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Cardioid

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Throughout mathematics, we study about a number of shapes and figures. There are lots of two dimensional and  three dimensional  (and even more) kinds studied in geometry. Cardioid is probably the important ones. A cardioid can be a two dimensional or perhaps plane figure that is studied in higher-level instructional math concepts.

The word "cardioid" originated from a Greek notion whose meaning has been "heart". It is known as so since this is a heart-shaped curve. The design of the cardioid can be considered the cross a component of an apple excluding its stalk.
This shape could be formed by tracing a region on the border of any circle which moves on top of another circle of same radius. On this posting, we definitely will learn more at length regarding cardioid, its equation in addition to some other concepts according to it.

Definition

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A cardioid is really a heart-shaped plane ascertain. It is the curve which is described as the locus with the point lying concerning the circumference of a circle that's rolling externally with no slip on the boundary of a different circle of very same radius.
A cardioid offers exactly 3 parallel tangents obtaining any particular gradient.
 
They have a new cusp (just to recall that your cusp is formed with the intersection of two branches with the curve). The duration with the passing through your cusp of cardioid is normally 4a, where a very do the radius with the circles.

Equation

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The equation of your cardioid is commonly represented in polar form. It may likewise be converted in Cartesian coordinate method:
Polar Equation in the Cardioid:

Its polar equation is distributed by:

r = a(1 + cos $\theta$)

Where, a = Radius in the tracing circle.

$\theta$ = Polar position.
Cartesian Equation in the Cardioid

Its Cartesian equation is distributed by:

(x$^2 $ + y$^2 $ + ax)$^2 $ = a$^2 $ (x$^2 $ + y$^2 $)

Whose parametric equations are as follows:

x = a new cos t (1 - cos t)

y = a sin t (1 - cos t).

Properties

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Important properties of cardioid are shown below:

1) The location enclosed by a cardioid could be computed from ones polar equation: A= 6 $\pi a^2$.

This is 6 times the main circles used within the construction with relocating circles or 1. 5 times the the main circle used within the construction with eliptical along with tangent collections.

2) The arc duration of a cardioid could be computed exactly, a new rarity for algebraic shape. The total time-span is actually

L = 16 a.

3) The best way to represent ones cardioid is by means of its polar situation:
r = 1 + cos $\theta$

4) A cardioid can even be a special case of any Pascal Snail. The cardioid will be the pedal curve using the circle regarding an argument with its area.

Examples

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Below are some examples on cardioid.
Example 1: Calculate the area and arc length of the cardioid which is given by the following equation

r = 5 (1 + cos θ).

Solution: Here, a = 5

The formula for area of cardioid is given by :

A = 5 x $\frac{22}{7}$ x 5$^{2}$

A = 5x 3.14 x 25

A = 392.5 sq unit.

The arc length of the cardioid is calculated by :

16 a

= 16 x 5

= 80 unit.

Example 2: Find the area of a cardioid given by the equation r = 5 (1 + cos $\theta$).

Solution: Here a = 5

The formula to find the area of the cardioid is given below.

A = 6 $\pi$ a$^{2}$

A = 6 $\pi$ 25

= 6 * 3.14 * 25

A = 471

Therefore, for the given equation the area of the cardioid is found to be A = 471 sq unit.