To find the surface area of the given triangular pyramid, we must calculate the area of each face (triangular and rectangular) and then sum them up.

1. Identify the faces:

• 1 rectangular base with dimensions $12 \, \text{cm} \times 12 \, \text{cm}$.
• 4 triangular faces, but note that two faces are congruent.

2. Calculate the area of the rectangular base:

$\text{Area of base} = \text{Length} \times \text{Width} = 12 \, \text{cm} \times 12 \, \text{cm} = 144 \, \text{cm}^2.$

3. Calculate the triangular face areas:

We need to calculate the areas of the triangular faces using the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}.$

a) Two congruent triangular faces (front and back):

Here, the base of the triangle is $12 \, \text{cm}$ and the height is $9 \, \text{cm}$: $\text{Area of each face} = \frac{1}{2} \times 12 \, \text{cm} \times 9 \, \text{cm} = 54 \, \text{cm}^2.$ For two such faces: $\text{Total area} = 2 \times 54 \, \text{cm}^2 = 108 \, \text{cm}^2.$

b) Two triangular side faces (slanted faces):

The slanted faces are triangles, each with a base of $12 \, \text{cm}$. To find the height of these triangles, we need to calculate the slant height.

From the diagram:

• The slant height forms a right triangle with the height $9 \, \text{cm}$ and half the base of $12 \, \text{cm}$ (which is $6 \, \text{cm}$).

Using the Pythagorean theorem: $\text{Slant height} = \sqrt{(9 \, \text{cm})^2 + (6 \, \text{cm})^2}.$ $\text{Slant height} = \sqrt{81 + 36} = \sqrt{117} \approx 10.8 \, \text{cm}.$

Now calculate the area of one slanted triangle: $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Slant height}.$ $\text{Area} = \frac{1}{2} \times 12 \, \text{cm} \times 10.8 \, \text{cm} = 64.8 \, \text{cm}^2.$ For two such faces: $\text{Total area} = 2 \times 64.8 \, \text{cm}^2 = 129.6 \, \text{cm}^2.$

4. Total surface area:

Now add up all the areas: $\text{Surface Area} = \text{Base area} + \text{Two front/back triangles} + \text{Two side triangles}.$ $\text{Surface Area} = 144 \, \text{cm}^2 + 108 \, \text{cm}^2 + 129.6 \, \text{cm}^2.$ $\text{Surface Area} = 381.6 \, \text{cm}^2.$

Final Answer:

The surface area of the pyramid is 381.6 cm².

Questions for Further Exploration:

1. How do you calculate the surface area of a pyramid with a different base shape?
2. How would the answer change if the pyramid had a pentagonal base?
3. What is the volume of this pyramid if given the height from the base to the apex?
4. How do you find the slant height of a triangular face in any pyramid?
5. Can you use trigonometry to verify the slant height instead of the Pythagorean theorem?

Tip: Always break down a complex 3D surface into simple 2D shapes when calculating surface area.