What is the formula for the slant height of a pyramid?

The formula for the slant height of a pyramid is: \( l = \sqrt{h^2 + \left(\frac{a}{2}\right)^2} \).

In this formula, \( l \) represents the slant height, \( h \) is the vertical height (or altitude) of the pyramid, and \( a \) is the length of one side of the base of the pyramid. This formula is derived from the Pythagorean theorem, which is used to find the length of the hypotenuse in a right-angled triangle.

To understand this better, imagine a right-angled triangle formed by the vertical height of the pyramid, half the base length, and the slant height. The vertical height \( h \) is one leg of the triangle, and half the base length \( \frac{a}{2} \) is the other leg. The slant height \( l \) is the hypotenuse of this right-angled triangle. By applying the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, we get:

\[ l^2 = h^2 + \left(\frac{a}{2}\right)^2 \]

Taking the square root of both sides gives us the formula for the slant height:

\[ l = \sqrt{h^2 + \left(\frac{a}{2}\right)^2} \]

This formula is particularly useful when you need to find the slant height for calculations involving the surface area or the lateral surface area of a pyramid. Remember, the slant height is different from the vertical height, as it measures the distance along the face of the pyramid from the base to the apex.