What is the surface area and lateral area of a regular triangular pyramid if the slant is 7 and the altitude is 5?

To find the surface area and lateral area of a regular triangular pyramid, we will need to use some basic formulas.

First, let's define the given values:
- Slant (s) = 7
- Altitude (h) = 5

The surface area of a regular triangular pyramid consists of two parts: the base area and the lateral area.

The lateral area is the sum of the areas of the four triangular faces of the pyramid.

The formula for the lateral area (LA) of a regular triangular pyramid is given by LA = (1/2) * perimeter * slant.

To find the perimeter of the base triangle, we need to know the length of one side. Since the pyramid is regular, all sides are of equal length.

From geometry, we know that the altitude of an equilateral triangle divides the base into two equal segments, creating a right triangle. So, if the altitude is 5, the base length (b) is twice the height of the right triangle, which is 2 * 5 = 10.

Next, we calculate the perimeter (P) of the base triangle. For an equilateral triangle, the perimeter is given by P = 3 * side length.

Since the side length is half the base length, we have: P = 3 * (10 / 2) = 3 * 5 = 15.

Now we can calculate the lateral area using the slant length: LA = (1/2) * P * s = (1/2) * 15 * 7 = 52.5.

To find the base area (A) of the regular triangular pyramid, we can use the formula for the area of an equilateral triangle: A = (sqrt(3) / 4) * side length^2.

Since the side length is 10, we have: A = (sqrt(3) / 4) * 10^2 = (sqrt(3) / 4) * 100 = 25 * sqrt(3).

Finally, to determine the surface area (SA), we add the lateral area and the base area: SA = LA + A = 52.5 + 25 * sqrt(3).

Therefore, the surface area of the regular triangular pyramid is 52.5 + 25 * sqrt(3), and the lateral area is 52.5.