Worksheet for calculating the surface area of triangular prisms with labeled dimensions.

Let's solve each of these triangular prism surface area problems step by step.

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🔷 Surface Area of a Triangular Prism Formula:

The surface area (SA) of a triangular prism is:

$$
\text{SA} = 2 \times (\text{Area of triangular base}) + (\text{Perimeter of triangle}) \times (\text{Height of prism})
$$

Or more clearly:

$$
\text{SA} = 2A_{\text{triangle}} + P_{\text{triangle}} \times h_{\text{prism}}
$$

Where:
- $ A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} $
- $ P_{\text{triangle}} = \text{sum of the three side lengths} $
- $ h_{\text{prism}} = \text{length of the prism (the distance between the two triangular bases)} $

We'll go through each problem one by one.

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🔹 Problem 1:

- Triangle: base = 12 in, height = 8 in
- Sides of triangle: 10 in, 13 in, 12 in
- Prism height (length): 13 in

Step 1: Area of triangle
$$
A = \frac{1}{2} \times 12 \times 8 = 48 \text{ in}^2
$$

Step 2: Perimeter of triangle
$$
P = 10 + 13 + 12 = 35 \text{ in}
$$

Step 3: Surface Area
$$
SA = 2(48) + 35 \times 13 = 96 + 455 = \boxed{551} \text{ in}^2
$$

✔ Answer: 551 in²

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🔹 Problem 2:

- Triangle: base = 5 ft, height = 17 ft
- Sides: 17 ft, 17 ft, 5 ft
- Prism height (length): 17 ft

Step 1: Area of triangle
$$
A = \frac{1}{2} \times 5 \times 17 = 42.5 \text{ ft}^2
$$

Step 2: Perimeter
$$
P = 17 + 17 + 5 = 39 \text{ ft}
$$

Step 3: SA
$$
SA = 2(42.5) + 39 \times 17 = 85 + 663 = \boxed{748} \text{ ft}^2
$$

✔ Answer: 748 ft²

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🔹 Problem 3:

- Triangle: base = 9 yd, height = 14 yd
- Sides: 15 yd, 15 yd, 9 yd
- Prism height (length): 14 yd

Step 1: Area
$$
A = \frac{1}{2} \times 9 \times 14 = 63 \text{ yd}^2
$$

Step 2: Perimeter
$$
P = 15 + 15 + 9 = 39 \text{ yd}
$$

Step 3: SA
$$
SA = 2(63) + 39 \times 14 = 126 + 546 = \boxed{672} \text{ yd}^2
$$

✔ Answer: 672 yd²

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🔹 Problem 4:

- Triangle: base = 12 yd, height = 10 yd
- Sides: 13 yd, 13 yd, 12 yd
- Prism height (length): 20 yd

Step 1: Area
$$
A = \frac{1}{2} \times 12 \times 10 = 60 \text{ yd}^2
$$

Step 2: Perimeter
$$
P = 13 + 13 + 12 = 38 \text{ yd}
$$

Step 3: SA
$$
SA = 2(60) + 38 \times 20 = 120 + 760 = \boxed{880} \text{ yd}^2
$$

✔ Answer: 880 yd²

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🔹 Problem 5:

- Triangle: base = 11 in, height = 8 in
- Sides: 11 in, 11 in, 12 in
- Prism height (length): 10 in

Wait — let’s check: The triangle has sides: 11 in, 11 in, and base = 12 in? But height is 8 in.

Check if it makes sense:
Using Pythagorean theorem:
From apex to base midpoint: half-base = 6 in
Then: $ \sqrt{11^2 - 6^2} = \sqrt{121 - 36} = \sqrt{85} \approx 9.2 $ → but given height is 8 in → inconsistency?

Wait — maybe the triangle is not isosceles? Let's look again.

Actually, the figure shows:
- Base = 12 in
- Height = 8 in
- Two other sides: 11 in and 11 in? Wait — no, the labels show 11 in on both slanted sides, and base 12 in.

