Circles

A circle with the center labeled c, the diameter labeled d, the tangent labeled t, and the secant labeled s.

Circles are a special type of ellipse. They have an infinite amount of symmetrical lines. If all of the radii drawn from the center of the circle to the point across the center are equidistant to the center, then the ellipse is a circle.

Properties

Circumference

The circumference of a circle is the distance around the circle. The formula for circumference is

$C = 2\pi \, r$, or

$C = \pi\,d$.

For instance, find the circumference of a wheel with a radius of 1.5 feet to the nearest hundredth.

$C = 2\pi \, r$

$C = 2\pi\,*1.5$

$C = (2*1.5)\pi\,$

$C = 3\pi\, \approx\, 9.4247779607693... \approx\, 9.42$

The wheel has a circumference of approximately 9.42 feet.

Use To Determine Speed

Circumference can be used to determine the speed of a rotating object. The example below shows how.

On a the vinyl record, the orange line represents a point on the edge of the record, and the red line represents a point on the edge of the label. The "vinyl record example" describes how to prove that the point on the edge of the record is faster than the point on the edge of the label.

Let's take a vinyl record and compare the speeds of points on the edges of the record itself and the label. The point on the edge of the record is 5 inches from the point on the edge of the label, which is 3 inches from the center of the record. The record makes 60 revolutions per minute. Which point goes faster?

Step 1. Determine the circumferences of the circles made by both points.

Point on record edge:

$C = \pi\,(2 * 8) in.$

$C = 16\pi\, in. \approx\, 50.3 in.$

Point on label edge:

$C = \pi\,(2 * 3) in.$

$C = 6\pi\, in. \approx\, 18.8 in.$

Step 2. Determine the average speeds of both points in inches per second to the nearest thousandth.

Point on record edge:

$S = \frac{d}{t}\,$

$S \approx\, \frac{50.3 in.}{60 sec.}\,$

$S \approx\, 0.838 in./sec.$

Point on label edge:

$S = \frac{d}{t}\,$

$S \approx\, \frac{18.8 in.}{60 sec.}\,$

$S \approx\, 0.313 in./sec.$

Step 3. Compare and conclude.

$0.838 in./sec. > 0.313 in./sec.$

The point on the record's edge is faster than the point on the label's edge.

Area

The area of a circle can be found by using this formula:

$A = \pi\,r^2$.

Find the area of the same wheel with the same radius of 1.5 feet to the nearest hundredth.

$A = \pi\,r^2$

$A = \pi\,*1.5^2$

$A = 2.25\pi\, \approx\, 7.068583470577035... \approx\, 7.07$

The area of the wheel is approximately 7.07 feet².

Trigonometric Ratios in Circles

Trigonometric ratios in circles are always measured in radians, rather than degrees for triangles.

The sine of an angle of 27° in radians:

$\sin{27}\, \approx\, 0.956375928404503... \approx\, 0.9564$

The sine of an angle of 27° in degrees:

$\sin{27}\, \approx\, 0.453990499739547... \approx\, 0.454$