Surface area of triangular prism, height is 5cm, lenth is 12cm, width is 10cm, and the other face is 13cm

To find the surface area of a triangular prism, we need to calculate the area of both the triangular bases and the rectangular faces.

Given:
- Height of the triangular face (\(h\)) = 5 cm
- Length of the prism (\(L\)) = 12 cm
- Width of the triangular base (\(w\)) = 10 cm
- The other side of the triangular face (\(s\)) = 13 cm

### Step-by-step solution:

1. **Area of the two triangular bases**: 
   The area of a triangle is given by the formula:
   \[
   \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}
   \]
   For the triangular base:
   \[
   A_{\text{triangle}} = \frac{1}{2} \times 10 \times 5 = 25 \, \text{cm}^2
   \]
   Since there are two triangular bases, the total area of both triangular bases is:
   \[
   2 \times A_{\text{triangle}} = 2 \times 25 = 50 \, \text{cm}^2
   \]

2. **Area of the three rectangular faces**:
   - The first rectangular face has dimensions \(L\) and the base of the triangle (\(w\)):
     \[
     A_1 = L \times w = 12 \times 10 = 120 \, \text{cm}^2
     \]
   - The second rectangular face has dimensions \(L\) and the height of the triangle (\(h\)):
     \[
     A_2 = L \times h = 12 \times 5 = 60 \, \text{cm}^2
     \]
   - The third rectangular face has dimensions \(L\) and the other side of the triangle (\(s\)):
     \[
     A_3 = L \times s = 12 \times 13 = 156 \, \text{cm}^2
     \]

3. **Total surface area of the triangular prism**:
   The total surface area is the sum of the areas of both triangular bases and the three rectangular faces:
   \[
   \text{Surface Area} = 2 \times A_{\text{triangle}} + A_1 + A_2 + A_3
   \]
   Substituting the values:
   \[
   \text{Surface Area} = 50 + 120 + 60 + 156 = 386 \, \text{cm}^2
   \]

Thus, the surface area of the triangular prism is **386 cm²**.

Would you like further details or have any questions? 

Here are 5 related questions to expand your understanding:
1. How does the surface area change if the length of the prism increases?
2. What if the triangular face was equilateral, how would the surface area change?
3. How would you calculate the volume of the same triangular prism?
4. How does the surface area of a rectangular prism compare to a triangular prism of the same height and length?
5. How can you calculate the surface area of more complex prisms?

**Tip**: Always double-check the dimensions to ensure you're using the correct ones for each face calculation.

Surface area of triangular prism, height is 5cm, length is 12cm, width is 10cm, and the other face is 13cm

Discover how to calculate the surface area of a triangular prism with a height of 5cm, a length of 12cm, and a base of 10cm. This step-by-step guide covers key geometry concepts and formulas suitable for Grades 6-8 students.

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