Question

Find the surface area of the triangular prism. In order to receive full credit, you must show your work and label your answer with the appropriate units.

Surface Area of Triangular Prisms | CK-12 Foundation

Answer 1

To find the surface area of a triangular prism, we need to calculate the area of the two triangular bases and the three rectangular lateral surfaces.

### Step 1: Calculate the Area of the Triangular Bases

Let's assume:
- The base of the triangle is \( b \)
- The height of the triangle is \( h \)

The area \( A_{triangle} \) of one triangular base is given by the formula:

\[
A_{triangle} = \frac{1}{2} \times b \times h
\]

Since there are two triangular bases, the total area of both bases is:

\[
A_{bases} = 2 \times A_{triangle} = 2 \times \left(\frac{1}{2} \times b \times h\right) = b \times h
\]

### Step 2: Calculate the Area of the Rectangular Lateral Surfaces

Assuming the length of the prism is \( L \), we have three rectangular lateral faces:
1. Rectangle corresponding to the base \( b \)
2. Rectangle corresponding to side \( a \) (the other two sides of the triangle would need their lengths denoted as \( a \) and \( c \) respectively)
3. Rectangle corresponding to side \( c \)

The areas of the rectangles are calculated as follows:
- Area of the rectangle with base \( b \):
\[
A_{b} = b \times L
\]
- Area of the rectangle with side \( a \):
\[
A_{a} = a \times L
\]
- Area of the rectangle with side \( c \):
\[
A_{c} = c \times L
\]

### Step 3: Calculate the Total Surface Area

The total surface area \( A_{total} \) of the prism is the sum of the areas of the bases and the lateral faces:

\[
A_{total} = A_{bases} + A_{b} + A_{a} + A_{c}
\]

Substituting in what we calculated:

\[
A_{total} = (b \times h) + (b \times L) + (a \times L) + (c \times L)
\]

\[
A_{total} = b \times h + L \times (b + a + c)
\]

### Final Units and Example Calculation

If we were given specific values, for example:
- \( b = 5 \) units
- \( h = 4 \) units
- \( a = 3 \) units
- \( c = 6 \) units
- \( L = 10 \) units

We can substitute these values into our equation:

**Area of the Bases:**
\[
A_{bases} = 5 \times 4 = 20 \text{ square units}
\]

**Area of the Rectangles:**
\[
A_{b} = 5 \times 10 = 50 \text{ square units}
\]
\[
A_{a} = 3 \times 10 = 30 \text{ square units}
\]
\[
A_{c} = 6 \times 10 = 60 \text{ square units}
\]

**Total Surface Area:**
\[
A_{total} = 20 + 50 + 30 + 60 = 160 \text{ square units}
\]

### Conclusion
Thus, the surface area of the triangular prism is

\[
\boxed{160 \text{ square units}}
\]