Question

Find the length of the height of the cone with diameter of 10 and slant length of 13

Answer 1

To find the height of a cone given its diameter and slant length, we can use the Pythagorean theorem.

1. First, determine the radius of the cone. The diameter is given as 10, so the radius \( r \) is:
\[
r = \frac{diameter}{2} = \frac{10}{2} = 5
\]

2. The slant length \( l \) (the hypotenuse of the right triangle) is given as 13.

3. In a right triangle, the relationship between the height \( h \), radius \( r \), and slant length \( l \) can be expressed as:
\[
l^2 = r^2 + h^2
\]

4. We can rearrange this formula to solve for the height \( h \):
\[
h^2 = l^2 - r^2
\]

5. Plugging in the values we have:
\[
h^2 = 13^2 - 5^2
\]
\[
h^2 = 169 - 25
\]
\[
h^2 = 144
\]

6. To find \( h \), take the square root of 144:
\[
h = \sqrt{144} = 12
\]

Thus, the height of the cone is \( \boxed{12} \).