Find the length of the height og the cone diameter: 10cm slant height:13 cm

To find the height of the cone, we can use the relationship between the radius, height, and slant height of the cone, which can be expressed using the Pythagorean theorem.

Given:
- Diameter of the cone = 10 cm
- Therefore, the radius \( r \) = \( \frac{10}{2} = 5 \) cm
- Slant height \( l \) = 13 cm

We can use the following formula derived from the Pythagorean theorem:

\[
l^2 = r^2 + h^2
\]

Where:
- \( l \) = slant height
- \( r \) = radius
- \( h \) = height

Substituting the known values into the equation:

\[
13^2 = 5^2 + h^2
\]

Calculating the squares:

\[
169 = 25 + h^2
\]

Now, subtract 25 from both sides:

\[
169 - 25 = h^2
\]

\[
144 = h^2
\]

Now, take the square root of both sides to find \( h \):

\[
h = \sqrt{144} = 12 \text{ cm}
\]

Thus, the height of the cone is **12 cm**.