Surface Area of a Triangular Prism Worksheets - Free Printable

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:

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

Or more clearly:

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

Where:
- $ A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} $
- $ P_{\text{triangle}} = \text{sum of the three sides of the triangle} $
- $ h $ = length (or height) 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 = 8 in
- Height = 6 in
- Sides: 8 in, 10 in, 6 in
Prism length (depth): 9 in

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

#### Step 2: Perimeter of triangle
$$
P = 8 + 10 + 6 = 24 \text{ in}
$$

#### Step 3: Lateral surface area (rectangular sides)
$$
P \times h = 24 \times 9 = 216 \text{ in}^2
$$

#### Step 4: Total Surface Area
$$
SA = 2(24) + 216 = 48 + 216 = \boxed{264} \text{ in}^2
$$

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✔ Problem 2:

Triangle:
- Base = 9 ft
- Height = 12 ft
- Sides: 9 ft, 15 ft, 12 ft
Prism length: 11 ft

#### Area of triangle
$$
A = \frac{1}{2} \times 9 \times 12 = 54 \text{ ft}^2
$$

#### Perimeter
$$
P = 9 + 15 + 12 = 36 \text{ ft}
$$

#### Lateral SA
$$
36 \times 11 = 396 \text{ ft}^2
$$

#### Total SA
$$
2(54) + 396 = 108 + 396 = \boxed{504} \text{ ft}^2
$$

---

✔ Problem 3:

Triangle:
- Base = 6 yd
- Height = 3 yd
- Sides: 5 yd, 5 yd, 6 yd
Prism length: 4 yd

#### Area
$$
A = \frac{1}{2} \times 6 \times 3 = 9 \text{ yd}^2
$$

#### Perimeter
$$
P = 5 + 5 + 6 = 16 \text{ yd}
$$

#### Lateral SA
$$
16 \times 4 = 64 \text{ yd}^2
$$

#### Total SA
$$
2(9) + 64 = 18 + 64 = \boxed{82} \text{ yd}^2
$$

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✔ Problem 4:

Triangle:
- Base = 5 ft
- Height = 12 ft
- Hypotenuse = 13 ft (since 5–12–13 is a right triangle)
- Other side = 11 ft (prism depth?)

Wait — the prism has:
- Triangle with sides: 5 ft, 12 ft, 13 ft → right triangle
- The length of the prism is 11 ft (the rectangle side labeled "11 ft")

So:

#### Area of triangle
$$
A = \frac{1}{2} \times 5 \times 12 = 30 \text{ ft}^2
$$

#### Perimeter
$$
P = 5 + 12 + 13 = 30 \text{ ft}
$$

#### Lateral SA
$$
30 \times 11 = 330 \text{ ft}^2
$$

#### Total SA
$$
2(30) + 330 = 60 + 330 = \boxed{390} \text{ ft}^2
$$

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✔ Problem 5:

Triangle:
- Base = 9 yd
- Height = 7 yd
- Sides: 9 yd, 12 yd, 15 yd? Wait — let’s see:

Actually, from diagram:
- Right triangle: legs 9 yd and 7 yd? But hypotenuse is given as 12 yd?

Wait — check:
Is $ 9^2 + 7^2 = 81 + 49 = 130 $, but $ 12^2 = 144 $. Not matching.

But the triangle has:
- One leg = 9 yd
- Other leg = ?
- Hypotenuse = 12 yd
- Height = 7 yd? That doesn’t fit.

Wait — actually, the triangle has:
- Base = 9 yd
- Height = 7 yd (perpendicular to base)
- Sides: 9 yd, 12 yd, and unknown?

Wait — perhaps it's a right triangle with legs 9 yd and 7 yd, and hypotenuse = ?

But hypotenuse should be $ \sqrt{9^2 + 7^2} = \sqrt{81+49} = \sqrt{130} \approx 11.4 $, but shown as 12 yd.

Hmm — maybe not a right triangle.

