To find the height of a cone, we can use the Pythagorean theorem. Given that the diameter \(d\) of the cone's base is 10 cm, we can first determine the radius \(r\):
\[ r = \frac{d}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \]
Now, we have the slant height \(l\) of the cone, which is given as 13 cm, and the radius \(r\) determined above. We can use the Pythagorean theorem, which states:
\[ l^2 = r^2 + h^2 \]
where \(h\) is the height of the cone we're looking for. Rearranging the formula to solve for \(h\):
\[ h^2 = l^2 - r^2 \]
Substituting in the known values:
\[ h^2 = 13^2 - 5^2 \] \[ h^2 = 169 - 25 \] \[ h^2 = 144 \]
Taking the square root of both sides:
\[ h = \sqrt{144} = 12 \text{ cm} \]
Thus, the height of the cone is \( \boxed{12} \) cm.
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