Question

Identify the radius of the following cone.

A cone is marked with a height of 12 centimeters and a radius of 5 centimeters, both shown as perpendicular dotted lines.
Solve. Show all of your work

Answer 1

To identify the radius of the cone as given in the problem, we start by noting that the radius is already provided: 5 centimeters. However, if needed, let’s clarify how these measurements relate to the cone's geometry.

### Given:
- Height (h) of the cone = 12 cm
- Radius (r) of the cone = 5 cm

### Formulae:
1. **Volume of a Cone**: \( V = \frac{1}{3} \pi r^2 h \)
2. **Surface Area of the Cone**:
- Lateral Area: \( A_L = \pi r l \) (where \( l \) is the slant height)
- Base Area: \( A_B = \pi r^2 \)
- Total Surface Area: \( A = A_L + A_B = \pi r l + \pi r^2 \)

### Slant Height Calculation:
The slant height \( l \) of a cone can be found using the Pythagorean theorem, considering the radius and height as the two shorter sides of a right triangle, with the slant height being the hypotenuse:

\[
l = \sqrt{r^2 + h^2}
\]

Substituting the values we have:

\[
l = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ cm}
\]

### Summary:
- The radius of the cone is already provided as **5 centimeters**.
- The calculated slant height is **13 centimeters**.

If you need any further information or other specific calculations related to the cone, please let me know!