To find the surface area of a triangular pyramid, we need to find the area of each face and then add them together.
First, let's find the area of the base. The base is a triangle, so we can use the formula for the area of a triangle: \(\text{{Area}} = \frac{1}{2} \times \text{{base}} \times \text{{height}}\).
Substituting in the values, we get: \(\text{{Area}} = \frac{1}{2} \times 3.9 \times 3.5 = 6.825\).
Now, let's find the area of each triangular face. Each face is a triangle with one side being the slant height and two sides being the edge length. To find the area of each face, we can use Heron's formula: \(\text{{Area}} = \sqrt{s(s-a)(s-b)(s-c)}\), where \(s\) is the semi-perimeter of the triangle and \(a\), \(b\), and \(c\) are the side lengths.
The semi-perimeter of each triangular face is equal to \(\frac{1}{2} \times (3.5 + 4.5 + 4.5) = 6.25\).
Substituting in the values, we get: \(\text{{Area}} = \sqrt{6.25(6.25-3.5)(6.25-4.5)(6.25-4.5)} = 11.261\).
Now, let's find the total surface area of the triangular pyramid. Each triangular face has the same area, so we need to multiply the area of one face by the number of faces, which is 3, since the pyramid has 3 triangular faces. The base is also a triangle, so we need to add its area to the total surface area.
Total surface area = (Area of one face) × 3 + Area of base = \(11.261 \times 3 + 6.825 = 40.604\).
Therefore, the surface area of the triangular pyramid is approximately 40.604 square units.
Find the surface area of a triangular pyramid with an edge length equaling 4.5, a slant height of 3.5, and a base length of 3.9.
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