Circles

This course is a part of Mathematics II, a 14-course Topic series from Khan Academy.

Explore, prove, and apply important properties of circles that have to do with things like arc length, radians, inscribed angles, and tangents.

This course contains 13 segments:

Circle basics

Make sure you're familiar with notation and key terms like radius, diameter, circumference, pi, tangent, secant, and major/minor arcs before you dive into the rest of the circles content.

Arc measure

Arc measure is equal to the arc's central angle. We'll explore this fact and solve some problems related to it.

Arc length (degrees)

Think about the relationship between central angle and arc length. This tutorial uses degrees not radians.

Introduction to radians

Most people know that you can measure angles with degrees, but only exceptionally worldly people know that radians can be an exciting alternative. As you'll see, degrees are somewhat arbitrary.

Arc length (radians)

Think about the relationships between arc measures, central angles, and arc length in radians.

Sectors

Learn how to find the area of a sector.

Inscribed angles

We'll now dig a bit deeper in our understanding of circles by looking at inscribed angles and related properties.

Inscribed shapes problem solving

Use properties of inscribed angles to prove properties of inscribed shapes, then apply these properties some fun problem solving!

Properties of tangents

Explore, prove, and apply properties of circles that involve tangents.

Area of inscribed triangle

This more advanced (and very optional) tutorial is fun to look at for enrichment. It builds to figuring out the formula for the area of a triangle inscribed in a circle!

Constructing regular polygons inscribed in circles

Have you ever wondered how people would draw a square, equilateral triangle or even hexagon before there were computers? Well, now you're going to do just that (ironically, with a computer). Using our virtual compass and straightedge, you'll construct several regular shapes (by inscribing them inside circles).

Constructing circumcircles & incircles

In our study of triangles, we spent a decent amount of time think about incenters (the intersections of the angle bisectors) and circumcenters (the intersections of the perpendicular bisectors). We'll now leverage this knowledge to actually construct circle inscribed and circumscribed about a triangle using only a compass and straightedge (actually virtual versions of them).

Constructing a line tangent to a circle

Learn how to construct tangents to circles with certain conditions using compass and straightedge. For example, draw the tangent to a given circle that passes through a given point.