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## High school geometry

# Arcs, ratios, and radians

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.## Reasoning about ratios

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

- Circle 2 is a dilation of circle 1.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.### 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.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.

### Why use radians instead of degrees?

Just like we choose different length units for different purposes, we can choose our angle measure units based on the situation as well.

Degrees can be helpful when we want to work with whole numbers, since several common fractions of a circle have whole numbers of degrees. Radians can simplify formulas, especially when we're finding arc lengths.

There are several other ways of measuring angles, too, such as simply describing the number of full turns or dividing a full turn into 100 equal parts. The most important thing is to make sure you've communicated which measurement you're using, so everyone understands how much of a rotation there is between the rays of the angle.

## Want to join the conversation?

- that gif about halfway down is new, weird, and interesting. I've never seen a gif on khan academy before.(16 votes)
- It's very helpful, in my opinion, too. Seeing the radius wrap around the circle to create the arc shows the idea clearly.(5 votes)

- I think that in the table above it would be clearer to say Fraction of a Circle instead of just Fraction, don't you agree?(15 votes)
- can someone reword what radians are plz(0 votes)
- A radian is another way to measure angles and arcs based on the idea that 1 radian is the length of the radius. Thus, you are converting line segment (radius) into an arc (radian).(11 votes)

- What is the measure (in radians) of central angle

θ in the circle(2 votes) - What is the length of arc

S

SS shown below?

A circle with a radius of eight centimeters. There are two radii that highlight a sector. The measure of the arc is S centimeters. The central angle measuring three radians.

3

33

8

cm

8cm8, start text, c, m, end text

S

SS

A circle with a radius of eight centimeters. There are two radii that highlight a sector. The measure of the arc is S centimeters. The central angle measuring three radians.

The angle in the figure is a central angle in radians.(0 votes) - What is the length of arc

S

SS shown below?

A circle with a radius of eight centimeters. There are two radii that highlight a sector. The measure of the arc is S centimeters. The central angle measuring three radians.(0 votes)