Let's solve the problem step by step using the given diagram and information.

Given:

• The pyramid has a square base and congruent triangular lateral faces.
• The side of the square base is 6 mm.
• The slant height of the triangular face is 2.3 mm.

(a) Find the area of the base of the pyramid.

The base is a square, and the area $A$ of a square is given by: $A = \text{side}^2$ Here, the side of the square is $6$ mm. $A = 6^2 = 36 \text{ mm}^2$ So, the area of the base is 36 mm².

(b) Find the area of one lateral face of the pyramid.

Each lateral face is a triangle, and the area $A_{\text{triangle}}$ of a triangle is given by: $A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}$ The base of the triangular face is the same as the side of the square base, which is 6 mm, and the height of the triangular face (slant height) is 2.3 mm. $A_{\text{triangle}} = \frac{1}{2} \times 6 \times 2.3 = \frac{1}{2} \times 13.8 = 6.9 \text{ mm}^2$ So, the area of one lateral face is 6.9 mm².

(c) Find the lateral surface area of the pyramid.

The lateral surface area includes only the four triangular faces. Since all the triangular faces are congruent, we can multiply the area of one triangular face by 4: $A_{\text{lateral}} = 4 \times 6.9 = 27.6 \text{ mm}^2$ So, the lateral surface area is 27.6 mm².

(d) Find the total surface area of the pyramid.

The total surface area is the sum of the lateral surface area and the area of the base: $A_{\text{total}} = A_{\text{lateral}} + A_{\text{base}} = 27.6 + 36 = 63.6 \text{ mm}^2$ So, the total surface area is 63.6 mm².

Summary:

• (a) Area of the base: $36 \, \text{mm}^2$
• (b) Area of one lateral face: $6.9 \, \text{mm}^2$
• (c) Lateral surface area: $27.6 \, \text{mm}^2$
• (d) Total surface area: $63.6 \, \text{mm}^2$

Would you like further details on any of the calculations or concepts?

Here are some related questions:

1. How would the surface area change if the side length of the base doubled?
2. What is the height of the pyramid from the center of the base to the apex?
3. How would the surface area change if only the slant height increased?
4. Can you find the volume of the pyramid given the base and height?
5. What is the ratio of the lateral surface area to the base area?

Tip: When solving for areas involving triangles, always ensure you're using the correct dimensions for the base and height specific to that triangle.