Printable math worksheet for calculating surface area of prisms and cylinders.

Let's solve each problem step by step to find the exact surface area of each shape. We'll use formulas for:

- Rectangular Prisms:
$$
SA = 2(lw + lh + wh)
$$

- Triangular Prisms:
$$
SA = \text{Base Area} \times 2 + \text{Lateral Area}
$$
(Lateral area = perimeter of base × height)

- Cylinders:
$$
SA = 2\pi r^2 + 2\pi rh
$$

We’ll go through each shape.

---

1) Rectangular Prism

Dimensions: 6 ft (length), 4 ft (width), 12 ft (height)

$$
SA = 2(lw + lh + wh) = 2(6×4 + 6×12 + 4×12) = 2(24 + 72 + 48) = 2(144) = \boxed{288} \text{ ft}^2
$$

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2) Triangular Prism

Base triangle: base = 6 yd, height = 3 yd
Side lengths: 6 yd, 5 yd, 5 yd
Height of prism = 11 yd

Step 1: Area of triangular bases
$$
A_{\text{base}} = \frac{1}{2} \times 6 \times 3 = 9 \text{ yd}^2
$$
Two bases: $ 2 \times 9 = 18 $

Step 2: Lateral area
Perimeter of base = 6 + 5 + 5 = 16 yd
Lateral area = perimeter × height = $ 16 \times 11 = 176 $

Total SA = $ 18 + 176 = \boxed{194} \text{ yd}^2 $

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3) Rectangular Prism

Dimensions: 8 in (length), 5 in (width), 14 in (height)

$$
SA = 2(8×5 + 8×14 + 5×14) = 2(40 + 112 + 70) = 2(222) = \boxed{444} \text{ in}^2
$$

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4) Triangular Prism (Right Triangle Base)

Base triangle: legs = 9 in and 12 in, hypotenuse = 15 in
Prism height = 12 in

Step 1: Area of triangular bases
$$
A = \frac{1}{2} \times 9 \times 12 = 54 \text{ in}^2
$$
Two bases: $ 2 \times 54 = 108 $

Step 2: Lateral area
Perimeter = 9 + 12 + 15 = 36 in
Lateral area = $ 36 \times 12 = 432 $

Total SA = $ 108 + 432 = \boxed{540} \text{ in}^2 $

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5) Cube

All sides = 9 ft

$$
SA = 6s^2 = 6 \times 9^2 = 6 \times 81 = \boxed{486} \text{ ft}^2
$$

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6) Cylinder

Radius = 10 yd, Height = 30 yd

$$
SA = 2\pi r^2 + 2\pi r h = 2\pi(10)^2 + 2\pi(10)(30) = 2\pi(100) + 2\pi(300) = 200\pi + 600\pi = 800\pi
$$
Exact answer: $ \boxed{800\pi} \text{ yd}^2 $

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7) Cylinder

Diameter = 2 m → radius = 1 m, Height = 11 m

$$
SA = 2\pi r^2 + 2\pi r h = 2\pi(1)^2 + 2\pi(1)(11) = 2\pi + 22\pi = 24\pi
$$
Exact answer: $ \boxed{24\pi} \text{ m}^2 $

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8) Rectangular Prism

Dimensions: 7 m (length), 4 m (width), 15 m (height)

$$
SA = 2(7×4 + 7×15 + 4×15) = 2(28 + 105 + 60) = 2(193) = \boxed{386} \text{ m}^2
$$

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9) Triangular Prism

Base triangle: base = 10 yd, height = 12 yd, side lengths: 13 yd, 13 yd, 10 yd
Prism height = 14 yd

Step 1: Area of triangular bases
$$
A = \frac{1}{2} \times 10 \times 12 = 60 \text{ yd}^2
$$
Two bases: $ 2 \times 60 = 120 $

Step 2: Lateral area
Perimeter = 13 + 13 + 10 = 36 yd
Lateral area = $ 36 \times 14 = 504 $

Total SA = $ 120 + 504 = \boxed{624} \text{ yd}^2 $

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✔ Final Answers:

1) $ \boxed{288} \text{ ft}^2 $
2) $ \boxed{194} \text{ yd}^2 $
3) $ \boxed{444} \text{ in}^2 $
4) $ \boxed{540} \text{ in}^2 $
5) $ \boxed{486} \text{ ft}^2 $
6) $ \boxed{800\pi} \text{ yd}^2 $
7) $ \boxed{24\pi} \text{ m}^2 $
8) $ \boxed{386} \text{ m}^2 $
9) $ \boxed{624} \text{ yd}^2 $

Let me know if you'd like a visual explanation or help with any specific one!