But if height is 8 in from vertex to base, then the right triangle formed has leg = 8, hypotenuse = 11, so half-base should be:

$$
\sqrt{11^2 - 8^2} = \sqrt{121 - 64} = \sqrt{57} \approx 7.55 \Rightarrow \text{full base} \approx 15.1 \text{ in}
$$

But base is labeled as 12 in — contradiction.

Wait — perhaps I misread.

Looking at image:
- Base = 12 in
- Height = 8 in
- Side lengths: 11 in and 11 in? No — wait, the two slanted sides are labeled 11 in and 11 in? But that can't be with base 12 and height 8.

Wait — actually, the diagram shows:
- One side is 11 in
- Another side is 11 in
- Base is 12 in
- Height from top to base is 8 in

But geometrically: If height is 8 in and base is 12 in, and two equal sides are 11 in, we can verify:

Split base into two parts: x and (12 - x)

Then using Pythagoras:
$$
x^2 + 8^2 = 11^2 \Rightarrow x^2 = 121 - 64 = 57 \Rightarrow x \approx 7.55
$$
Then other part: 12 - 7.55 = 4.45

Now for other side: $ (4.45)^2 + 8^2 = ? $ → $ 19.8 + 64 = 83.8 $, sqrt ≈ 9.15 ≠ 11 → not matching.

So likely, the two equal sides are not both 11 in, or the height is not perpendicular.

Wait — looking closely: the triangle has:
- Base = 12 in
- Height = 8 in (perpendicular)
- One side = 11 in
- Other side = 11 in? Or is one side 11 in and the other different?

Wait — actually, the figure shows one side labeled 11 in, and another side labeled 11 in, and base 12 in.

But this only works if it's an isosceles triangle with equal sides 11 in, base 12 in.

Then height from apex to base should be:

$$
h = \sqrt{11^2 - (6)^2} = \sqrt{121 - 36} = \sqrt{85} \approx 9.22 \text{ in}
$$

But the height is labeled 8 in — contradiction.

Ah! So maybe the height is 8 in, and the base is 12 in, and the sides are not both 11 in?

Wait — the diagram shows:
- Base = 12 in
- Height = 8 in
- One side = 11 in
- Other side = 11 in? Or is it just one side labeled?

Wait — looking again: the triangle has:
- Bottom side = 12 in
- Left side = 11 in
- Right side = 11 in
- Height from top to base = 8 in

This cannot be — unless the height is not to the base.

Wait — maybe the height is not to the base of 12 in, but the label "8 in" is the height from the apex to the base.

But if both sides are 11 in and base is 12 in, then height must be √(11² - 6²) = √85 ≈ 9.22 in.

But it's labeled as 8 in → conflict.

Alternatively, maybe the height is 8 in, and the base is 12 in, and the two other sides are unknown, but we don’t need them — we can use base and height to compute area.

Yes! We don’t need the side lengths for area — we have base and height.

So for Problem 5:
- Base = 12 in
- Height = 8 in
- So area of triangle = $ \frac{1}{2} \times 12 \times 8 = 48 \text{ in}^2 $

Now, what are the side lengths of the triangle? We need them for perimeter.

We know:
- Base = 12 in
- Two other sides: one is labeled 11 in, the other is also labeled 11 in? Yes — both slanted sides are 11 in.

So even though height doesn't match perfectly, perhaps the drawing is approximate.

Assume:
- Sides: 11 in, 11 in, 12 in
- Height = 8 in (but actual should be ~9.22), so maybe typo? Or maybe height is not from apex?

Wait — maybe the height is not to the base, but to a different side?

No — the diagram shows a perpendicular from the top vertex to the base, labeled 8 in.

So unless the triangle is not isosceles, but we have two 11-in sides and base 12 in, then height must be √(11² - 6²) = √85 ≈ 9.22 in.

But it says 8 in → so inconsistency.

Possibility: The height is 8 in, and the base is 12 in, and the other sides are not both 11 in — but the diagram shows one side labeled 11 in and another labeled 11 in?

Wait — rechecking image description: It says “11 in” on both slanted sides, and base 12 in.

But height is 8 in.

This is impossible.

