Surface area calculation for a triangular prism with given dimensions.

To find the surface area of a triangular prism, we need to calculate the areas of all its faces and add them together.

A triangular prism has:
- Two triangular bases
- Three rectangular lateral faces

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Step 1: Identify the dimensions

From the image:
- The triangle base has:
- Base = 6 mm
- Height = 4 mm (this is the height of the triangle, shown as a dashed line)
- The length of the prism (the distance between the two triangular bases) = 9 mm
- The side of the triangle that is 5 mm appears to be one of the other sides of the triangle (likely the hypotenuse or another leg).

Let’s assume the triangle is a right triangle since there's a right angle symbol. So:
- Legs: 6 mm and 4 mm
- Hypotenuse: given as 5 mm? Wait — let’s check if this makes sense.

Wait! If the triangle has legs 6 mm and 4 mm, then by the Pythagorean theorem:

$$
\text{Hypotenuse} = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} \approx 7.21 \text{ mm}
$$

But in the diagram, it says 5 mm on the slanted side. That suggests that maybe the triangle is not a right triangle with legs 6 and 4?

Wait — actually, look carefully:

- The height of the triangle is 4 mm, drawn perpendicular from the 6 mm base.
- The side of the triangle opposite the height is labeled 5 mm — so that might be the slant side.

So the triangle has:
- Base = 6 mm
- Height = 4 mm (perpendicular to the base)
- One of the other sides = 5 mm

We can use this to confirm the triangle is valid.

Let’s suppose the triangle has:
- Base = 6 mm
- Height = 4 mm (from the apex to the base)
- Then, the two other sides can be found using Pythagoras.

But we are told one of the sides is 5 mm.

Let’s suppose the triangle is split into two right triangles by the height.

So, the height (4 mm) splits the base (6 mm) into two parts. Let’s say it splits it into $ x $ and $ 6 - x $. Then:

For each right triangle:
$$
x^2 + 4^2 = 5^2 \Rightarrow x^2 + 16 = 25 \Rightarrow x^2 = 9 \Rightarrow x = 3
$$

So yes, the height divides the base into two equal parts of 3 mm each. So the triangle is isosceles with two sides of 5 mm and base 6 mm, and height 4 mm.

Perfect!

So now we know:
- Triangle: base = 6 mm, height = 4 mm
- Prism length = 9 mm

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Step 2: Area of the two triangular bases

Area of one triangle:
$$
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12 \text{ mm}^2
$$

Two bases:
$$
2 \times 12 = 24 \text{ mm}^2
$$

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Step 3: Areas of the three rectangular faces

The three rectangles correspond to the three sides of the triangle, each extended along the length of the prism (9 mm).

So:
1. Rectangle with width = 6 mm (base), length = 9 mm → area = $ 6 \times 9 = 54 $
2. Rectangle with width = 5 mm, length = 9 mm → area = $ 5 \times 9 = 45 $
3. Another rectangle with width = 5 mm, length = 9 mm → area = $ 5 \times 9 = 45 $

Wait — but we have two sides of 5 mm, so yes, two identical rectangles.

So total lateral surface area:
$$
54 + 45 + 45 = 144 \text{ mm}^2
$$

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Step 4: Total Surface Area

Add the two triangular bases and the three rectangular faces:

$$
\text{Total SA} = 24 + 144 = 168 \text{ square millimeters}
$$

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✔ Final Answer:

168 square millimeters

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✔ Answer: `168` square millimeters