Lateral: $3×12 + 4×12 + 5×12 = 36+48+60 = 144$
- Total: $12 + 144 = \boxed{156\ \text{ft}^2}$

3. Cylinder (r=3 ft, h=4 ft):
$$
2\pi r^2 + 2\pi r h = 2\pi(9) + 2\pi(12) = 18\pi + 24\pi = \boxed{42\pi\ \text{ft}^2}
$$

4. Rectangular Prism (8×6×5 m):
$$
2(8×6 + 8×5 + 6×5) = 2(48+40+30) = 2×118 = \boxed{236\ \text{m}^2}
$$

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