To find the total surface area of this prism, we need to calculate the area of each of its faces and sum them up.

Step 1: Calculate the Area of the Triangular Faces

The prism has two triangular faces (front and back), which are isosceles triangles with a base of $24 \, \text{cm}$ and height of $5 \, \text{cm}$.

The area of one triangular face is: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 24 \times 5 = 60 \, \text{cm}^2$ Since there are two triangular faces: $\text{Total area of triangular faces} = 2 \times 60 = 120 \, \text{cm}^2$

Step 2: Calculate the Area of the Rectangular Faces

The prism has three rectangular faces along its length (the 24 cm side).

1. First Rectangle (left side):

   • Dimensions: $13 \, \text{cm} \times 24 \, \text{cm}$
   • Area = $13 \times 24 = 312 \, \text{cm}^2$

2. Second Rectangle (bottom):

   • Dimensions: $24 \, \text{cm} \times 10 \, \text{cm}$
   • Area = $24 \times 10 = 240 \, \text{cm}^2$

3. Third Rectangle (right side):

   • Dimensions: $13 \, \text{cm} \times 24 \, \text{cm}$
   • Area = $13 \times 24 = 312 \, \text{cm}^2$

Total area of the rectangular faces: $312 + 240 + 312 = 864 \, \text{cm}^2$

Step 3: Add All Areas Together

$\text{Total Surface Area} = 120 + 864 = 984 \, \text{cm}^2$

Final Answer

The total surface area of the prism is: $\boxed{984 \, \text{cm}^2}$

Would you like a further breakdown or have any questions?

1. How do you calculate the surface area of a different shaped prism?
2. How would the calculation change if the height of the triangle was different?
3. How does the surface area formula apply to other 3D shapes like cylinders?
4. Can we apply a similar method for volume calculation of this prism?
5. How would you find the surface area if the cross-section was not isosceles?

Tip: Always check if a 3D shape has symmetrical faces to simplify surface area calculations.