Surface Area of a Torus Formula

The surface area of a torus, a donut-shaped 3D figure, depends on its major radius and minor radius. The major radius is the distance from the center of the torus to the center of the tube, while the minor radius is the radius of the tube itself. Here’s the formula for calculating the surface area of a torus, along with detailed explanations and examples.

Formula for the Surface Area of a Torus

If the major radius (distance from the center of the torus to the center of the tube) is \( R \) and the minor radius (radius of the tube) is \( r \),

The surface area \( S \) of the torus can be calculated with the formula:

\( S = 4 \pi^2 R r \)

In this formula:

  • \( R \) is the major radius, or the distance from the center of the torus to the center of the tube
  • \( r \) is the minor radius, or the radius of the circular tube
  • \( \pi \) (Pi) is approximately equal to 3.14159

Detailed Explanation of the Formula

The formula \( S = 4 \pi^2 R r \) calculates the surface area of a torus by taking the circumference of the circular cross-section of the tube and multiplying it by the circumference of the path traveled by the center of the tube.

  • The circumference of the circular cross-section of the tube is \( 2 \pi r \), where \( r \) is the minor radius.
  • The circumference of the circular path around the center of the torus is \( 2 \pi R \), where \( R \) is the major radius.
  • Multiplying these two circumferences gives the surface area: \( S = 4 \pi^2 R r \).

Example: Calculating the Surface Area of a Torus

Problem: Find the surface area of a torus with a major radius \( R = 8 \, \text{cm} \) and a minor radius \( r = 3 \, \text{cm} \).

Solution:

  1. Write down the formula: \( S = 4 \pi^2 R r \).
  2. Substitute \( R = 8 \) and \( r = 3 \): \( S = 4 \pi^2 \times 8 \times 3 \).
  3. Multiply: \( S = 96 \pi^2 \).
  4. Approximate \( \pi^2 \approx 9.8696 \): \( S \approx 96 \times 9.8696 = 947.88 \, \text{cm}^2 \).

The surface area of the torus is approximately \( 947.88 \, \text{cm}^2 \).

This example demonstrates how to calculate the surface area of a torus using the formula \( S = 4 \pi^2 R r \) with given values for the major and minor radii.