Circles

Circles come with a set of standard terms for describing their parts. Use the diagram below as a reference.

Any time you see a circle problem, you’ll likely use at least one of these two equations.

Let’s use these equations in an example.

What must be the radius of a circle if both the area and the circumference of the circle are equal in terms of units?

First, interpret what “in terms of units” means here. Circumference is a 1-dimensional measurement (for example, $X$ meters), while area is a 2-dimensional measurement (for example, $X$ square meters). The question is telling you to compare the numerical values only, ignoring the fact that the units are different.

Most circles do not have equal numerical values for area and circumference. This question asks for the specific radius where they match. Set the area formula equal to the circumference formula and solve for $r$.

$$\begin{array}{rl}\text{area}& =\text{circumference}\\ \pi r^2& =2r\pi \\ r^2& =2r\\ r& =2\end{array}$$

The only circle whose area equals its circumference (numerically) has a radius of $2$.

The central angle theorem (CAT)

The central angle theorem (CAT) is a rule that often turns a hard-looking circle problem into a straightforward one.

It says:

• Pick two points on the circle.
• The angle formed at the center of the circle by those two points is twice the angle formed by the same two points and any third point on the circle.

It’s much easier to see than to read:

In the diagram above, points B and F are the centers of their respective circles. According to the central angle theorem, we can write:

• $2\ast \angle \text{CAD}=\angle \text{CBD}$
• $2\ast \angle \text{GEH}=\angle \text{GFH}$

The key idea is that the third point (like A or E) can be anywhere on the circle, and the 2-to-1 relationship still holds.

This also works when the central angle is $180^{\circ }$. In that case, the inscribed angle is $90^{\circ }$. This matters because it means:

• Any right triangle inscribed in a circle has its hypotenuse as the diameter of the circle.

In the figures below, B and F are the centers of their circles.

Let’s see how this works in an example.

Points A and B are the centers of their respective circles, and both circles have the same radius.

Quantity A: $y$
Quantity B: $x$

Try to solve this using the central angle theorem.

Start with $y$. By the central angle theorem, the central angle is twice the inscribed angle. Since the inscribed angle is $44^{\circ }$, you get:

• $y=2(44^{\circ })=88^{\circ }$

Now solve for $x$. The arrow shape inside the circle forms a quadrilateral, so the sum of its interior angles is:

• $(4-2)\cdot 180^{\circ }=360^{\circ }$

You’re given two interior angles of $24^{\circ }$. To find the other two:

• One interior angle is the large obtuse angle at the center. A full circle is $360^{\circ }$, so that angle is $360-x$.
• The last interior angle is at the rightmost point on the circle. It’s an inscribed angle that intercepts the same arc as the central angle $x$, so by CAT it equals $x/2$.

Now add the four interior angles and set the sum equal to $360^{\circ }$:

$$\begin{array}{rl}24+24+(360-x)+x/2& =360\\ 24+24+360-x+x/2& =360\\ 24+24-x+x/2& =0\\ 48-x+x/2& =0\\ 48-x/2& =0\\ 48& =x/2\\ 96& =x\end{array}$$

Now compare the quantities:

• $y=88$
• $x=96$

So Quantity B is greater than Quantity A.