Circles FAQ

There are many places where we use properties of circles in the real world. For example, architects often use arcs and sectors when designing buildings, and engineers use circles when designing gears and other mechanical parts. Additionally, understanding the properties of circles can help us solve problems in geometry, trigonometry, and calculus.

The radius of a circle is the distance from the center of the circle to any point on the circle. The diameter is twice the radius, or the distance across the circle through the center.

A tangent line is a line that touches a circle at just one point. A secant line is a line that cuts through a circle at two points.

Radians are a unit of measurement for angles.

One radian is the angle measure that we turn to travel one radius length around the circumference of a circle.

We often use radians in geometry because they make working certain formulas easier.

Arc length is the distance along a curved line, like on the edge of a circle.

On a circle, we need to know the central angle of the arc $(\theta)$left parenthesis, theta, right parenthesis and the radius of the circle $(r)$left parenthesis, r, right parenthesis to find arc length.

If $\theta$theta is in degrees:

If $\theta$theta is in radians:

A sector is a part of a circle, defined by two radii (the plural of radius) and the arc between them. Think of it as "pie slice" of the circle.

We need to know the central angle of the sector $(\theta)$left parenthesis, theta, right parenthesis and the radius of the circle $(r)$left parenthesis, r, right parenthesis to find the sector's area.

If $\theta$theta is in degrees:

If $\theta$theta is in radians:

Note that we could simplify the formula that uses radians a bit:

An inscribed angle is an angle that has its vertex on the circumference of a circle.

An inscribed shape is a shape that is drawn inside a circle such that all of its vertices are touching the circumference of the circle.