To find the height of the cone, we can use the relationship between the radius, height, and slant height of the cone. The slant height (l) forms a right triangle with the height (h) and the radius (r) of the base of the cone.
The given data is:
- Diameter of the cone = 10 cm, so the radius (r) = diameter / 2 = 10 cm / 2 = 5 cm
- Slant height (l) = 13 cm
Using the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + h^2 \]
Calculating \(13^2\) and \(5^2\):
\[ 169 = 25 + h^2 \]
Now, rearranging the equation to solve for \(h^2\):
\[ h^2 = 169 - 25 \]
\[ h^2 = 144 \]
Taking the square root of both sides to find \(h\):
\[ h = \sqrt{144} \]
\[ h = 12 \text{ cm} \]
Thus, the height of the cone is 12 cm.