This is a square based pyramid. The angle between the sloping edge and the plane ABCD is 55°. This means the angle DBV is 55°. The square base length is 20cm. Calculate the surface area of the pyramid.

To calculate the surface area of a square-based pyramid, we need to find both the area of the base and the area of the four triangular faces.

### Step-by-Step Solution

1. **Determine the Area of the Base:**

   The base of the pyramid is a square with a side length of 20 cm.

   \[
   \text{Area of the base} = \text{side}^2 = 20^2 = 400 \, \text{cm}^2
   \]

2. **Find the Height of the Triangular Faces (Slant Height):**

   The angle between the sloping edge \( BV \) and the base plane \( ABCD \) is given as 55°. Let's use triangle \( DBV \) to find the slant height \( BV \). Here, \( DB \) is the diagonal of the square base.

   - Calculate the length of the diagonal \( DB \) of the square base:

   \[
   DB = \sqrt{20^2 + 20^2} = \sqrt{400 + 400} = \sqrt{800} = 20\sqrt{2} \, \text{cm}
   \]

   - Since angle \( DBV \) is 55°, we can use trigonometry (cosine) to find \( BV \) (the slant height):

   \[
   \cos(55°) = \frac{DB / 2}{BV}
   \]

   Rearranging to solve for \( BV \):

   \[
   BV = \frac{DB / 2}{\cos(55°)} = \frac{20\sqrt{2} / 2}{\cos(55°)} = \frac{10\sqrt{2}}{\cos(55°)}
   \]

   - Calculate \( BV \) using \(\cos(55°) \approx 0.5736\):

   \[
   BV = \frac{10\sqrt{2}}{0.5736} \approx \frac{14.14}{0.5736} \approx 24.66 \, \text{cm}
   \]

3. **Calculate the Area of One Triangular Face:**

   Each triangular face of the pyramid has a base length equal to the side of the square and a height equal to the slant height \( BV \).

   \[
   \text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 20 \times 24.66 = 10 \times 24.66 = 246.6 \, \text{cm}^2
   \]

4. **Calculate the Total Surface Area of the Pyramid:**

   The total surface area includes the area of the base and the areas of the four triangular faces.

   \[
   \text{Total surface area} = \text{Area of the base} + 4 \times \text{Area of one triangular face}
   \]

   \[
   \text{Total surface area} = 400 + 4 \times 246.6 = 400 + 986.4 = 1386.4 \, \text{cm}^2
   \]

### Final Answer:

The surface area of the pyramid is **1386.4 cm²**.

Would you like more details on any part of the solution, or do you have any questions?

Here are some related questions to consider:
1. How do you find the height of the pyramid if only given the slant height and the angle?
2. What is the volume of this pyramid?
3. How does the surface area change if the angle between the sloping edge and the base changes?
4. How would you calculate the surface area if the base were not a square but a rectangle?
5. What is the effect of changing the base side length on the surface area?

**Tip:** When solving problems involving pyramids, always determine whether you need the height or the slant height, as they are different and affect different calculations.

Learn how to calculate the surface area of a square-based pyramid with a 55° sloping edge angle. This problem involves finding the area of the base square and four triangular faces using trigonometric functions. Suitable for high school students.

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