Surface area of triangular prisms worksheet with four problems requiring calculation of surface area using given dimensions.

Here are the step-by-step solutions for finding the surface area of each triangular prism.

To find the total surface area, we need to calculate the area of every flat shape in the "net" (the unfolded pattern) and add them all together. Each prism has:
1. Two identical triangles (the bases).
2. Three rectangles (the sides). Note that two of these rectangles are usually identical.

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Problem 1

Step 1: Find the area of the two triangles.
* The base of the triangle is $3\text{ cm}$.
* The height of the triangle is $2\text{ cm}$.
* Area of one triangle = $\frac{1}{2} \times \text{base} \times \text{height}$
$$0.5 \times 3 \times 2 = 3\text{ cm}^2$$
* Since there are two triangles: $3 \times 2 = \mathbf{6\text{ cm}^2}$

Step 2: Find the area of the three rectangles.
* Middle Rectangle: Length is $6\text{ cm}$, width is $3\text{ cm}$.
$$6 \times 3 = 18\text{ cm}^2$$
* Top and Bottom Rectangles: These are identical. Length is $6\text{ cm}$, width is $2.5\text{ cm}$.
$$6 \times 2.5 = 15\text{ cm}^2$$
Since there are two of them: $15 \times 2 = \mathbf{30\text{ cm}^2}$

Step 3: Add everything together.
$$6 + 18 + 30 = 54$$

Surface Area = $54\text{ cm}^2$

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Problem 2

Step 1: Find the area of the two triangles.
* Base = $6\text{ in}$, Height = $4\text{ in}$.
* Area of one triangle = $0.5 \times 6 \times 4 = 12\text{ in}^2$
* Two triangles: $12 \times 2 = \mathbf{24\text{ in}^2}$

Step 2: Find the area of the three rectangles.
* Middle Rectangle: Length is $14\text{ in}$, width is $6\text{ in}$.
$$14 \times 6 = 84\text{ in}^2$$
* Side Rectangles: These are identical. Length is $14\text{ in}$, width is $5\text{ in}$.
$$14 \times 5 = 70\text{ in}^2$$
Since there are two of them: $70 \times 2 = \mathbf{140\text{ in}^2}$

Step 3: Add everything together.
$$24 + 84 + 140 = 248$$

Surface Area = $248\text{ in}^2$

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Problem 3

Step 1: Find the area of the two triangles.
* Base = $10\text{ cm}$, Height = $12\text{ cm}$.
* Area of one triangle = $0.5 \times 10 \times 12 = 60\text{ cm}^2$
* Two triangles: $60 \times 2 = \mathbf{120\text{ cm}^2}$

Step 2: Find the area of the three rectangles.
* Middle Rectangle: Length is $13\text{ cm}$, width is $10\text{ cm}$.
$$13 \times 10 = 130\text{ cm}^2$$
* Side Rectangles: These are identical. Length is $13\text{ cm}$, width is $10\text{ cm}$ (based on the side length labeled at the top).
$$13 \times 10 = 130\text{ cm}^2$$
Since there are two of them: $130 \times 2 = \mathbf{260\text{ cm}^2}$

Step 3: Add everything together.
$$120 + 130 + 260 = 510$$

Surface Area = $510\text{ cm}^2$

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Problem 4

Step 1: Find the area of the two triangles.
* Base = $12\text{ cm}$, Height = $8\text{ cm}$.
* Area of one triangle = $0.5 \times 12 \times 8 = 48\text{ cm}^2$
* Two triangles: $48 \times 2 = \mathbf{96\text{ cm}^2}$

Step 2: Find the area of the three rectangles.
* Middle Rectangle: Length is $15\text{ cm}$, width is $12\text{ cm}$.
$$15 \times 12 = 180\text{ cm}^2$$
* Side Rectangles: These are identical. Length is $15\text{ cm}$, width is $10\text{ cm}$.
$$15 \times 10 = 150\text{ cm}^2$$
Since there are two of them: $150 \times 2 = \mathbf{300\text{ cm}^2}$

Step 3: Add everything together.
$$96 + 180 + 300 = 576$$

Surface Area = $576\text{ cm}^2$

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Final Answer:
1) 54 cm²
2) 248 in²
3) 510 cm²
4) 576 cm²