Wait — look again:
The triangle has:
- Base = 9 yd
- Height = 7 yd (from top vertex perpendicular to base)
- One side = 12 yd (hypotenuse-like)
- Another side = 11 yd?

Wait — label says: “9 yd”, “12 yd”, “11 yd” — so all three sides are 9, 11, 12 yd.

And the height from the 9-yd base is 7 yd.

So we can use:
- Base = 9 yd
- Height = 7 yd
→ So area is valid.

#### Area of triangle
$$
A = \frac{1}{2} \times 9 \times 7 = 31.5 \text{ yd}^2
$$

#### Perimeter
$$
P = 9 + 11 + 12 = 32 \text{ yd}
$$

#### Prism length = 20 yd (side labeled “20 yd”)

#### Lateral SA
$$
32 \times 20 = 640 \text{ yd}^2
$$

#### Total SA
$$
2(31.5) + 640 = 63 + 640 = \boxed{703} \text{ yd}^2
$$

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✔ Problem 6:

Triangle:
- Base = 12 m
- Height = 10 m
- Sides: 12 m, 17 m, 17 m? Wait — no.

Wait: triangle has:
- Base = 12 m
- Height = 10 m
- Side = 17 m (slanted side)
- Other side = ?
But the figure shows:
- Two equal sides: 17 m
- Base = 12 m
- Height = 10 m (drawn from apex to base)

So yes, it's an isosceles triangle.

Sides: 17 m, 17 m, 12 m
Height = 10 m → confirms: $ \frac{1}{2} \times 12 \times 10 = 60 $ → matches area

#### Area of triangle
$$
A = \frac{1}{2} \times 12 \times 10 = 60 \text{ m}^2
$$

#### Perimeter
$$
P = 17 + 17 + 12 = 46 \text{ m}
$$

#### Prism length = 17 m (the side labeled "17 m" on the rectangle)

Wait — the rectangle has one side 17 m, and other side 12 m — so the length of the prism is 17 m?

Yes — the rectangular face has dimensions 17 m × 12 m → so the prism length is 17 m.

#### Lateral SA
$$
46 \times 17 = 782 \text{ m}^2
$$

#### Total SA
$$
2(60) + 782 = 120 + 782 = \boxed{902} \text{ m}^2
$$

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✔ Problem 7:

Triangle:
- Base = 10 yd
- Height = 12 yd
- Sides: 13 yd, 13 yd, 10 yd?
- Height drawn = 12 yd → yes, since $ \frac{1}{2} \times 10 \times 12 = 60 $

Also, slant side = 13 yd → check: $ \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 $ → yes, right triangle

So triangle has sides: 10 yd, 13 yd, 13 yd

Wait — no: base 10 yd, height 12 yd, so the two equal sides are 13 yd each.

So sides: 13 yd, 13 yd, 10 yd

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

#### Perimeter
$$
P = 13 + 13 + 10 = 36 \text{ yd}
$$

#### Prism length = 11 yd (labeled on the side)

#### Lateral SA
$$
36 \times 11 = 396 \text{ yd}^2
$$

#### Total SA
$$
2(60) + 396 = 120 + 396 = \boxed{516} \text{ yd}^2
$$

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✔ Problem 8:

Triangle:
- Base = 12 m
- Height = 5 m
- Hypotenuse = 13 m → 5–12–13 triangle

So sides: 5 m, 12 m, 13 m

#### Area
$$
A = \frac{1}{2} \times 12 \times 5 = 30 \text{ m}^2
$$

#### Perimeter
$$
P = 5 + 12 + 13 = 30 \text{ m}
$$

#### Prism length = 13 m (labeled on the rectangle)

Wait — the rectangle has sides 13 m and 12 m → so the length is 13 m?

Yes — the prism extends 13 m along the direction of the 13-m side.

So lateral SA = perimeter × length = $ 30 \times 13 = 390 \text{ m}^2 $

#### Total SA
$$
2(30) + 390 = 60 + 390 = \boxed{450} \text{ m}^2
$$

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✔ Problem 9:

Triangle:
- Base = 13 ft
- Height = 12 ft
- Sides: 13 ft, 15 ft, 13 ft? Wait — no

Wait:
- Base = 13 ft
- Height = 12 ft
- One side = 15 ft
- Other side = 13 ft?

