If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## High school geometry

### Course: High school geometry>Unit 8

We can use the constant of proportionality between the arc length and the radius of a sector as a way to describe an angle measure, because all sectors with the same angle measure are similar.

## Dilated circles and sectors

All circles are similar, because we can map any circle onto another using just rigid transformations and dilations. Circles are not all congruent, because they can have different radius lengths.
A sector is the portion of the interior of a circle between two radii. Two sectors must have congruent central angles to be similar.
An arc is the portion of the circumference of a circle between two radii. Likewise, two arcs must have congruent central angles to be similar.
Circle B and its sector are dilations of circle A and its sector with a scale factor of 3.
Which properties of circle B are the same as in circle A?
PropertySame or different
Area of the sector
Central angle measure of the sector
Length of the arc defined by the sector
Ratio of the circle's circumference to its radius
Ratio of the arc's length to the radius

When we studied right triangles, we learned that for a given acute angle measure, the ratio start fraction, start text, o, p, p, o, s, i, t, e, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, space, l, e, n, g, t, h, end text, end fraction was always the same, no matter how big the right triangle was. We call that ratio the sine of the angle.
Something very similar happens when we look at the ratio start fraction, start text, a, r, c, space, l, e, n, g, t, h, end text, divided by, start text, r, a, d, i, u, s, space, l, e, n, g, t, h, end text, end fraction in a sector with a given angle. For each claim below, try explaining the reason to yourself before looking at the explanation.
The sectors in these two circles have the same central angle measure.

### Claims

1. Circle 2 is a dilation of circle 1.
2. If the scale factor from circle 1 to circle 2 is start color #7854ab, k, end color #7854ab, then r, start subscript, 2, end subscript, equals, start color #7854ab, k, end color #7854ab, r, start subscript, 1, end subscript.
3. The arc length in circle 1 is start color #bc2612, ell, start subscript, 1, end subscript, end color #bc2612, equals, start color #bc2612, start fraction, theta, divided by, 360, degree, end fraction, dot, 2, pi, r, start subscript, 1, end subscript, end color #bc2612.
4. By the same reasoning, the arc length in circle 2 is start color #a75a05, ell, start subscript, 2, end subscript, end color #a75a05, equals, start color #a75a05, start fraction, theta, divided by, 360, degree, end fraction, dot, 2, pi, r, start subscript, 2, end subscript, end color #a75a05.
5. By substituting, we can rewrite that as start color #a75a05, ell, start subscript, 2, end subscript, end color #a75a05, equals, start fraction, theta, divided by, 360, degree, end fraction, dot, 2, pi, start color #7854ab, k, end color #7854ab, r, start subscript, 1, end subscript.
6. So start color #a75a05, ell, start subscript, 2, end subscript, end color #a75a05, equals, start color #7854ab, k, end color #7854ab, start color #bc2612, ell, start subscript, 1, end subscript, end color #bc2612.
7. In conclusion, start fraction, start color #bc2612, ell, start subscript, 1, end subscript, end color #bc2612, divided by, r, start subscript, 1, end subscript, end fraction, equals, start fraction, start color #a75a05, ell, start subscript, 2, end subscript, end color #a75a05, divided by, r, start subscript, 2, end subscript, end fraction.

### Conclusion

The ratio of arc length to radius length is the same in any two sectors with a given angle, no matter how big the circles are!

## A new ratio and new way of measuring angles

For any angle, we can imagine a circle centered at its vertex. The radian measure of the angle equals the ratio start fraction, start text, a, r, c, space, l, e, n, g, t, h, end text, divided by, start text, r, a, d, i, u, s, end text, end fraction. The angle has the same radian measure no matter how big the circle is.
Complete the table with the measure in degrees and the value of the ratio start fraction, start text, a, r, c, space, l, e, n, g, t, h, end text, divided by, start text, r, a, d, i, u, s, end text, end fraction for each fraction of a circle.
FractionCentral angle measure (degrees)Central angle measure (radians) theta, equals, start fraction, start text, a, r, c, space, l, e, n, g, t, h, end text, divided by, start text, r, a, d, i, u, s, end text, end fraction
start fraction, 1, divided by, 2, end fraction
degree
theta, equals
start fraction, 1, divided by, 3, end fraction
degree
theta, equals
start fraction, 1, divided by, 4, end fraction
degree
theta, equals

### More ways of describing radians

One radian is the angle measure that we turn to travel one radius length around the circumference of a circle.
So radians are the constant of proportionality between an arc length and the radius length.
\begin{aligned}\theta&=\dfrac{\text{arc length}}{\text{radius}}\\\\ \theta \cdot \text{radius}&=\text{arc length} \end{aligned}
It takes 2, pi radians (a little more than 6 radians) to make a complete turn about the center of a circle. This makes sense, because the full circumference of a circle is 2, pi, r, or 2, pi radius lengths.