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