Let's solve the problem described in the uploaded image step by step.

Problem Breakdown:

• Given:
  • The lateral faces of a square pyramid are equilateral triangles with side length $10 \, \text{cm}$.
  • The pyramid is square-based.

(a) What is the slant height of the pyramid?

• In a square pyramid, the slant height is the perpendicular height of the triangular lateral faces from the base edge to the apex.

Since the lateral face is an equilateral triangle, the slant height is the altitude of this triangle.

For an equilateral triangle with side $a$, its altitude $h$ is given by:

$h = \frac{\sqrt{3}}{2} a$

Substituting $a = 10 \, \text{cm}$:

$h = \frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3} \approx 8.66 \, \text{cm}.$

Thus, the slant height of the pyramid is $8.66 \, \text{cm}$.

(b) What is the lateral surface area and total surface area?

1. Lateral Surface Area (LSA):

   • A square pyramid has 4 triangular lateral faces.
   • The area of one triangular face is:

   $A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{altitude}$

   Using $\text{base} = 10 \, \text{cm}$ and $\text{altitude} = 8.66 \, \text{cm}$:

   $A_{\text{triangle}} = \frac{1}{2} \times 10 \times 8.66 = 43.3 \, \text{cm}^2$

   For 4 triangular faces:

   $\text{LSA} = 4 \times 43.3 = 173.2 \, \text{cm}^2$

2. Total Surface Area (TSA):

   • The total surface area includes the lateral surface area and the base area.
   • The base of the pyramid is a square with side $10 \, \text{cm}$, so its area is:

   $A_{\text{base}} = \text{side}^2 = 10^2 = 100 \, \text{cm}^2$

   Thus, the total surface area is:

   $\text{TSA} = \text{LSA} + \text{Base Area} = 173.2 + 100 = 273.2 \, \text{cm}^2$

Final Answers:

(a) Slant height of the pyramid: $8.66 \, \text{cm}$

(b) Lateral surface area: $173.2 \, \text{cm}^2$, Total surface area: $273.2 \, \text{cm}^2$

Would you like further explanation or a breakdown of the steps?

Related Questions:

1. What is the formula for the altitude of an equilateral triangle?
2. How do you calculate the volume of a square pyramid with the given slant height?
3. If the side of the base changes, how does it affect the lateral surface area?
4. Can you derive the formula for the total surface area of a square pyramid?
5. What happens if the lateral faces are not equilateral triangles?

Tip:

Always confirm the type of triangle (equilateral, isosceles, etc.) when working on geometry problems to ensure accurate altitude calculations.