How to Calculate the Surface Area and Volume of a Torus
In Mathematics, a torus is a doughnut-shaped object such as an O ring. It is a surface of an object formed by revolving a circle in three-dimensional space about an axis that lies in the same plane as the circle. If the axis of revolution does not touch the circle, the surface forms a ring shape known as ring torus or simply torus if the ring shape is implicit. As the distance from the axis of revolution minimizes, then the ring torus transforms into a horn torus.
Real -World objects that approximately look like a torus shape are swim rings or inner tubes. Eyeglass lenses that combine cylindrical or spherical corrections are defined as toric lenses.
Torus Equation
Let the radius of torus from the centre of the circle to the centre of the torus tube be 'c' and the radius of the tube be ‘a'. Then the torus equation in cartesian coordinate is given as:
\[(c=\sqrt{x^{2}-y^{2}}+z^{2})=a^{2}\]
Torus equation in parametric form is given as:
x = (c + a cos 𝜈) cos u
y = (c + a cos 𝜈) cos u
x = a sin 𝜈
For u 𝜈 [ 0, 2 λ]
The Three Types of a Torus, Known as Standard Tori are Possible, Depending on the Relative Size of a and c.
The Ring Torus is formed when c > a
The Horn Torus is formed when c = a, which is tangent itself at the point (0,0,0).
The self - intersecting Spindle Torus is formed when c < a
If no specification is given, then torus shape is simply considered as Ring torus.
The three standard torus images are given below, where the first image shows ring torus, the second image shows horn torus.
Surface Area of Torus
To calculate the surface area of ring torus, consider the inner radius as r and outer radii as R
Surface Area of Torus = 2πr + 2πR
= 4 x π² x R x r
Surface Area of Torus Formula = 4 x π² x R x r
Similarly, volume of Torus is calculated as
V = 2 π² Rr²
Volume of Torus Formula = V = 2 π² Rr²
Facts to Remember
The value of in volume and surface area of the torus is constant and approximately equals 3.14 or 22/7
Two or more than two torus is known as tori.
Solved Example
1. What is the surface area of torus shape which has inner radius equal to 5 mm and outer radius equal to 10 mm?
Solution:
Given,
Outer Radius = 7 mm
Inner Radius = 3 mm
As we know,
Surface Area of Torus = 4 x π² x R x r
Substituting the values in the above equation, we get
= 4 x π² x 7 x 3
= 84 x π²
= 84 x (3.14)2
= 828.2 mm2
Therefore, the surface area of the torus shape is 828.2 mm2
2. What is the volume of torus shape which has an inner radius equal to 7 cm and outer radius equal to 28 cm? ( π = 22/7)
Solution:
Given,
Outer Radius = 28 cm
Inner Radius = 7 cm
As we know,
Volume of Torus = 2 π² Rr²
Substituting the values in the above equation, we get
= 2 x π² x 28 x 72
= 2 x \[(\frac{22}{7})^{2}\] x 28 x 72
= 2 x \[\frac{484}{49}\] x 28 x 72
= 2 x \[\frac{484}{49}\] x 28 x 49
= 27,104 cm3
Therefore, the volume of the torus is 27,104 cm3
FAQs on Torus in Mathematics: Definition, Properties & Examples
1. Is a donut a torus?
Yes, a donut is a classic example of a torus in geometry. A torus is defined as a three-dimensional shape formed by revolving a circle around an axis outside the circle. Donuts and bagels both show this torus shape.
2. What does torus mean?
Torus means a ring-shaped surface or solid in mathematics. It comes from the Latin word for “cushion.” In geometry, a torus is created by rotating a circle in 3D space, resulting in a shape similar to a lifebuoy or donut.
3. What is a torus in human anatomy?
In human anatomy, a torus refers to a natural, rounded bony protrusion. Common examples include the
- torus palatinus (roof of the mouth)
- torus mandibularis (inside the lower jaw)
4. What is so special about the torus?
A torus is special in mathematics because it has unique properties. It is a surface with a hole, described as genus 1. Its geometry allows interesting applications in
- topology
- physics
- engineering
5. How is the surface area of a torus calculated?
The surface area of a torus is calculated with the formula $A = 4\pi^2 Rr$, where $R$ is the distance from the center of the tube to the center of the torus, and $r$ is the radius of the tube.
6. Where do we see tori in real life?
Tori (plural of torus) appear in many real-world objects, such as
- donuts
- swim rings
- bagels
- O-rings
7. What is the equation of a torus?
The standard torus equation in 3D Cartesian coordinates is $\left(\sqrt{x^2 + y^2} - R\right)^2 + z^2 = r^2$, where $R$ is the major radius (distance from center) and $r$ is the minor radius (tube radius).
8. What is the volume formula of a torus?
The volume of a torus is given by the formula $V = 2\pi^2 R r^2$, where $R$ is the distance from the center of the tube to the torus center, and $r$ is the tube’s radius. Both values must be positive.
9. Can a torus exist in higher dimensions?
Yes, in mathematics, a torus can be generalized to higher dimensions. For example, a 3-torus is a three-dimensional analog which has important uses in
- topology
- physics
10. How is a torus different from a sphere?
A torus and a sphere are different shapes. A sphere has no holes and every point is the same distance from its center. A torus, on the other hand, has a ring shape with a central hole, so they have different topologies.
11. What are some uses of tori in engineering?
Engineers use tori for
- O-rings for sealing fluids
- magnetic coils in fusion reactors
- designs of pneumatic tubes
12. What is meant by a solid torus?
A solid torus is the three-dimensional object enclosed by a torus surface. Imagine filling a donut with material—every point inside is part of the solid torus, not just the outer surface. In topology, solid tori are important in understanding 3D spaces.
