If you knew that the slant height of the cone = 13 cm, and the radius of the cone = 5 cm, how would you find the height of the cone?

Question

The image depicts a pink cone with labels for its dimensions. 'S' represents the slant height, 'H' represents the height, and 'R' represents the radius of the circular base. The cone is oriented vertically, with the base at the bottom and the apex at the top, and it is drawn with a shadow effect to give it a three-dimensional appearance. This type of diagram is often used in geometry to illustrate the properties and formulas related to cones, such as calculating volume or surface area. The formula for the volume of a cone is ⅓ 𝜋r²h.

(4 points)

Step 1
Step 2
Step 3
Step 4

Responces:
Set up the equation 5^(2) +h^(2) = 13^(2).
Square 5 and 13 to get 25 + h^(2) = 169.
subtract 25 from both sides to get h^(2) = 144.
take the square root of both sides h = 12 cm.

Answer 1

To find the height of the cone using the given slant height and radius, you can follow these steps:

Step 1
Set up the equation using the Pythagorean theorem, since the radius, height, and slant height form a right triangle. In this case, the formula is:

\[ R^2 + H^2 = S^2 \]

where:

• \( R = 5 \) cm (radius),
• \( H \) is the height we want to find,
• \( S = 13 \) cm (slant height).

So, we can write:

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

Step 2
Square the values of the radius and slant height:

\[ 25 + H^2 = 169 \]

(Here, \( 5^2 = 25 \) and \( 13^2 = 169 \).)

Step 3
Subtract \( 25 \) from both sides to isolate \( H^2 \):

\[ H^2 = 169 - 25 \] \[ H^2 = 144 \]

Step 4
Take 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.