A **Truncated Cone Volume Calculator** is a **specialized tool** designed to compute the **volume** of a truncated cone, also known as a **frustum**.

- A truncated cone with
**R = 5 cm**,**r = 3 cm**, and**h = 10 cm** - A larger frustum with
**R = 15 m**,**r = 8 m**, and**h = 20 m** - A small truncated cone with
**R = 2 inches**,**r = 1 inch**, and**h = 4 inches**

This unique shape is formed when a plane parallel to the base cuts off the top portion of a cone, resulting in a structure with **two circular bases** of different sizes.

This calculator requires input values such as:

- The
**radius**of the**larger base**(R) - The
**radius**of the**smaller base**(r) - The
**height**of the truncated cone (h)

With these measurements, the calculator applies the appropriate formula to determine the **volume** of the truncated cone.

## Truncated Cone Volume Calculator

Larger Radius (R) | Smaller Radius (r) | Height (h) | Volume (V) | Conversion Equation |
---|---|---|---|---|

5 cm | 3 cm | 10 cm | 314.16 cm³ | V = (1/3) × π × 10 × (5² + 3² + 5 × 3) |

2 m | 1 m | 5 m | 26.18 m³ | V = (1/3) × π × 5 × (2² + 1² + 2 × 1) |

8 in | 6 in | 12 in | 1,809.56 in³ | V = (1/3) × π × 12 × (8² + 6² + 8 × 6) |

10 ft | 7 ft | 15 ft | 3,436.12 ft³ | V = (1/3) × π × 15 × (10² + 7² + 10 × 7) |

1.5 m | 0.5 m | 3 m | 3.93 m³ | V = (1/3) × π × 3 × (1.5² + 0.5² + 1.5 × 0.5) |

**Related Tools**

## Truncated Cone Volume Formula

The formula for calculating the **volume** of a truncated cone is:

**V = (1/3) × π × h × (R² + r² + R × r)**

Where:

**V**is the**volume****h**is the**height**of the truncated cone**R**is the**radius**of the larger base**r**is the**radius**of the smaller base**π**is approximately**3.14159**

This formula is derived from the difference between the volumes of two full cones—the larger cone minus the smaller cone that was **“cut off.”**

### Example:

Let’s calculate the volume of a truncated cone with **R = 6 cm**, **r = 4 cm**, and **h = 8 cm**.

$$ V = (1/3) × π × 8 × (6² + 4² + 6 × 4) $$ $$ V = (1/3) × 3.14159 × 8 × (36 + 16 + 24) $$ $$ V = 8.37757 × 76 $$ $$ V ≈ 636.70 \, \text{cm}³ $$

## How to Find the Volume of a Truncated Cone?

To find the volume of a truncated cone, follow these **simple steps**:

**Measure the Dimensions:**Determine the**radius**of the larger base (**R**), the**radius**of the smaller base (**r**), and the**height**(**h**) of the truncated cone.**Apply the Formula:**Use the formula**V = (1/3) × π × h × (R² + r² + R × r)**.**Calculate:**Plug in the values and compute the result. Use a**calculator**for precision.**Check Units:**Ensure your final answer is in**cubic units**(e.g., cm³, m³, in³).

To calculate the volume of a truncated cone using the formula:

$$ V = \frac{1}{3} \pi h \left( r_1^2 + r_1 r_2 + r_2^2 \right) $$

**Given Values**

**Larger base radius ($$ r_1 $$)**: 5 units**Smaller base radius ($$ r_2 $$)**: 3 units**Height ($$ h $$)**: 10 units

Substituting the values into the formula:

$$ V = \frac{1}{3} \pi \times 10 \left( 5^2 + 5 \times 3 + 3^2 \right) $$

Calculating the terms inside the parentheses:

$$ 5^2 = 25 $$ $$ 5 \times 3 = 15 $$ $$ 3^2 = 9 $$

Now, summing these values:

$$ 25 + 15 + 9 = 49 $$

Substituting back into the volume formula:

$$ V = \frac{1}{3} \pi \times 10 \times 49 $$

Calculating the volume:

$$ V = \frac{490}{3} \pi \approx 490.87 \text{ cubic units} $$

The volume of the truncated cone is approximately **490.87 cubic units**.

## What is the Formula for the Volume of a Truncated Cylinder?

The formula for the volume of a cylinder (truncated or not) is:

**V = π × r² × h**

Where:

**V**is the**volume****r**is the**radius**of the base**h**is the**height**of the cylinder

It’s important to note that a **truncated cylinder** is different from a truncated cone. A truncated cylinder has the **same radius** for both bases.

## How to Calculate Truncated Cone Surface Area?

The formula is:

**SA = π × (R² + r²) + π × (R + r) × s**

Where:

**SA**is the**surface area****R**is the**radius**of the larger base**r**is the**radius**of the smaller base**s**is the**slant height**

To find the **slant height** (**s**), use the **Pythagorean theorem**: $$ s² = h² + (R – r)² $$

The **surface area** of a truncated cone includes the area of both circular bases and the **lateral surface area**.

## What is Truncated Cone Volume?

**Truncated Cone Volume** refers to the **amount of space** enclosed within a truncated cone. It represents the **capacity** of the frustum to contain a substance.

The volume is expressed in **cubic units**, such as cubic centimeters (cm³), cubic meters (m³), or cubic inches (in³).

**Understanding truncated cone volume** is crucial in various fields, including:

**Engineering:**Designing containers, tanks, or parts**Architecture:**Creating conical structures or elements**Manufacturing:**Producing conical products or molds**Science:**Analyzing geological formations or conducting experiments

Is there a way to calculate the height if we only know the Volume, base radius, and the slant angle?

Surely you can, follow the below steps please:

To calculate the height of a cone when you know the volume, base radius, and slant angle, you can follow these steps:

Volume of a Cone: The volume V of a cone is given by the formula: V = (1/3) pi r^2 * h where r is the radius of the base and h is the height.

Relationship Between Height, Radius, and Slant Height: The slant height l, radius r, and height h are related by the Pythagorean theorem: l^2 = r^2 + h^2 which can be rearranged to find the height: h = sqrt(l^2 – r^2)

Finding Slant Height from the Slant Angle: If you know the slant angle θ, you can express the slant height in terms of the height: l = h / cos(θ)

Steps to Calculate Height

Calculate Slant Height: From the slant angle, you can calculate the slant height l using: l = h / cos(θ)

Substitute in Volume Formula: Rearranging the volume formula gives: h = (3V) / (pi r^2)

Combine Equations: Substitute l into the height equation: h = sqrt((h / cos(θ))^2 – r^2)

Solve for Height: This leads to a quadratic equation in terms of h. You can solve this equation to find the height h.

Example Calculation

If you have a volume V, base radius r, and slant angle θ:

Calculate the height using: h = (3V) / (pi r^2)

Find the slant height using: l = h / cos(θ)

Use the Pythagorean relationship to confirm or adjust h as needed.

This approach allows you to find the height of the cone using the known volume, base radius, and slant angle.