Recall that a circle is a special case of an ellipse. What does this mean?
See solution.
Practice makes perfect
Because we want to find the connection between the equations for the area of a circle and the area of an ellipse, let's start by identifying the variables in the formula for the area of an ellipse.
Area of an Ellipse: π a b
Here, a is half of the length of the major axis and b is half of the length of the minor axis.
In a previous exercise, we showed that a circle is a special case of an ellipse. That is, if the lengths of the major and minor axes of an ellipse are the same, then it is a circle. Let r be the length to which both a and b are equal.
Now, we substitute r for a and b in the formula for the area of an ellipse.
π ab ⟶ π ( r)( r) ⟶ π r^2
This is the formula for the area of a circle with a radius r!
Area of a Circle: π r^2
Therefore, we can say that the formula for the area of a circle is a special case of the formula for the area of an ellipse.
Area of an Ellipse: π a b
Area of a Circle: π r^2
