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104 LearnersLast updated on September 4, 2025
Surface Area of Torus
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.
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.
Mistake 1
Incorrect substitution of π
Students might use incorrect values for π. Always use accurate values like 22/7 or 3.14 for computations involving π.
Mistake 2
Neglecting the full formula
Some students might forget to use the complete formula for the surface area of a torus, which is 4π²rR. Ensure that both π² and the product of r and R are included in the calculation.
Mistake 3
Confusing volume with surface area
Students might mistake the formula for volume with that for surface area. Always ensure you are using the correct formula for surface area, which is 4π²rR, rather than volume.
Mistake 4
Solved Examples of Surface Area of Torus
Find the surface area of a torus with a tube radius of 5 cm and a distance from the center of the tube to the center of the torus of 12 cm.
Surface Area = 1193.76 cm²
Problem 1
Use the formula: Surface Area = 4π²rR = 4 × 3.14² × 5 × 12 = 4 × 9.86 × 5 × 12 = 4 × 591.6 = 2366.4 cm²
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
Problem 2
Use the formula: Surface Area = 4π²rR = 4 × 3.14² × 6 × 8 = 4 × 9.86 × 6 × 8 = 4 × 473.28 = 1893.12 cm²
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
Problem 3
Use the formula: Surface Area = 4π²rR = 4 × 3.14² × 7 × 9 = 4 × 9.86 × 7 × 9 = 4 × 620.28 = 2481.12 cm²
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
Problem 4
Use the formula: Surface Area = 4π²rR = 4 × 3.14² × 3 × 10 = 4 × 9.86 × 3 × 10 = 4 × 295.8 = 1183.2 cm²
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
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?
2.How is a torus formed?
3.What unit is surface area measured in?
4.Can a torus have different tube and central radii?
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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: She has songs for each table which helps her to remember the tables