Wait — perhaps the height is not to the base, but to a different side?

No — the perpendicular is drawn from the top to the base.

So either:
- The height is wrong, or
- The side lengths are wrong.

But since we are told to find surface area, and we have base and height, we can use base and height to compute area, regardless of side lengths.

But for perimeter, we need the three side lengths.

So assume:
- Base = 12 in
- Two other sides = 11 in and 11 in → perimeter = 11 + 11 + 12 = 34 in
- Height of prism = 10 in

Then:
- Area of triangle = $ \frac{1}{2} \times 12 \times 8 = 48 \text{ in}^2 $
- Perimeter = 34 in
- SA = $ 2 \times 48 + 34 \times 10 = 96 + 340 = \boxed{436} \text{ in}^2 $

Even though geometry is inconsistent, we proceed with given values.

✔ Answer: 436 in²

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🔹 Problem 6:

- Triangle: base = 8 ft, height = 5 ft
- Sides: 5 ft, 5 ft, 8 ft
- Prism height (length): 14 ft

Step 1: Area
$$
A = \frac{1}{2} \times 8 \times 5 = 20 \text{ ft}^2
$$

Step 2: Perimeter
$$
P = 5 + 5 + 8 = 18 \text{ ft}
$$

Step 3: SA
$$
SA = 2(20) + 18 \times 14 = 40 + 252 = \boxed{292} \text{ ft}^2
$$

✔ Answer: 292 ft²

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🔹 Problem 7:

- Triangle: base = 6 ft, height = 2 ft
- Sides: 4 ft, 4 ft, 6 ft
- Prism height (length): 10 ft

Wait — base = 6 ft, height = 2 ft, so area = $ \frac{1}{2} \times 6 \times 2 = 6 \text{ ft}^2 $

Sides: 4 ft, 4 ft, 6 ft → perimeter = 4 + 4 + 6 = 14 ft

Prism length = 10 ft

SA = $ 2(6) + 14 \times 10 = 12 + 140 = \boxed{152} \text{ ft}^2 $

✔ Answer: 152 ft²

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🔹 Problem 8:

- Triangle: base = 16 yd, height = 12 yd
- Sides: 17 yd, 17 yd, 16 yd
- Prism height (length): 17 yd

Step 1: Area
$$
A = \frac{1}{2} \times 16 \times 12 = 96 \text{ yd}^2
$$

Step 2: Perimeter
$$
P = 17 + 17 + 16 = 50 \text{ yd}
$$

Step 3: SA
$$
SA = 2(96) + 50 \times 17 = 192 + 850 = \boxed{1042} \text{ yd}^2
$$

✔ Answer: 1042 yd²

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🔹 Problem 9:

- Triangle: base = 8 in, height = 6 in
- Sides: 10 in, 10 in, 8 in
- Prism height (length): 12 in

Step 1: Area
$$
A = \frac{1}{2} \times 8 \times 6 = 24 \text{ in}^2
$$

Step 2: Perimeter
$$
P = 10 + 10 + 8 = 28 \text{ in}
$$

Step 3: SA
$$
SA = 2(24) + 28 \times 12 = 48 + 336 = \boxed{384} \text{ in}^2
$$

✔ Answer: 384 in²

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✔ Final Answers:

| Problem | Surface Area |
|--------|--------------|
| 1 | 551 in² |
| 2 | 748 ft² |
| 3 | 672 yd² |
| 4 | 880 yd² |
| 5 | 436 in² |
| 6 | 292 ft² |
| 7 | 152 ft² |
| 8 | 1042 yd² |
| 9 | 384 in² |

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📝 Notes:

- For Problem 5, there is a geometric inconsistency (height should be ~9.22 in if sides are 11 in and base 12 in), but since the problem gives height = 8 in, we used it.
- Always double-check whether the height is to the base — here it appears to be.
- In all cases, we used:
- Area of triangle = $ \frac{1}{2} \times \text{base} \times \text{height} $
- Perimeter = sum of three sides
- SA = $ 2A + P \times L $

Let me know if you'd like a visual explanation or diagram help!