Circles and Ellipses Part 1 : What are Circles and Ellipses?

Photo by Patrick McManaman on Unsplash

What is a circle?

A circle is a shape that describes the set of points (locus) that are a certain constant distance from a center point.

The distance from the center point to the circle is called the radius (r). The diameter (d) is any straight line segment that passes through the center with endpoints that lie on the circle. The length of the diameter is equal to twice the length of the radius (\(d = 2r\)). The circumference (C) is equal to the distance all the way around the circle. A unique property all circles share is that the ratio of every circle’s circumference to its diameter is constant. We call this constant pi (\( \pi \)), which is an irrational number that is usually approximated as 3.14. Irrational numbers can’t be written as a ratio (fraction) of two integers; they are non-repeating and non-terminating. Because the relationship between a circle’s circumference (C) and its diameter (d) is an irrational number, it is impossible to form a circle that has integers for BOTH the circumference and diameter at the same time. The area (A) of a circle is the amount of space enclosed by the circle’s circumference. 

What is an ellipse?

An ellipse is a regular oval shape surrounding two focal points (the foci).

The ellipse itself is the set of all points such that the sum of the distance from the two foci to each point on the ellipse is a constant. The straight line that passes through the two foci of an ellipse is called the major axis; the major axis also happens to be the longest diameter of the ellipse. The line perpendicular to the major axis that passes through the center of the ellipse is called the minor axis. The eccentricity of an ellipse is the measure of how elongated it is. In calculations, it is convenient to use the distance from the center to the ellipse along both the major and minor axes. We call these two distances the semi-major and semi-minor axes because they are equal to half of the length of the major and minor axis, respectively.

Ellipses can be more elongated vertically than horizontally as well. Regardless of how it is oriented, the semi-major axis is always the longer of the two axes.

A circle is a particular kind of ellipse with an eccentricity of \(0\). This is true because both foci are the same point (the center point of the circle). This means that the distance from the center to each focus is \(0\) . The semi-major and semi-minor axes are therefore equivalent and are equal to the radius of the circle. 

Properties of circles

Conventionally, circles are divided into 360 equal angle measures, called degrees (\(^\circ\)). 

Degrees in a circle are useful for calculating or measuring what fraction of the total circle is taken up by an arc or a sector. An arc is a portion of the circumference of a circle, and a sector is the corresponding wedge of a circle swept out by an arc. 

Another useful angle measure is the radian. A radian is defined as the central angle created by an arc of length r, the radius, on the circle. Therefore, just as there are 360 degrees in a circle, there are \(2\pi\) radians in a circle. In other words, \(360^\circ = 2\pi\) radians. 

To convert from degrees to radians (or vice versa), use the following formula: \(x^\circ*\frac{2\pi}{360^\circ}=y_{radians}\)

For any sector of a circle, the central angle, radius, and corresponding arc length are related by the following equation:

Arc Length: \(\theta_{radians} = \frac{S}{r}\)   or   \( S =2\pi r \frac{\theta_{degrees}}{360^\circ}\) 

Sector Area: \(A=\pi r^2 \frac{\theta_{degrees}}{360^\circ} \)

where S is arc length and r is the radius. 

 

Equation of a circle

A circle can be described using the following equation:

\( (x-h)^2 + (y-k)^2 = r^2 \)

The reason this is true is much easier to understand if we also think about triangles. Let’s look at the following triangle ABC: 

The hypotenuse of this triangle is \(\overline{AC}\). Its length can be found using the Pythagorean Theorem,  \(a^2 + b^2 = c^2 \) (more on that here). Let’s consider a circle centered around point A that also passes through point C. 

In this case, \(\overline{AC}\) is equal to the radius of the circle. To calculate how long the radius is, we can solve the equation \(\overline{AB}^2 + \overline{BC}^2 = \overline{AC}^2 \). Now, let’s imagine that the circle is graphed on a normal xy coordinate plane (Cartesian plane) with its center at the origin (0,0). 

The coordinates of point C are \((\overline{AB}, \overline{BC})\). In other words, \((\overline{AB}\) and \(\overline{BC}\) are the x and y coordinates of point C, respectively. That will allow us to substitute (X, Y) for \((\overline{AB}, \overline{BC})\). This means that if \(\overline{AC}\) = the radius of the circle, then the circle is defined by all points (x, y) such that \( x^2 + y^2 = \overline{AC}^2 = r^2\).  Therefore, a circle centered at the origin can be described using the equation \(x^2 + y^2 = r^2\).

What if the circle isn’t centered at the origin but instead at some point (h, k) where h is the x coordinate and k is the y coordinate? The form of the equation remains the same, but we need to shift each point on the circle the same amount as we shifted the center point from the origin. 

If the circle is centered at (h, k), to get it back to the origin (and our original equation), you would need to move it in the (-h, -k) direction. Therefore, the equation for a circle with a center point at  (h,k) is \((x-h)^2 + (y-k)^2 = r^2 \) (if the circle is centered at the origin, (h, k) is (0, 0)). 

Equation of an ellipse

The equation of an ellipse, centered at the origin, is below:

 \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

This equation looks suspiciously similar to the equation for a circle! Let’s look at the equation for a circle again, with one change: we will divide the entire equation by \(r^2\). (If you need a refresher on dividing equations by a particular term, including why a term divided by itself equals 1, check out our Arithmetic unit!)

 \(x^2+y^2 = r^2 \, \, \rightarrow \, \,  \frac{x^2}{r^2} + \frac{y^2}{r^2} = 1 \)

Because an ellipse has more than one diameter length, it technically doesn’t have one radius! That means that we need to do something a bit different to make the equation work. Here is the equation for an ellipse again:

 \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

\(r^2\) has been replaced by \(a^2\) for the major axis (that’s the longer one), and \(b^2 \) for the minor axis. 

Equation of foci 

Another equation you will occasionally see on the ACT® is that for the foci of an ellipse: 

\(a^2 – b^2 = c^2\). 

 The foci are the two points around which the ellipse is drawn. (As a side note, “foci” is the plural of the word “focus.”)  

This equation might remind you of the Pythagorean Theorem with one key difference: \(b^2\) is subtracted from \(a^2\) instead of added to it. “a” is technically the hypotenuse of the triangle below: 

You might also notice that the semi-major axis is the same length as the hypotenuse of \(\triangle ABC\) ! You won’t need to know the details, but basically, because a is the hypotenuse instead of c (which is the hypotenuse in the Pythagorean Theorem), we rearrange the Pythagorean Theorem and get \(a^2 – b^2 = c^2\) as the equation for the foci of an ellipse. 

The coordinates for the foci of an ellipse centered at the origin are \((\pm \, c, 0)\), with c being the distance from the center of the ellipse to each focus.