To find the slant height of the cone using the Pythagorean theorem, we can visualize the cone's dimensions:
- The height \( h \) of the cone is given as 6 cm.
- The diameter \( d \) of the cone is given as 13 cm, which means the radius \( r \) is \( d/2 = 13/2 = 6.5 \) cm.
In a right triangle formed by the height, radius, and slant height of the cone, we can apply the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \]
Where \( l \) is the slant height, \( r \) is the radius, and \( h \) is the height. Plugging in the values:
\[ l^2 = (6.5)^2 + (6)^2 \] \[ l^2 = 42.25 + 36 \] \[ l^2 = 78.25 \] \[ l = \sqrt{78.25} \approx 8.8 \text{ cm} \]
Thus, the slant height must be approximately 8.8 cm.