https://file.aiquickdraw.com/mathbot/file/1726680798072hwinsbyn.jpg

To calculate the surface area of the triangular prism in the image, we need to calculate the areas of all the faces: two triangular faces and three rectangular faces.

### 1. **Area of the triangular face:**
The triangle is a right-angled triangle, so we can use the formula for the area of a triangle:

\[
\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]

Here:
- The base = 27 mm
- The height = 18 mm

\[
\text{Area of triangle} = \frac{1}{2} \times 27 \times 18 = 243 \, \text{mm}^2
\]

Since there are two triangular faces, the total area for the triangular faces is:

\[
\text{Total area of triangular faces} = 2 \times 243 = 486 \, \text{mm}^2
\]

### 2. **Area of the three rectangular faces:**

Each rectangular face is determined by the sides of the triangular face and the length of the prism (21 mm).

- **First rectangle (base 27 mm × length 21 mm):**

\[
\text{Area} = 27 \times 21 = 567 \, \text{mm}^2
\]

- **Second rectangle (height 18 mm × length 21 mm):**

\[
\text{Area} = 18 \times 21 = 378 \, \text{mm}^2
\]

- **Third rectangle (hypotenuse 9 mm × length 21 mm):**

The hypotenuse is given as 9 mm, so:

\[
\text{Area} = 9 \times 21 = 189 \, \text{mm}^2
\]

### 3. **Total surface area of the prism:**
Adding all the areas together:

\[
\text{Total surface area} = 486 \, (\text{triangular faces}) + 567 \, (\text{first rectangle}) + 378 \, (\text{second rectangle}) + 189 \, (\text{third rectangle})
\]

\[
\text{Total surface area} = 486 + 567 + 378 + 189 = 1620 \, \text{mm}^2
\]

Thus, the total surface area of the triangular prism is **1620 mm²**.

Would you like further details or have any questions?

Here are 5 related questions you might want to explore:
1. How would the surface area change if the length of the prism was doubled?
2. What is the volume of this triangular prism?
3. How can we calculate the lateral surface area (only the rectangular faces) of the prism?
4. How does the hypotenuse influence the surface area calculation?
5. What are some practical applications of surface area calculations for prisms?

**Tip:** When dealing with prisms, always break the surface area into individual face areas, especially when shapes are non-uniform!

Calculate the surface area of the triangular prism below.

Learn how to calculate the surface area of a triangular prism using a right-angled triangle base. This detailed solution breaks down the calculations for each face of the prism. Ideal for students in Grades 6-8 studying geometry.

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