https://file.aiquickdraw.com/mathbot/file/1729881222906cvk8oafp.jpg

Let's break down the problem based on the image you uploaded, which contains two parts related to a cube and a pyramid formed inside the cube.

The cube \(ABCDEFGH\) has an edge length of 6 cm.

### Part (a)
We need to find the surface area of the pyramid \(ABCDE\). The pyramid's base is the square \(ABCD\), and its apex is point \(E\).

1. **Base Area**: 
   Since the base is the square \(ABCD\), with side length \(AB = 6 \, \text{cm}\), the area of the base is:
   \[
   A_{\text{base}} = AB^2 = 6^2 = 36 \, \text{cm}^2
   \]

2. **Slant Heights**:
   We need to calculate the areas of the triangular sides \(ABE\), \(BCE\), \(CDE\), and \(DAE\).

   For each triangular face, we use the base and the slant height to find the area. First, calculate the slant height \(AE\), which is the diagonal from point \(A\) to point \(E\):
   \[
   AE = \sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2} \, \text{cm}
   \]
   Each triangular face shares an edge of the square base and a slant height that we need to compute.

   - Triangular face \(ABE\): Area \(= \frac{1}{2} \times AB \times AE = \frac{1}{2} \times 6 \times 6\sqrt{2}\)
   - Similar calculations apply for other triangular faces.

### Part (b)
We need to express the vector \(\overrightarrow{EC}\) using the vectors \(\overrightarrow{AB}\), \(\overrightarrow{AD}\), and \(\overrightarrow{AE}\).

1. Identify the positions of the points:
   - \( \overrightarrow{AB} \) represents the vector along one edge.
   - \( \overrightarrow{AD} \) represents another edge of the base.
   - \( \overrightarrow{AE} \) represents the vector from the base to the apex.

2. Use vector addition or linear combinations to express \( \overrightarrow{EC} \) in terms of the given vectors.

Would you like me to expand the explanation further or provide more details on the specific steps for part (b)?

Here are five related questions you could explore:
1. How do you calculate the height of the pyramid if needed?
2. How can you verify that the diagonals in a square are perpendicular?
3. What is the formula for calculating the volume of this pyramid?
4. How would the surface area change if the cube's side length was different?
5. Can you express other vectors (like \( \overrightarrow{EB} \)) in terms of the base vectors?

**Tip**: For vector problems, always sketch the vectors in the coordinate plane to better visualize their directions and relationships.

Az ABCDEFGH kocka élhossza 6 cm. a) Számítsa ki az ábrán látható ABCDE gúla felszínét! b) Fejezze ki az EC vektort az AB, az AD és az AE vektorok segítségével!

Calculate the surface area of the pyramid ABCDE inside a cube with a 6 cm edge. Learn how to express vector EC using the vectors AB, AD, and AE. This problem involves geometric and vector algebra concepts, ideal for students in Grades 9-12.

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