# Circles

Contents

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

15 exercises available

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

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

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

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

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

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

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!

Learn about the standard form to represent a circle with an equation. For example, the equation (x-1)^2+(y+2)^2=9 is a circle whose center is (1,-2) and radius is 3.

Learn how to analyze an equation of a circle that is not given in the standard form. For example, find the center of the circle whose equation is x^2+y^2+4x-5=0.

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).

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).

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.