Torus surface area and volume formula with derivation and examples
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 Geometry and Its Key Formulas
1. What is a torus in mathematics?
A torus is a three-dimensional surface formed by rotating a circle around an axis in the same plane that does not cut the circle, creating a doughnut-shaped object. In geometry and topology, it is defined as a surface of revolution with a hole in the middle. A common example is a ring doughnut or an inner tube. The torus is an important shape in solid geometry and topology because of its unique properties, such as having genus 1 (one hole).
2. What is the formula for the volume of a torus?
The volume of a torus is given by the formula V = 2π²Rr², 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.
- R = major radius (center of hole to center of tube)
- r = minor radius (radius of circular tube)
3. What is the surface area of a torus?
The surface area of a torus is calculated using A = 4π²Rr, where R is the major radius and r is the minor radius.
- R = distance from center of torus to center of tube
- r = radius of the tube
4. What do R and r mean in the torus formula?
In torus formulas, R is the major radius and r is the minor radius.
- Major radius (R): distance from the center of the hole to the center of the tube.
- Minor radius (r): radius of the circular cross-section (tube).
5. How do you derive the volume of a torus?
The volume of a torus is derived using Pappus's Theorem, which multiplies the area of the circle by the distance its centroid travels.
- Area of generating circle = πr²
- Distance traveled by centroid = 2πR
- Volume = πr² × 2πR = 2π²Rr²
6. What is the difference between a torus and a sphere?
The main difference is that a torus has a hole while a sphere has no hole.
- A sphere is perfectly round with genus 0.
- A torus is doughnut-shaped with genus 1.
- A sphere has only one radius, while a torus has two radii (R and r).
7. What are the properties of a torus?
A torus has several important geometric and topological properties.
- It is a surface of revolution.
- It has genus 1 (one hole).
- It has no edges or vertices.
- It is symmetric about its central axis.
8. What is the parametric equation of a torus?
The parametric equations of a torus are x = (R + r cosθ) cosφ, y = (R + r cosθ) sinφ, and z = r sinθ.
- θ and φ range from 0 to 2π.
- R = major radius.
- r = minor radius.
9. Can you give an example of calculating the volume of a torus?
Yes, to calculate the volume of a torus, use V = 2π²Rr².
- Let R = 4 units and r = 1 unit.
- Substitute into formula: V = 2π²(4)(1²).
- V = 8π² cubic units.
10. Where is a torus used in real life?
A torus appears in many real-world applications, especially in science and engineering.
- Doughnuts and lifebuoys (basic geometric examples).
- Magnetic confinement devices like tokamaks in nuclear fusion.
- Inner tubes and rubber rings.
- Topology and advanced mathematics research.
