The exact volume of a right circular cone with a height of 12 cm and a base radius of 6 cm is 144π cm³.
Understanding Cone Volume: A Detailed Calculation
Calculating the volume of a right circular cone is a fundamental concept in geometry, essential for various applications from engineering to design. A right circular cone is a three-dimensional geometric shape with a circular base and a single vertex directly above the center of the base.
The Volume Formula for a Cone
The volume ($V$) of any cone, including a right circular cone, is determined by its base area and its perpendicular height. The universally accepted formula is:
$$V = \frac{1}{3} \pi r^2 h$$
Where:
- $V$ represents the volume of the cone.
- $\pi$ (Pi) is a mathematical constant approximately equal to 3.14159. For exact answers, it is often left as the symbol $\pi$.
- $r$ is the radius of the circular base.
- $h$ is the height of the cone (the perpendicular distance from the base to the apex).
For more detailed information on cones and their properties, you can refer to Wikipedia's article on Cones.
Given Parameters
In this specific problem, we are provided with the following dimensions for the right circular cone:
- Height ($h$): 12 cm
- Base Radius ($r$): 6 cm
Step-by-Step Volume Calculation
Let's substitute these values into the volume formula to find the exact volume of the cone.
-
Write down the formula:
$V = \frac{1}{3} \pi r^2 h$ -
Substitute the given values for radius ($r$) and height ($h$):
$V = \frac{1}{3} \pi (6 \text{ cm})^2 (12 \text{ cm})$ -
Calculate the square of the radius ($r^2$):
$6^2 = 36 \text{ cm}^2$ -
Substitute the calculated $r^2$ back into the equation:
$V = \frac{1}{3} \pi (36 \text{ cm}^2) (12 \text{ cm})$ -
Simplify the terms by multiplying the numerical values:
To simplify, we can multiply $36$ by $12$ and then divide by $3$, or divide $12$ by $3$ first:
$V = \pi \times 36 \text{ cm}^2 \times \left(\frac{12}{3}\right) \text{ cm}$
$V = \pi \times 36 \text{ cm}^2 \times 4 \text{ cm}$ -
Perform the final multiplication:
$V = 144\pi \text{ cm}^3$
The unit for volume is cubic centimeters (cm³), as we are multiplying three length dimensions (cm × cm × cm).
Summary of Cone Dimensions and Volume
To make the information easily digestible, here's a summary table:
| Parameter | Value | Unit |
|---|---|---|
| Radius ($r$) | 6 | cm |
| Height ($h$) | 12 | cm |
| Volume ($V$) | $144\pi$ | cm³ |
Practical Implications
Understanding how to calculate cone volume is crucial in fields like:
- Architecture and Construction: Estimating materials needed for conical structures (e.g., roofs, silos).
- Manufacturing: Designing and producing items with conical shapes (e.g., funnels, certain types of packaging).
- Hydraulics: Calculating the capacity of conical tanks or the flow rates through conical sections.
By leaving $\pi$ in the answer, we provide the exact mathematical value, avoiding any rounding errors that would occur if we substituted an approximate decimal value for $\pi$.
