https://file.aiquickdraw.com/mathbot/file/1731701123190jmrc32pi.jpg

To determine the volume of the cone shown in the diagram, we can use the volume formula for a cone:

\[
V = \frac{1}{3} \pi r^2 h
\]

where:
- \( r \) is the radius of the base of the cone,
- \( h \) is the height of the cone.

### Given Values
From the diagram:
- The radius \( r = 5 \, \text{cm} \).
- The slant height \( s = 13 \, \text{cm} \).
- The horizontal distance from the center to the edge is \( 12 \, \text{cm} \), which represents the height \( h \) of the cone.

### Calculation
1. Substitute the values into the formula:
   \[
   V = \frac{1}{3} \pi (5)^2 (12)
   \]
2. Calculate \( r^2 \):
   \[
   5^2 = 25
   \]
3. Calculate \( V \):
   \[
   V = \frac{1}{3} \pi \cdot 25 \cdot 12
   \]
4. Simplify:
   \[
   V = \frac{1}{3} \pi \cdot 300
   \]
5. Further calculation:
   \[
   V = 100 \pi \approx 314.2 \, \text{cm}^3
   \]

### Answer
The closest answer to the volume is:
**(A) 314.2 cm³**

Would you like a further breakdown of the steps or have any questions?

Here are some related questions to consider:
1. What is the formula for the surface area of a cone, given its slant height?
2. How would the volume change if the radius doubled but the height remained the same?
3. Can the Pythagorean theorem verify the height in this cone diagram?
4. What would be the effect on volume if both radius and height were halved?
5. How does the volume formula of a cone relate to that of a cylinder?

**Tip:** Remember, the volume of a cone is always one-third of a cylinder with the same base and height.

The diagram shows a cone and its dimensions. Which measurement is closest to the volume of the cone in cubic centimeters?

Learn how to calculate the volume of a cone with a radius of 5 cm and height of 12 cm using the volume formula V = (1/3)πr^2h. This geometry problem also demonstrates the use of the Pythagorean theorem to find missing dimensions. Suitable for Grades 9-10.

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