The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This essential trains for: SAT-I, SAT-II, GMAT, AMC-8, Math Kangaroo 7-8, Math Kangaroo 9-10.

Definition: A circle is the locus (i.e. set of points) of all the points in a plane that are equally far from a fixed point called 'center'.

Note: In 3D space, this is the definition of a sphere.

Definitions:

figure

The circumference is the length of the arc that comprises the whole circle. It is similar to the notion of 'perimeter' for polygons.

Experimentally, we can measure the circumference of a circle by winding a thread around the circle until the two ends meet, unwind the thread and measure its length with a ruler. The thread must be inelastic and of zero width (a completely abstract thread).

The circumference of the circle is directly proportional (i.e. varies directly) with the radius of the circle:

equation

Concentric circles are circles that have the same center:

figure

The length of an arc that subtends a central angle is also directly proportional to the measure of the angle it subtends.

To find the length of an arc when given the central angle that subtends it and the radius of the circle think this way: if the angle is 360° then we have a complete circle and the length of the arc is the same as the circumference. Since the angle is different, apply direct proportionality:

AngleArc length
equationequation
equationequation

equation

equation

From the formula above we see that the arc length is also proportional to the radius. In the figure, the red arc length L is proportional to the red radius R. The blue arc length l is proportional to the blue radius l. Note that they both subtend the same angle .

figure

The area is a measure (in given units) of the surface enclosed by a circular contour.

Together with the points inside it, a circle forms a disc.

The area of a circle varies quadratically as the radius:

equation

A sector of a circle is a slice shaped figure that is delimited by the circle and two different radii:

figure

The area of a sector varies directly as the central angle:

figure

To compute the area of a sector think like this: "if the central angle is 360° we have a complete circle and the area of the sector is the same as the area of the whole circle." Since the angle is different, apply direct proportionality:

AngleArea
equationequation
equationequation

equation

equation

A segment of a circle is the area enclosed between the circle and a chord.

figure

The area of a segment can be calculated using:

equation

Common tangent theorem Two segments that have a common end while their other ends are points of tangent contact to the same circle, have equal lengths.

figure

This is because the triangles PMO and PNO are congruent (they have a right angle and two congruent pairs of edges: the radius and the segment PO which is common to both).

equation

equation

For tangent circles it is important to immediately mark on the figure that:

The two centers and the point of contact are collinear (i.e. on the same line).

figure

This happens for interior tangent circles as well:

figure

In problems, this establishes the important fact that:

equation

Example:

Compute the length of the segment MN if you know the radii of the two tangent circles are R and r:

figure

Careful! The magenta segment is the hypotenuse - apply the Pythagorean theorem correctly.

equation

equation

equation

equation