Circles

What shape could be more straightforward than a circle? It’s got a radius, a circumference, an area, and just two associated formulas:

$C = 2 π r$
$A = π r^{2}$

“What could be easier than that?” many people ask.

Ah, but that’s only a tiny piece of the story with circles. In this course you’ll find 17 (!) different helpful facts and formulas related to circles, covering ground like:

central angles
inscribed angles
chords
relationships between arcs subtended by crossing lines
the Power of a Point theorem

So… do you really know circles well enough to avoid common AMC traps?

One example

If you have a right triangle inscribed in a circle with radius $5$ , what’s the length of the triangle’s hypotenuse?

(spoiler)

It’s $10$ , because any right triangle inscribed in a circle has the diameter of the circle as its hypotenuse.

More generally, the inscribed angle (in this case, the right angle of the triangle) has half the measurement of the central angle (in this case , the $180$ -degree “angle” in the middle of the hypotenuse).

Other related content

Definitions

chord: A line segment whose (distinct) endpoints are both on the circumference of a circle. (So, a diameter is a chord that happens to go through the circle’s center.)
secant: A line containing a chord.
tangent: A line touching a circle at exactly one point. I.e. a line that “just grazes” the circumference.
subtend: When the rays of an angle cut into a circle, making a finite arc, the angle is said to subtend that arc.

When two chords or two secants cross, they create an angle that is related to the lengths of the arcs they subtend on the circumference.

Also, when two chords cross, the “subchords” created by the crossing point have lengths that are related to each other.

These facts are all covered in the quizzes for this section.