Question

What is the formula for finding the surface area of a regular triangular pyramid

Answer 1

The surface area \( S \) of a regular triangular pyramid (also known as a tetrahedron when all faces are equilateral triangles) can be calculated using the formula:

\[
S = B + \frac{1}{2} P l
\]

where:
- \( B \) is the area of the base,
- \( P \) is the perimeter of the base,
- \( l \) is the slant height of the triangular faces.

For a regular triangular pyramid where the base is an equilateral triangle with side length \( a \):

1. **Area of the base** (\( B \)):
\[
B = \frac{\sqrt{3}}{4} a^2
\]

2. **Perimeter of the base** (\( P \)):
\[
P = 3a
\]

3. **Slant height** (\( l \)):
To find the slant height, you can use the Pythagorean theorem. The height \( h \) of the pyramid can be found using:
- The height of the base triangle (using \( h_{base} = \frac{\sqrt{3}}{2} a \)).
- The height of the pyramid is generally a different variable, which can be designated as \( h_p \).

The slant height is given by:
\[
l = \sqrt{h_p^2 + \left(\frac{a}{2}\right)^2}
\]

Thus, substituting these values into the surface area formula yields:
\[
S = \frac{\sqrt{3}}{4} a^2 + \frac{1}{2} (3a) l
\]
This will give you the total surface area of the regular triangular pyramid. Make sure to calculate \( l \) using the appropriate height of the pyramid based on your specific case to achieve the accurate surface area.