Surface Area of Torus
2026-02-21 20:35 Diff
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Last updated on September 4, 2025

A torus is a 3-dimensional shape that resembles a doughnut, featuring a circular path that revolves around an axis in its plane. The surface area of a torus is the total area covered by its outer surface. In this article, we will learn about the surface area of a torus.

What is the Surface Area of a Torus?

The surface area of a torus is the total area occupied by the boundary or surface of a torus. It is measured in square units.

A torus is a 3D shape formed by revolving a circle around an axis that is coplanar with the circle.

The torus has a ring shape with a hole in the middle, similar to a doughnut. It consists of a circular tube, and its surface area includes the entire outer boundary of this tube.

Surface Area of a Torus Formula

The surface area of a torus depends on the radii of the two circles involved: the radius of the tube (r) and the distance from the center of the tube to the center of the torus (R).

The formula for the surface area of a torus is given as: Surface Area = 4π²rR square units

Here, r is the radius of the tube of the torus. R is the distance from the center of the tube to the center of the torus.

Examples of Surface Area of a Torus

Let's consider examples involving different dimensions of a torus to calculate its surface area.

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Example 1: Surface Area of a Torus

Find the surface area of a torus with a tube radius (r) of 3 cm and a distance (R) from the center of the tube to the center of the torus of 5 cm.

Using the formula: Surface Area = 4π²rR = 4 × 3.14² × 3 × 5 = 4 × 9.86 × 3 × 5 = 4 × 147.9 = 591.6 cm²

Example 2: Surface Area of a Torus

Calculate the surface area of a torus where the tube radius (r) is 2 cm and the distance (R) from the center of the tube to the center of the torus is 7 cm.

Surface Area = 4π²rR = 4 × 3.14² × 2 × 7 = 4 × 9.86 × 2 × 7 = 4 × 137.04 = 548.16 cm²

Example 3: Surface Area of a Torus

A torus has a tube radius (r) of 4 cm and a distance (R) from the center of the tube to the center of the torus of 10 cm. Find its surface area.

Surface Area = 4π²rR = 4 × 3.14² × 4 × 10 = 4 × 9.86 × 4 × 10 = 4 × 394.4 = 1577.6 cm²

Confusion between radii

Students sometimes confuse the tube radius (r) with the distance (R) from the center of the tube to the center of the torus. Remember, r is the radius of the circular cross-section of the tube, and R is the larger radius from the center of the torus to the center of the tube.

Problem 1

Use the formula: Surface Area = 4π²rR = 4 × 3.14² × 5 × 12 = 4 × 9.86 × 5 × 12 = 4 × 591.6 = 2366.4 cm²

Okay, lets begin

Calculate the surface area of a torus where the tube radius is 6 cm and the distance from the center of the tube to the center of the torus is 8 cm.

Explanation

Surface Area = 1184.64 cm²

Well explained 👍

Problem 2

Use the formula: Surface Area = 4π²rR = 4 × 3.14² × 6 × 8 = 4 × 9.86 × 6 × 8 = 4 × 473.28 = 1893.12 cm²

Okay, lets begin

Find the surface area of a torus with a tube radius of 7 cm and a distance from the center of the tube to the center of the torus of 9 cm.

Explanation

Surface Area = 2484.36 cm²

Well explained 👍

Problem 3

Use the formula: Surface Area = 4π²rR = 4 × 3.14² × 7 × 9 = 4 × 9.86 × 7 × 9 = 4 × 620.28 = 2481.12 cm²

Okay, lets begin

A torus has a tube radius of 3 cm and a distance from the center of the tube to the center of the torus of 10 cm. What is its surface area?

Explanation

Surface Area = 1183.2 cm²

Well explained 👍

Problem 4

Use the formula: Surface Area = 4π²rR = 4 × 3.14² × 3 × 10 = 4 × 9.86 × 3 × 10 = 4 × 295.8 = 1183.2 cm²

Okay, lets begin

The surface area of a torus is 2000 cm², and the tube radius is 4 cm. Find the distance from the center of the tube to the center of the torus.

Explanation

Distance R = 10.12 cm

Well explained 👍

It is the total area that covers the outer surface of the torus, calculated using the formula 4π²rR.

1.What are the two radii involved in the calculation of a torus's surface area?

The tube radius (r) and the distance (R) from the center of the tube to the center of the torus.

2.How is a torus formed?

A torus is formed by rotating a circle around an external axis coplanar with the circle.

3.What unit is surface area measured in?

Surface area is always measured in square units like cm², m², or in².

4.Can a torus have different tube and central radii?

Yes, the tube radius (r) and the central radius (R) can vary, affecting the overall shape and surface area of the torus.

Common Mistakes and How to Avoid Them in the Surface Area of a Torus

Students often make mistakes while calculating the surface area of a torus, leading to incorrect answers. Below are some common mistakes and how to avoid them.

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Seyed Ali Fathima S

About the Author

Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.

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