Arc Length
Arc Length
The definition of radian measure s = rθ
The unit circle
An angle of 1 radian
Proof of the theorem
IT IS CONVENTIONAL to let the letter s symbolize the length of an arc, which is called arc length. We say in geometry that an arc "subtends" an angle θ; literally, "stretches under."
Now the circumference of a circle is an arc length. And the ratio of the circumference to the diameter is the basis of radian measure. That ratio is the definition of π. π | = | C
D | . |
Since D = 2r, then π | = | C
2r | or, C r | = | 2π | . |
That ratio -- 2π -- of the circumference of a circle to the radius, is called the radian measure of 1 revolution, which are four right angles at the center. The circumference subtends those four right angles.
Radian measure of θ = | s r |
Thus the radian measure is based on ratios -- numbers -- that are actually found in the circle. The radian measure is a real number that names the ratio of a curved line to a straight, of an arc to the radius. For, the ratio of s to r does determine a unique central angle θ. Theorem. | | In any circles the same ratio of arc length to radius | | | determines a unique central angle that the arcs subtend. |
Proportionally,
if and only if θ1 = θ2.
We will prove this theorem below. Example 1. If s is 4 cm, and r is 5 cm, then the number | 4
5 | , i.e. | s r | , is the | radian measure of the central angle.
At that central angle, the arc is four fifths of the radius.
Example 2. An angle of .75 radians means that the arc is three fourths of the radius. s = .75r
Example 3. In a circle whose radius is 10 cm, a central angle θintercepts an arc of 8 cm.
a) What is the radian measure of that angle?
Answer. According to the definition: θ = | s r | = | 8
10 | = .8 |
b) At that same central angle θ, what is the arc length if the radius is
b) 5 cm?
Answer. For a given central angle, the ratio of arc to radius
The definition of radian measure s = rθ
The unit circle
An angle of 1 radian
Proof of the theorem
IT IS CONVENTIONAL to let the letter s symbolize the length of an arc, which is called arc length. We say in geometry that an arc "subtends" an angle θ; literally, "stretches under."
Now the circumference of a circle is an arc length. And the ratio of the circumference to the diameter is the basis of radian measure. That ratio is the definition of π. π | = | C
D | . |
Since D = 2r, then π | = | C
2r | or, C r | = | 2π | . |
That ratio -- 2π -- of the circumference of a circle to the radius, is called the radian measure of 1 revolution, which are four right angles at the center. The circumference subtends those four right angles.
Radian measure of θ = | s r |
Thus the radian measure is based on ratios -- numbers -- that are actually found in the circle. The radian measure is a real number that names the ratio of a curved line to a straight, of an arc to the radius. For, the ratio of s to r does determine a unique central angle θ. Theorem. | | In any circles the same ratio of arc length to radius | | | determines a unique central angle that the arcs subtend. |
Proportionally,
if and only if θ1 = θ2.
We will prove this theorem below. Example 1. If s is 4 cm, and r is 5 cm, then the number | 4
5 | , i.e. | s r | , is the | radian measure of the central angle.
At that central angle, the arc is four fifths of the radius.
Example 2. An angle of .75 radians means that the arc is three fourths of the radius. s = .75r
Example 3. In a circle whose radius is 10 cm, a central angle θintercepts an arc of 8 cm.
a) What is the radian measure of that angle?
Answer. According to the definition: θ = | s r | = | 8
10 | = .8 |
b) At that same central angle θ, what is the arc length if the radius is
b) 5 cm?
Answer. For a given central angle, the ratio of arc to radius