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To calculate the surface area of this triangular prism, let's break the problem down step by step:

1. **Identify the faces of the prism**:
   - The prism has two triangular faces (front and back), which are right-angled triangles.
   - There are three rectangular faces that connect these triangular faces: one from the base, one from the height, and one from the hypotenuse.

### Step 1: Calculate the area of the triangular faces
Each triangular face is a right-angled triangle, where:
- The base = 27 mm
- The height = 6 mm

The area of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
\[
A = \frac{1}{2} \times 27 \, \text{mm} \times 6 \, \text{mm} = 81 \, \text{mm}^2
\]
Since there are two triangular faces, the total area of the triangular faces is:
\[
2 \times 81 = 162 \, \text{mm}^2
\]

### Step 2: Calculate the areas of the rectangular faces
There are three rectangular faces:
1. **Rectangle 1** (along the base of the triangle): 
   - Length = 36 mm (the depth of the prism)
   - Width = 27 mm (the base of the triangle)
   \[
   A_1 = 36 \, \text{mm} \times 27 \, \text{mm} = 972 \, \text{mm}^2
   \]

2. **Rectangle 2** (along the height of the triangle):
   - Length = 36 mm (the depth)
   - Width = 6 mm (the height of the triangle)
   \[
   A_2 = 36 \, \text{mm} \times 6 \, \text{mm} = 216 \, \text{mm}^2
   \]

3. **Rectangle 3** (along the hypotenuse):
   First, calculate the hypotenuse using the Pythagorean theorem:
   \[
   \text{Hypotenuse} = \sqrt{(27 \, \text{mm})^2 + (6 \, \text{mm})^2} = \sqrt{729 + 36} = \sqrt{765} \approx 27.66 \, \text{mm}
   \]
   Now, the area of this rectangle is:
   \[
   A_3 = 36 \, \text{mm} \times 27.66 \, \text{mm} \approx 995.76 \, \text{mm}^2
   \]

### Step 3: Add up all the areas
Now, sum the areas of all the faces:
\[
\text{Total surface area} = 162 \, \text{mm}^2 + 972 \, \text{mm}^2 + 216 \, \text{mm}^2 + 995.76 \, \text{mm}^2
\]
\[
\text{Total surface area} \approx 2345.8 \, \text{mm}^2
\]

Thus, the surface area of the prism is approximately **2345.8 mm²**.

Would you like further clarification or details on any part of the solution? Here are some related questions to expand on this:

1. How do you calculate the surface area for prisms with different shapes (e.g., trapezoidal)?
2. Can you explain how the Pythagorean theorem applies to calculating the hypotenuse?
3. How would the surface area change if the depth of the prism increased?
4. What is the difference between lateral surface area and total surface area?
5. How does surface area relate to volume for this type of prism?

**Tip**: Always double-check the units when working with surface area and volume problems, as this can affect the final result!

Calculate the surface area of a right-angled triangular prism with the following dimensions: base = 27 mm, height = 6 mm, depth = 36 mm.

Learn how to calculate the surface area of a right-angled triangular prism with dimensions of base 27 mm, height 6 mm, and depth 36 mm. This problem involves applying the Pythagorean theorem, calculating triangle areas, and finding the surface area of rectangular faces. Suitable for Grades 9-10.

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