Question
If you knew that the slant height of the cone = 13cm, and the radius of the cone = 5cm, how would you find the height of the cone?
Answers
GPT-4o mini
To find the height of the cone when you know the slant height and the radius, you can use the Pythagorean theorem. The relationship between the radius (r), height (h), and slant height (l) of the cone can be expressed as follows:
\[
l^2 = r^2 + h^2
\]
Where:
- \( l \) is the slant height,
- \( r \) is the radius, and
- \( h \) is the height.
Given:
- \( l = 13 \, \text{cm} \)
- \( r = 5 \, \text{cm} \)
Now, you can rearrange the formula to solve for the height \( h \):
1. Substitute the values into the equation:
\[
13^2 = 5^2 + h^2
\]
2. Calculate \( 13^2 \) and \( 5^2 \):
\[
169 = 25 + h^2
\]
3. Subtract \( 25 \) from both sides:
\[
169 - 25 = h^2
\]
\[
144 = h^2
\]
4. Take the square root of both sides to find \( h \):
\[
h = \sqrt{144} = 12 \, \text{cm}
\]
Thus, the height of the cone is \( 12 \, \text{cm} \).
\[
l^2 = r^2 + h^2
\]
Where:
- \( l \) is the slant height,
- \( r \) is the radius, and
- \( h \) is the height.
Given:
- \( l = 13 \, \text{cm} \)
- \( r = 5 \, \text{cm} \)
Now, you can rearrange the formula to solve for the height \( h \):
1. Substitute the values into the equation:
\[
13^2 = 5^2 + h^2
\]
2. Calculate \( 13^2 \) and \( 5^2 \):
\[
169 = 25 + h^2
\]
3. Subtract \( 25 \) from both sides:
\[
169 - 25 = h^2
\]
\[
144 = h^2
\]
4. Take the square root of both sides to find \( h \):
\[
h = \sqrt{144} = 12 \, \text{cm}
\]
Thus, the height of the cone is \( 12 \, \text{cm} \).