Let's break down the problem step by step using the information provided in the image.

Given:

• The base is an equilateral triangle with side length $s = 5 \, \text{mm}$.
• The height of the pyramid (slant height of the lateral face) is $6 \, \text{mm}$.
• The height of the triangular base (height of the equilateral triangle) is $4.3 \, \text{mm}$.

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

The base is an equilateral triangle. The formula to calculate the area of an equilateral triangle is:

$A_{\text{base}} = \frac{s^2 \sqrt{3}}{4}$

Substitute $s = 5 \, \text{mm}$:

$A_{\text{base}} = \frac{5^2 \sqrt{3}}{4} = \frac{25 \sqrt{3}}{4} \approx 10.83 \, \text{mm}^2$

So, the area of the base is approximately 10.83 mm².

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

The lateral faces are congruent triangles with base $s = 5 \, \text{mm}$ and height $6 \, \text{mm}$ (slant height). The area of a triangle is:

$A_{\text{lateral}} = \frac{1}{2} \times \text{base} \times \text{height}$

Substitute the values:

$A_{\text{lateral}} = \frac{1}{2} \times 5 \times 6 = 15 \, \text{mm}^2$

So, the area of one lateral face is 15 mm².

(c) Use the net to find the lateral surface area of the pyramid (without the base).

There are three lateral faces, all of which are congruent. The total lateral surface area is:

$A_{\text{lateral total}} = 3 \times A_{\text{lateral}} = 3 \times 15 = 45 \, \text{mm}^2$

So, the lateral surface area of the pyramid is 45 mm².

(d) Use the net to find the total surface area of the pyramid.

The total surface area includes the lateral surface area and the base area:

$A_{\text{total}} = A_{\text{lateral total}} + A_{\text{base}} = 45 + 10.83 \approx 55.83 \, \text{mm}^2$

So, the total surface area of the pyramid is approximately 55.83 mm².

Do you need further details or explanations on any of these steps?

Here are 5 related questions:

1. How would the surface area change if the base side length increased?
2. What would the lateral surface area be if the slant height increased?
3. Can you calculate the volume of this pyramid?
4. How do you find the height of the equilateral triangle using its side length?
5. What if the pyramid's base was a different shape (e.g., square)?

Tip: When dealing with geometric shapes, always double-check the formulas for area and volume based on the shape’s properties (e.g., equilateral triangle, square, etc.).