# 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:

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.

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, Î¸) 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 Î¸ is the angle that is created, counterclockwise, between the polar axis and the line segment from the point to the origin.

Therefore, the polar form of an equation has variables r and Î¸, and is satisfied by the points (r, Î¸) 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 ± acosÎ¸.
• The equation of a vertical cardioid is r = a ± asinÎ¸.

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 + 2cosÎ¸
• r = 3 + 3sinÎ¸

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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