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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.

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 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.

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.

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 mm^{2}

Therefore, the surface area of the torus shape is 828.2 mm^{2}

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 7^{2}

= 2 x \[(\frac{22}{7})^{2}\] x 28 x 7^{2}

= 2 x \[\frac{484}{49}\] x 28 x 7^{2}

= 2 x \[\frac{484}{49}\] x 28 x 49

= 27,104 cm^{3}

Therefore, the volume of the torus is 27,104 cm^{3}

FAQ (Frequently Asked Questions)

1. What Does Torus Mean in Geometry?

Ans. In Geometry, the torus is defined as the doughnut-shaped, three -dimensional figure formed when the circle is rotated about a line in its plane, but not rotated through its axis. The word 'torus' is received from the Latin word meaning bulge. The plural form of the torus is tori. The term torus is sometimes referred to as an anchor.

The most common example of the torus is an inner tube of a tire. Imagine, a circle lies in a plane, such that its diameter is perpendicular to its straight line. The figure that is formed, is a hollow circular tube known as a torus.

2. What is a Solid Torus?

Ans. A solid torus is formed by rotating a disc, rather than rotating a circle around its axis. A solid torus is a torus plus the volume inside the torus. Real-world objects that approximately look like solid torus are O rings, ring doughnuts, non-inflatable lifebuoys, and bagels.

3. How to Calculate the Volume and Surface Area of Torus Shape?

Ans. The volume and surface area of the torus can be calculated if we know the radius of the circle, and the radius of the torus itself, which is the furthest portion of the circle from the line about which it is rotated. If the radius of the torus is given as 'r' and the radius of Circle is given an 'R', then the surface area of the torus is given by, whereas the volume of the torus is given by 2 π² Rr² where the value of pi is constant, and approximately equals to 3.14.