Find the surface area of a triangular pyramid with an edge length equaling 4.5, a slant height of 3.5, and a base height of 3.9.
To find the surface area of a triangular pyramid, we first need to find the area of each face and then add them together.
The base of the pyramid is a triangle, so we can find its area using the formula for the area of a triangle: 1/2 * base * height.
The base of the pyramid has a height of 3.9 and a base length equal to one of the edges, which is 4.5. Therefore, the area of the base is 1/2 * 4.5 * 3.9 = 8.775.
Now, let's find the lateral area of the pyramid. The lateral area is the sum of the areas of the triangular faces.
Each triangular face of the pyramid has the same dimensions: two equal sides equal to the slant height of 3.5, and a base equal to one of the edges, which is 4.5.
We can use Heron's formula to find the area of a triangle with side lengths 3.5, 3.5, and 4.5. Heron's formula states that the area of a triangle with side lengths a, b, and c is given by:
area = √(s(s-a)(s-b)(s-c))
where s is the semiperimeter, calculated as (a + b + c) / 2.
In this case, a = b = 3.5, and c = 4.5. Substituting these values into Heron's formula, we get:
s = (3.5 + 3.5 + 4.5) / 2 = 11.5 / 2 = 5.75
area = √(5.75(5.75 - 3.5)(5.75 - 3.5)(5.75 - 4.5))
= √(5.75 * 2.25 * 2.25 * 1.25)
= √(44.265625)
≈ 6.652
Since there are 4 triangular faces on a pyramid and they are all congruent, the total lateral area of the pyramid is 4 * 6.652 = 26.608.
Finally, we can calculate the surface area of the pyramid by adding the area of the base and the lateral area:
surface area = base area + lateral area
= 8.775 + 26.608
= 35.383
Therefore, the surface area of the triangular pyramid is approximately 35.383 square units.