Cardioid in Math: Definition, Equation & Examples

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  • 0:04 Cardioid
  • 0:51 Equation of a Cardioid
  • 2:01 Examples of Cardioids
  • 4:09 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will cover a neat shape studied in upper-level mathematics called a cardioid. We will look at the basic shape, how it is constructed, its equation in polar form, and various examples of these equations and corresponding cardioids.

Cardioid

Suppose you've just sat down to have an apple as your afternoon snack. As you cut the apple in half, take a moment to look at the resulting shape of the apple's cross section like the one appearing here in this image:


Cross Section of an Apple Cut in Half
cardioid1


You probably never gave much thought to the shape of an apple cut in half, but in mathematics, this shape has a special name: a cardioid.

Cardioids are fascinating shapes that are studied in higher-level mathematics. One way to remember the name of this shape is that notice that it kind of resembles the shape of a heart. The word 'cardioid' is similar to the word 'cardiac,' which means relating to the heart.

Technically speaking, a cardioid shape can be created by following the path of a point on a circle as the circle rolls around another fixed circle, with both of the circles having the same radius.


Creating a Cardioid
cardioid2


Let's explore these neat shapes a bit further!

Equation of a Cardioid

When it comes to equations of cardioids, polar form is usually used for simplicity. The polar form of an equation involves polar coordinates instead of rectangular (meaning x and y) coordinates.

Polar coordinates are points (r, θ) that are plotted on a polar coordinate system in which r is the length of the line segment connecting the point to the origin and θ is the angle that is created, counterclockwise, between the polar axis and the line segment from the point to the origin.


Polar Coordinates
cardioid3


Therefore, the polar form of an equation has variables r and θ, and is satisfied by the points (r, θ) that make the equation true.

Okay, now that the explanation is out of the way, let's take a look at the equation of a cardioid in polar form. There are two possibilities: a horizontal cardioid and a vertical cardioid. If the radius of the circle that creates the cardioid is a, then we have the following:

  • The equation of a horizontal cardioid is r = a ± acosθ.
  • The equation of a vertical cardioid is r = a ± asinθ.


Horizontal and Vertical Cardioids
cardioid4


Let's take a look at an example of each of these types of cardioids, their equations, and their graphs.

Examples of Cardioids

Consider the following two equations in polar form:

  • r = 2 + 2cosθ
  • r = 3 + 3sinθ

First, let's see what type of information we can get from the equations without looking at their graphs.

Notice here that both of the equations are of the form of a cardioid equation, so we know the general shapes of their graphs. Now, one of these equations represents a horizontal cardioid, and one represents a vertical cardioid. Can you tell which is which?

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