Wait — from diagram:
- Triangle has base 13 ft
- Height 12 ft
- One side = 15 ft
- Other side = 13 ft?

But 13–12–15? Let’s check if it’s a right triangle: $ 12^2 + 13^2 = 144 + 169 = 313 $, $ 15^2 = 225 $ → no.

Wait — perhaps it’s not a right triangle.

But the height is 12 ft from apex to base, so area is:

$$
A = \frac{1}{2} \times 13 \times 12 = 78 \text{ ft}^2
$$

Now, what are the side lengths?

From the diagram:
- One side = 15 ft
- Other side = 13 ft
- Base = 13 ft?

Wait — base is 13 ft, and one side is 15 ft, and the other side is labeled 13 ft?

No — wait:
The triangle has:
- Base = 13 ft
- One leg = 13 ft (left side)
- Other leg = 15 ft (right side)
- Height = 12 ft (from apex to base)

So sides: 13 ft, 15 ft, 13 ft → no, that would make two sides 13 ft, but base also 13 ft?

Wait — base = 13 ft, and both other sides are 13 ft and 15 ft? So sides: 13, 13, 15?

Wait — no: base = 13 ft, and the two other sides are 13 ft and 15 ft?

But then it’s scalene.

But height is 12 ft → so area = $ \frac{1}{2} \times 13 \times 12 = 78 \text{ ft}^2 $

Now, perimeter: sum of all sides = 13 + 15 + 13 = 41 ft?

Wait — the two non-base sides are 13 ft and 15 ft? And base is 13 ft?

But in diagram, the left side is labeled 13 ft, the right side is 15 ft, base is 13 ft → yes.

So sides: 13 ft, 15 ft, 13 ft → total = 41 ft

But wait — is the prism length 13 ft or 15 ft?

Look at the rectangle: it has one side 13 ft, and the other side is 13 ft → so the prism length is 13 ft?

Wait — the rectangle is labeled 13 ft and 13 ft? No — it says "13 ft" and "13 ft"? Wait — no.

Wait — the diagram shows:
- Rectangle with sides: 13 ft and 13 ft? Or is one side 13 ft and the other is the prism length?

Wait — the triangle has base 13 ft, and the prism extends 13 ft along the base direction.

But the prism length is the distance between the two triangular faces.

Looking at the rectangle: one side is 13 ft (same as base), and the other side is labeled 13 ft? Wait — no.

Wait — the rectangle is labeled “13 ft” and “13 ft”? Actually, the vertical side is labeled “13 ft”, and the horizontal is “13 ft”?

No — the prism length is the side that connects the two triangles.

From the diagram: the side connecting the two triangles is labeled 13 ft.

So the prism length is 13 ft.

#### Area of triangle
$$
A = \frac{1}{2} \times 13 \times 12 = 78 \text{ ft}^2
$$

#### Perimeter of triangle
Sides: 13 ft, 15 ft, 13 ft → total = $ 13 + 15 + 13 = 41 \text{ ft} $

#### Lateral SA
$$
41 \times 13 = 533 \text{ ft}^2
$$

#### Total SA
$$
2(78) + 533 = 156 + 533 = \boxed{689} \text{ ft}^2
$$

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✔ Final Answers:

| Problem | Surface Area |
|--------|--------------|
| 1 | 264 in² |
| 2 | 504 ft² |
| 3 | 82 yd² |
| 4 | 390 ft² |
| 5 | 703 yd² |
| 6 | 902 m² |
| 7 | 516 yd² |
| 8 | 450 m² |
| 9 | 689 ft² |

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✔ Summary of Method:
For each triangular prism:
1. Find area of triangular base: $ \frac{1}{2} \times \text{base} \times \text{height} $
2. Multiply by 2 for both bases.
3. Add the lateral surface area: perimeter of triangle × length of prism.
4. Sum them.

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