Find the exact surface area of each prism in this printable math worksheet.

Let's solve each problem step by step to find the exact surface area of each prism. We'll go through all nine figures, identifying their shapes and applying the correct formulas.

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🔷 Surface Area Formulas:

1. Cylinder:
$$
SA = 2\pi r^2 + 2\pi r h
$$

2. Rectangular Prism (Box):
$$
SA = 2(lw + lh + wh)
$$

3. Triangular Prism:
$$
SA = 2 \times (\text{Base Area}) + \text{Perimeter of Base} \times \text{Height}
$$

4. Prism with Trapezoidal Base:
Use:
$$
SA = 2 \times (\text{Area of Base}) + \text{Sum of Areas of Lateral Faces}
$$

5. General Rule:
For any prism:
$$
SA = 2 \times (\text{Base Area}) + \text{Lateral Surface Area}
$$

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Now let’s solve each one:

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1) Cylinder

- Radius $ r = 3 $ cm
- Height $ h = 11 $ cm

$$
SA = 2\pi r^2 + 2\pi r h = 2\pi(3)^2 + 2\pi(3)(11) = 2\pi(9) + 2\pi(33) = 18\pi + 66\pi = 84\pi \text{ cm}^2
$$

✔ Answer: $ \boxed{84\pi} $ cm²

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2) Rectangular Prism

- Dimensions: 11 m × 3 m × 3 m

$$
SA = 2(lw + lh + wh) = 2[(11)(3) + (11)(3) + (3)(3)] = 2[33 + 33 + 9] = 2(75) = 150 \text{ m}^2
$$

✔ Answer: $ \boxed{150} $ m²

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3) Triangular Prism

- Triangle base: 10 mm, 13 mm, 14 mm
- Height of triangle: 12 mm (given as perpendicular height)
- Length of prism: 14 mm (the side along the length)

First, find area of triangular base:
$$
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 12 = 60 \text{ mm}^2
$$

So two bases: $ 2 \times 60 = 120 $

Now lateral faces: three rectangles:
- One: $ 10 \times 14 = 140 $
- One: $ 13 \times 14 = 182 $
- One: $ 14 \times 14 = 196 $

Wait — but the length of the prism is 14 mm, and the triangle sides are 10, 13, 14? Let's check.

Actually, the triangle has sides 10, 13, 14, and the prism length is 14 mm (labeled on the side). So the lateral faces have areas:

- $ 10 \times 14 = 140 $
- $ 13 \times 14 = 182 $
- $ 14 \times 14 = 196 $

Total lateral area = $ 140 + 182 + 196 = 518 $

Add two bases: $ 120 $

$$
SA = 518 + 120 = 638 \text{ mm}^2
$$

✔ Answer: $ \boxed{638} $ mm²

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4) Rectangular Prism

- Dimensions: 3 ft × 5 ft × 8 ft

$$
SA = 2(lw + lh + wh) = 2[(3)(5) + (3)(8) + (5)(8)] = 2[15 + 24 + 40] = 2(79) = 158 \text{ ft}^2
$$

✔ Answer: $ \boxed{158} $ ft²

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5) Triangular Prism

- Triangle base: 12 cm, 13 cm, 13 cm (isosceles)
- Height of triangle: 11 cm (perpendicular from apex to base)
- Prism length: 12 cm (the length along the prism)

Area of triangle base:
$$
A = \frac{1}{2} \times 12 \times 11 = 66 \text{ cm}^2
$$

Two bases: $ 2 \times 66 = 132 $

Lateral faces:
- Rectangle 1: $ 12 \times 12 = 144 $
- Rectangle 2: $ 13 \times 12 = 156 $
- Rectangle 3: $ 13 \times 12 = 156 $

Lateral area: $ 144 + 156 + 156 = 456 $

Total SA: $ 132 + 456 = 588 \text{ cm}^2 $

✔ Answer: $ \boxed{588} $ cm²

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6) Rectangular Prism

- Dimensions: 8 ft × 8 ft × 11 ft

$$
SA = 2(lw + lh + wh) = 2[(8)(8) + (8)(11) + (8)(11)] = 2[64 + 88 + 88] = 2(240) = 480 \text{ ft}^2
$$

✔ Answer: $ \boxed{480} $ ft²

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7) Trapezoidal Prism

- Trapezoid base: parallel sides 8 cm and 5 cm, height = 6 cm
- Prism length = 8 cm

Step 1: Area of trapezoid base
$$
A = \frac{1}{2}(b_1 + b_2)h = \frac{1}{2}(8 + 5)(6) = \frac{1}{2}(13)(6) = 39 \text{ cm}^2
$$

Two bases: $ 2 \times 39 = 78 $

Step 2: Lateral faces

We need to find the areas of the four rectangular sides.

