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
