Open In App

How to calculate the Surface Area and Volume of a Torus?

Improve
Improve
Like Article
Like
Save
Share
Report

The formula for calculating Surface Area of Torus is 2π2 × R × r, where R is the radius of the circular axis of the torus and r is the radius of the tube (cross-sectional radius). The formula for calculating Volume of Torus is 2π2 × R × r2 where R is the radius of the circular axis of the torus and r is the radius of the tube (cross-sectional radius). A torus is a three-dimensional geometric form with a ring or doughnut-like appearance. A circle that does not cross the circle is created by rotating it around an axis in three dimensions. This produces a form with a hole in the middle. A variety of natural and man-made environments, including doughnuts, inner tubes, some pieces of equipment, and even the structure of galaxies, include torus-shaped features. In this article we are going to discuss what is Torus shape, its properties, surface area and volume formula of Torus and some FAQs on it.

What is Torus?

An object with a shape which resembles that of a donut, such as an O ring, is regarded as a torus in mathematics. It’s an object’s surface created by rotating a circle in tri space around an axis in the very same dimension as the circle. The surface produces a ringed shape known as a ringed torus or just torus if such axis of rotation does not contact the circle. 

The torus consist of a inner radius denoted by ‘r’ which is the inner width of the tube shape, while a outer radius ‘R’ is the distance from center to the outer edge.

A general visualization on Torus shape

Floating tubes used in swimming can be considered as example of real-world objects that resemble a torus form. Toric lenses are eyeglass lenses that incorporate cylindrical and spherical adjustments.

Properties

  • A torus is created by rotating a tight circle (radius r) across a line drawn by a larger circle (radius R).
  • A torus cannot be considered as a polyhedron because it does not have edges, faces or vertices.
  • Torus has only two radii and no other dimensions like height, length or breadth.

Surface Area of Torus

A = 2Ï€2Rr

where R is the outer radius and r is the inner radius

Volume of a Torus

V = 2Ï€2Rr2

where R is the outer radius and r is the inner radius

Sample Problems

Problem 1. Find the surface area of a torus whose inner and outer radii are 5 and 10 units respectively. Also, calculate its volume.

Solution:

Given: r = 5 units and R = 10 units

Since, A = 4Ï€2Rr

A = 4 x 3.14 x 3.14 x 5 x 10

A = 1971.92 sq. units

Now, V = 2Ï€2Rr2

V = 2 x 3.14 x 3.14 x 10 x 5 x 5

V = 4929.8 cu. units

Problem 2. Find the surface area of a torus whose inner and outer radii are 7 and 9 units respectively.

Solution:

Given: r = 7 units and R = 9 units

Since, A = 4Ï€2Rr

A = 4 x 3.14 x 3.14 x 7 x 9

A = 2484.61 sq. units

Problem 3. Find the surface area of a torus whose inner and outer radii are 8 and 9 units respectively.

Solution:

Given: r = 8 units and R = 9 units

Since, A = 4Ï€2Rr

A = 4 x 3.14 x 3.14 x 8 x 9

A = 2839.56 sq. units

Problem 4. Find the surface area of a torus whose inner and outer radii are 2 and 6 units respectively. Also, calculate its volume.

Solution:

Given: r = 2 units and R = 6 units

Since, A = 4Ï€2Rr

A = 4 x 3.14 x 3.14 x 2 x 6

A = 473.26 sq. units

Now, V = 2Ï€2Rr2

V = 2 x 3.14 x 3.14 x 6 x 2 x 2

V = 473.26 cu. units

Problem 5. Find the volume of a torus whose inner and outer radii are 4 and 11 units respectively.

Solution:

Given: r = 4 units and R = 11 units

Since, V = 2Ï€2Rr2

V = 2 x 3.14 x 3.14 x 11 x 4 x 4

V = 3470.57 cu. units


Last Updated : 19 Mar, 2024
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads