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High school geometry

Course: High school geometry > Unit 8

Lesson 4: Introduction to radians
Math
High school geometry
Circles
Introduction to radians
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Arcs, ratios, and radians

Google Classroom
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.
A circle with two radii marked and labeled. The outside edge of the circle between the radii is labeled arc. The area of the circle between the radii is labeled sector.
Circle B and its sector are dilations of circle A and its sector with a scale factor of 3.
Two circles. The circle on the left has the center labeled A. The circle on the right has the center labeled B. One fourth of both circles are shaded.
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
Radius length
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

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.
Two circles. The circle on the left is labeled circle one. The circle on the right is labeled circle two. Circle one is smaller than circle two. In circle one, a radius length is labeled R one, and arc length is labeled L one. The central angle measure of the arc in circle one is theta. In circle two, a radius length is labeled R two, and arc length is labeled L two. The central angle measure of the arc in circle two is theta.

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
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degree		theta, equals
start fraction, 1, divided by, 3, end fraction
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degree		theta, equals
start fraction, 1, divided by, 4, end fraction
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
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.
A circle. There are two radii that form a central angle. The arc length is shown to be equal to the length of the radius.
So radians are the constant of proportionality between an arc length and the radius length.
θ=arc lengthradiusθ⋅radius=arc 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.
A circle broken into seven sectors. Six of the sectors have a central angle measure of one radian and an arc length equal to length of the radius of a circle. The seventh sector is a smaller sector. The seven sectors represent the little more than six radians that it takes to make a complete turn around the center of a circle.

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.

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