The trapezoid has:
- Two parallel sides: 8 cm and 5 cm
- Height: 6 cm
- Non-parallel sides: labeled as 5 cm and 5 cm? Wait — looking at diagram:

From image:
- Left leg: 5 cm
- Right leg: 5 cm
- Top: 5 cm
- Bottom: 8 cm
- Height: 6 cm

So it's an isosceles trapezoid.

The lateral faces:
1. Bottom rectangle: $ 8 \times 8 = 64 $
2. Top rectangle: $ 5 \times 8 = 40 $
3. Left side: $ 5 \times 8 = 40 $
4. Right side: $ 5 \times 8 = 40 $

Total lateral area: $ 64 + 40 + 40 + 40 = 184 $

Total SA: $ 78 + 184 = 262 \text{ cm}^2 $

✔ Answer: $ \boxed{262} $ cm²

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8) Trapezoidal Prism

- Trapezoid base: parallel sides 13 m and 8 m, height = 6 m
- Prism length = 8 m

Area of trapezoid base:
$$
A = \frac{1}{2}(13 + 8)(6) = \frac{1}{2}(21)(6) = 63 \text{ m}^2
$$

Two bases: $ 2 \times 63 = 126 $

Lateral faces:
- Bottom: $ 13 \times 8 = 104 $
- Top: $ 8 \times 8 = 64 $
- Left: $ 8 \times 8 = 64 $
- Right: $ 10 \times 8 = 80 $? Wait — the slanted side is labeled 10 m?

Yes — right non-parallel side is 10 m.

Left side: not labeled? But from diagram, left side is 8 m (vertical), and top is 8 m, bottom 13 m.

Wait — we see:
- Vertical leg: 6 m (height)
- Slanted sides: 8 m and 10 m?
But in the diagram, the left side is labeled 8 m (but that’s vertical?), wait.

Looking again:
- The height of the trapezoid is 6 m (vertical)
- The non-parallel sides are labeled: 8 m and 10 m
- The top is 8 m, bottom is 13 m

So the lateral faces are:
1. Bottom: $ 13 \times 8 = 104 $
2. Top: $ 8 \times 8 = 64 $
3. Left slant: $ 8 \times 8 = 64 $
4. Right slant: $ 10 \times 8 = 80 $

Wait — is the length of the prism 8 m? Yes, the horizontal dimension.

So lateral faces:
- Bottom: $ 13 \times 8 = 104 $
- Top: $ 8 \times 8 = 64 $
- Left: $ 8 \times 8 = 64 $
- Right: $ 10 \times 8 = 80 $

Total lateral area: $ 104 + 64 + 64 + 80 = 312 $

Add bases: $ 126 $

Total SA: $ 126 + 312 = 438 \text{ m}^2 $

✔ Answer: $ \boxed{438} $ m²

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9) Cylinder

- Radius $ r = 3 $ m
- Height $ h = 9 $ m

$$
SA = 2\pi r^2 + 2\pi r h = 2\pi(9) + 2\pi(3)(9) = 18\pi + 54\pi = 72\pi \text{ m}^2
$$

✔ Answer: $ \boxed{72\pi} $ m²

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✔ Final Answers:

| Problem | Surface Area |
|--------|--------------|
| 1) | $ 84\pi $ cm² |
| 2) | 150 m² |
| 3) | 638 mm² |
| 4) | 158 ft² |
| 5) | 588 cm² |
| 6) | 480 ft² |
| 7) | 262 cm² |
| 8) | 438 m² |
| 9) | $ 72\pi $ m² |

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