The question aims to find the **volume** of the given **cone** with** 14 inches** in **height** and **10 inches in diameter,** as shown in Figure 1.

The question depends on the **cone geometry.** A **cone** is a** 3D solid shape** with a **circular body** at the bottom and a **triangular shape** towards the **top.** The **volume** of the **cone** can be calculated by the formula given below:

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

Here,

\[ r = Radius\ of\ the\ circular\ body\ of\ the\ cone \]

\[ h = Height\ of\ the\ cone \]

## Expert Answer

The given information about the question is as follows:

\[ Diameter\ d = 10\ in \]

\[ Height\ h = 14\ in \]

To find the **radius** of the **cone,** we **divide** the **diameter** in **half** to calculate its **radius,** which is given as:

\[ r = \dfrac{ d }{ 2 } \]

\[ r = \dfrac{ 10 }{ 2 } \]

\[ r = 5\ in \]

The formula for the **volume** of the **cone** is given as:

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

Substituting the values, we get:

\[ V = \dfrac{ 1 }{ 3 } \pi \times (5)^2 \times 14 \]

\[ V = \dfrac{ 1 }{ 3 } \pi \times 25 \times 14 \]

\[ V = \dfrac{ 1 }{ 3 } 350 \pi \]

\[ V = 116.67 \pi \]

\[ V = 366.52 {in}^3 \]

## Numerical Result

The **volume** of the **cone** with **10 inches** in **diameter** and** 14 inches** in **height** is calculated to be:

\[ V = 366.52 {in}^3 \]

## Example

Find the **volume** of the given **cone** below with **12 inches** in **diameter** and **10 inches in height.**

\[ Height\ h = 10\ in \]

\[ Diameter\ d = 12\ in \]

In order to calculate the **radius** of the **cone,** we divide the **diameter** in **half** to calculate its **radius,** which is given as:

\[ r = \dfrac{ d }{ 2 } \]

\[ r = \dfrac{ 12 }{ 2 } \]

\[ r = 6\ in \]

The formula for the **volume** of the **cone** is given as:

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

Substituting the values, we get:

\[ V = \dfrac{ 1 }{ 3 } \pi \times (6)^2 \times 10 \]

\[ V = \dfrac{ 1 }{ 3 } \pi \times 36 \times 10 \]

\[ V = \dfrac{ 1 }{ 3 } 360 \pi \]

\[ V = 120\pi \]

\[ V = 377 {in}^3 \]

*Images/Mathematical Drawings are created with Geogebra.*