Torus Volume Calculator

Instantly calculate the volume of a torus with precision using our Torus Volume Calculator at examples.com. Fast and accurate results every time.

How to use Torus Volume Calculator

Step 1: Enter the Major Radius (R)

Input the major radius (R) of the torus in the designated field. This is the radius from the center of the torus to the center of the tube.

Step 2: Enter the Minor Radius (r)

Type in the minor radius (r), which is the radius of the tube itself, in the corresponding input field.

Step 3: Calculate Volume

Click the “Calculate” button. The calculator will apply the formula for the volume of a torus to compute the result.

Step 4: View the Results

The volume of the torus will be displayed in cubic meters or the unit specified, showing the space the torus occupies.

How to find Torus Volume Calculator

Step 1: Input the Major Radius

• Enter the value for the major radius (R) in the field labeled “Major radius (R).” In this example, the major radius is set to 2 meters.

Step 2: Input the Minor Radius

• Enter the value for the minor radius (r) in the field labeled “Minor radius (r).” Here, the minor radius is 1 meter.

Step 3: Calculate the Volume

• Click the “Calculate” button. The calculator uses the formula for the volume of a torus, 𝑉=(𝜋𝑟²)(2𝜋𝑅), to compute the volume.

Step 4: View the Results

• The calculated volume will appear in the designated area, as shown as “Volume in Meter³.” In the provided example, the volume is 39.47842 cubic meters.+

Torus Volume Formula

The volume formula for a torus, 𝑉=(𝜋𝑟²)(2𝜋𝑅), effectively describes how the volume of the torus is computed based on its geometric dimensions. Here’s a breakdown of how this formula works:

• 𝑟 (Minor Radius): This is the radius of the circular tube that makes up the torus. It defines the size of the cross-sectional circle of the tube.
• 𝑅 (Major Radius): This is the distance from the center of the tube to the center of the torus. It represents the radius of the central circle around which the tube is revolved.

Formula Components:

1. 𝜋𝑟²: Calculates the area of the circular cross-section of the tube.
2. 2𝜋𝑅: Represents the circumference of the central circle around which the tube’s cross-section revolves.

Examples of Torus Volume Calculator

Example 1:

• Minor Radius (r): 2 cm
• Major Radius (R): 5 cm

Calculation: 𝑉=(𝜋×2²)(2𝜋×5)=(4𝜋)(10𝜋)=40𝜋² cubic cm=125.66 cubic cm

Example 2:

• Minor Radius (r): 1 cm
• Major Radius (R): 10 cm

Calculation: 𝑉=(𝜋×1²)(2𝜋×10)=(𝜋)(20𝜋)=20𝜋2 cubic cm=62.83 cubic cm

Example 3:

• Minor Radius (r): 3 cm
• Major Radius (R): 7 cm

Calculation: 𝑉=(𝜋×3²)(2𝜋×7)=(9𝜋)(14𝜋)=126𝜋² cubic cm=395.84 cubic cm

Example 4:

• Minor Radius (r): 0.5 cm
• Major Radius (R): 4 cm

Calculation: 𝑉=(𝜋×0.5²)(2𝜋×4)=(0.25𝜋)(8𝜋)=2𝜋² cubic cm=6.28 cubic cm

Example 5:

• Minor Radius (r): 6 cm
• Major Radius (R): 8 cm

Calculation: 𝑉=(𝜋×6²)(2𝜋×8)=(36𝜋)(16𝜋)=576𝜋² cubic cm=1809.56 cubic cm

How is a Torus Formed?

A torus is formed by revolving a circle in three-dimensional space around an axis coplanar with the circle, creating a doughnut-shaped surface.

What is the Equation of a Torus?

The standard equation for a torus in Cartesian coordinates is (√𝑥²+𝑦²−𝑅)²+𝑧²=𝑟², where 𝑅 is the major radius and r is the minor radius.

What are the units used in the torus volume formula?

The units for 𝑟 and 𝑅 should both be in length units (e.g., meters, centimeters). The resulting volume 𝑉 will then be in cubic units (e.g., cubic meters, cubic centimeters).

Why do we use 𝜋 in the torus volume formula?

The constant 𝜋 appears in the formula because the volume calculation involves circular rotations and the area calculation of circular cross-sections, both of which inherently involve 𝜋.

Can a torus have the same major and minor radii?

Yes, a torus can have equal major and minor radii, although it will appear more like a ring where the diameter of the tube is equal to the diameter of the whole torus.

Does the orientation of the torus affect its volume?

No, the orientation of the torus does not affect its volume. The volume depends solely on the sizes of the radii, not on its orientation in space.