Surface Area Worksheet 6th Grade - Free Printable

Let's solve each problem step by step to find the surface area of each triangular prism.

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🔷 Surface Area of a Triangular Prism Formula:

A triangular prism has:
- 2 triangular bases
- 3 rectangular faces

So,
> Surface Area = 2 × (Area of triangle) + (Sum of areas of 3 rectangles)

We'll use:
- Area of triangle = $ \frac{1}{2} \times \text{base} \times \text{height} $
- Area of rectangle = $ \text{length} \times \text{width} $

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## ✔ Problem 1:

Given:
- Triangle base = 3 cm
- Height of triangle = 2 cm
- Rectangles:
- Two identical rectangles with dimensions: 6 cm × 2.5 cm
- One rectangle: 6 cm × 3 cm (the base)

But wait — let’s look carefully at the net:

- The two triangles are isosceles with base = 3 cm and height = 2 cm.
- The three rectangles:
- Two rectangles are 6 cm × 2.5 cm → these are the sides
- One rectangle is 6 cm × 3 cm → this is the base

Wait — actually, from the net:
- The two triangles have base 3 cm and height 2 cm
- The three rectangles:
- Two rectangles: 6 cm × 2.5 cm → side faces
- One rectangle: 6 cm × 3 cm → bottom face

But note: The length of the prism is 6 cm, so all rectangles have one dimension = 6 cm.

Now, compute:

Step 1: Area of one triangle

$$
\text{Area} = \frac{1}{2} \times 3 \times 2 = 3 \text{ cm}^2
$$
Two triangles:
$$
2 \times 3 = 6 \text{ cm}^2
$$

Step 2: Areas of rectangles

- Rectangle 1: 6 cm × 2.5 cm = 15 cm²
- Rectangle 2: 6 cm × 2.5 cm = 15 cm² → same as above
- Rectangle 3: 6 cm × 3 cm = 18 cm²

Total rectangle area:
$$
15 + 15 + 18 = 48 \text{ cm}^2
$$

Total Surface Area:

$$
6 + 48 = \boxed{54} \text{ cm}^2
$$

✔ Answer for 1): 54 cm²

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## ✔ Problem 2:

Given:
- Triangle base = 6 in
- Height of triangle = 4 in
- Rectangles:
- One rectangle: 14 in × 6 in
- Two rectangles: 14 in × 5 in

From the net:
- The two triangles are isosceles with base 6 in, height 4 in
- The length of the prism is 14 in
- The rectangles:
- One rectangle connects the base: 14 in × 6 in
- Two side rectangles: 14 in × 5 in (since the equal sides of the triangle are 5 in)

Wait — the triangle has base 6 in, height 4 in, and two equal sides labeled 5 in.

Yes — it's a 3-4-5 right triangle scaled: base 6, height 4, hypotenuse? Let's verify:

If base = 6, height = 4, then the equal sides should be:
$$
\sqrt{(3)^2 + (4)^2} = \sqrt{9+16} = \sqrt{25} = 5 \text{ in}
$$
Yes, correct.

So, the three rectangular faces:
- One rectangle: 14 in × 6 in (along base)
- Two rectangles: 14 in × 5 in (along the slanted sides)

Step 1: Area of one triangle

$$
\frac{1}{2} \times 6 \times 4 = 12 \text{ in}^2
$$
Two triangles:
$$
2 \times 12 = 24 \text{ in}^2
$$

Step 2: Rectangle areas

- Base rectangle: 14 × 6 = 84 in²
- Two side rectangles: 14 × 5 = 70 in² each → 2 × 70 = 140 in²

Total rectangle area:
$$
84 + 140 = 224 \text{ in}^2
$$

Total Surface Area:

$$
24 + 224 = \boxed{248} \text{ in}^2
$$

✔ Answer for 2): 248 in²

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## ✔ Problem 3:

Given:
- Triangle base = 10 cm
- Height of triangle = 10 cm
- The two equal sides of the triangle are 12 cm
- Length of prism = 13 cm

Rectangles:
- One rectangle: 13 cm × 10 cm (base)
- Two rectangles: 13 cm × 12 cm (sides)

Step 1: Area of one triangle

$$
\frac{1}{2} \times 10 \times 10 = 50 \text{ cm}^2
$$
Two triangles:
$$
2 \times 50 = 100 \text{ cm}^2
$$

Step 2: Rectangle areas

- Base rectangle: 13 × 10 = 130 cm²
- Two side rectangles: 13 × 12 = 156 cm² each → 2 × 156 = 312 cm²

Total rectangle area:
$$
130 + 312 = 442 \text{ cm}^2
$$

Total Surface Area:

$$
100 + 442 = \boxed{542} \text{ cm}^2
$$

✔ Answer for 3): 542 cm²

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## ✔ Problem 4:

Given:
- Triangle base = 12 cm
- Height of triangle = 8 cm
- The two equal sides of the triangle: not directly given, but we can see from the net:
- The rectangles show: one rectangle 15 cm × 12 cm
- Two rectangles: 15 cm × 10 cm

So the triangle has:
- Base = 12 cm
- Height = 8 cm
- Equal sides = 10 cm (from the rectangles)

Let’s verify: Is a triangle with base 12 cm, height 8 cm, and equal sides 10 cm possible?

Split the base into two 6 cm parts. Then each half-triangle has legs 6 and 8 → hypotenuse:
$$
\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm}
$$
Yes! So the equal sides are 10 cm.

Prism length = 15 cm

Step 1: Area of one triangle

$$
\frac{1}{2} \times 12 \times 8 = 48 \text{ cm}^2
$$
Two triangles:
$$
2 \times 48 = 96 \text{ cm}^2
$$

Step 2: Rectangle areas

- Base rectangle: 15 × 12 = 180 cm²
- Two side rectangles: 15 × 10 = 150 cm² each → 2 × 150 = 300 cm²

Total rectangle area:
$$
180 + 300 = 480 \text{ cm}^2
$$

Total Surface Area:

$$
96 + 480 = \boxed{576} \text{ cm}^2
$$

✔ Answer for 4): 576 cm²

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✔ Final Answers:

| Problem | Surface Area |
|--------|--------------|
| 1) | 54 cm² |
| 2) | 248 in² |
| 3) | 542 cm² |
| 4) | 576 cm² |

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