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Surface area of cubes with one rectangular prism on top

The image shows a structure composed of a cube with a rectangular prism (in the shape of a triangular prism on top, resembling a house).

### To calculate the surface area of this figure:

#### Step 1: Surface area of the cube
The cube has sides of length 10 cm.

- The surface area \( A_{\text{cube}} \) of a cube is calculated using the formula:
\[
A_{\text{cube}} = 6 \times \text{side}^2 = 6 \times 10^2 = 600 \, \text{cm}^2
\]
This is the total surface area of the cube if it were by itself.

#### Step 2: Surface area of the triangular prism on top
The top portion is a rectangular prism that has two slanted triangular faces and three rectangular faces. Let’s calculate each component:

1. **Area of the triangular faces**: 
   The height of the triangle is given as 12 cm, and the base is 10 cm (same as the side of the cube).
   
   The area of one triangle is:
   \[
   A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 12 = 60 \, \text{cm}^2
   \]
   There are two triangular faces, so the total area for both triangles is:
   \[
   2 \times 60 = 120 \, \text{cm}^2
   \]

2. **Area of the rectangular sides**:
   - The base of the prism has the same length as the side of the cube, which is 10 cm, and the height of the rectangular face equals the slant height of the triangle. Using Pythagoras' theorem:
   \[
   \text{Slant height} = \sqrt{(10/2)^2 + 12^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \, \text{cm}
   \]
   - The rectangular faces have areas: 
     - Two side faces: \( 13 \times 10 = 130 \, \text{cm}^2 \), so for both sides: \( 2 \times 130 = 260 \, \text{cm}^2 \).
     - One bottom face (which lies on top of the cube): \( 10 \times 10 = 100 \, \text{cm}^2 \) (but this part is already counted within the surface area of the cube and should be subtracted later).

#### Step 3: Total surface area
Now, we sum the areas of all faces, but subtract the overlapping area between the prism and cube (which is the top face of the cube, i.e., 100 cm²).

\[
A_{\text{total}} = 600 \, (\text{cube}) + 120 \, (\text{triangles}) + 260 \, (\text{rectangles}) - 100 \, (\text{overlap}) = 880 \, \text{cm}^2
\]

Thus, the total surface area of the combined figure is **880 cm²**.

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Would you like more details, or do you have any specific questions?

Here are 5 related questions you can explore:
1. What would the volume of the figure be?
2. How does the surface area change if the dimensions of the cube increase?
3. What happens to the surface area if the height of the triangular prism changes?
4. How is surface area calculated for a general rectangular prism?
5. Can you derive a formula to calculate the surface area of any cube-based structure?

**Tip:** Always subtract the overlapping areas between connected shapes when calculating the surface area of composite figures.

Learn how to calculate the surface area of a composite figure made up of a cube and a triangular prism on top. This problem involves key geometry concepts, including surface area formulas and Pythagoras' theorem. Suitable for students in Grades 6-8.

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