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### Course: High school geometry>Unit 8

Lesson 5: Arc length (from radians)

# Arc length from subtended angle: radians

Sal finds the length of an arc using the radius and the radian measure of the angle subtended by the arc. Created by Sal Khan.

## Want to join the conversation?

• Why exactly is the degree measure corresponding to the length of its arc?
• That is how radian is defined. One radian is the angle where the arc length equals the radius.

You probably know that the circumference of a circle is 2πr.
And 2π radians = 360°. That is not a coincidence.
• Can't you use proportion to solve this instead of what Sal did?
• Yes you could solve this problem using proportions. angle/ 2pi = arc/circumference
0.4/2pi =?/2pi.5. Sals method helps us think about what a radian actually is by definition...and it is faster.
• So this formula is true right?
, ie, radian = length of arc/radius
so, length of arc = radian * radius
• What does radii mean?
• You can also say "radiuses," but radii is probably better.
• why did he multiply 0.4 radii by 5(length of radius)? what do you get from that? i m still not sure what radian is...so confused T_T
• He multiplied them in order to get the length of the arc.

A radian is a unit for angles, similar to degrees. However, radians measure the amount of radii that is formed when you draw an arc or a part of a circle. This is based on the fact that the radius can fit 2pi(~6.28) times a circle, or a full rotation. There are 2pi radii in a full circle, or 2pi radians.

• How did he get 2 from 0.4 and 5?
• What is an arch
(1 vote)
• An arch is a structure that spans a space and resolves forces into compressive stress. But that is not important to our context.

An arc is a portion of a circle or other curvilinear shape.

Interestingly enough, most arches are in the shape of an arc.
• Could this method have also worked?

1 radian = 180/pi degrees

1 radian * 0.4 = 180/pi * 0.4 = 72/pi degrees

72/pi degrees/ 360 = 1/5 pi (Arc length formula: n/360 * 2 pi r)

Then,
2 pi r = 10 pi, so 1/5 pi * 10 pi = 2 units.

2 units is the answer
• Why would you ever want to convert radians to degrees first? The equation for arc length in radians is so much simpler:
`l = θr`
`l = 0.4*5 = 2`
In fact, the equation of arc-length in degrees actually converts the degrees into radians with the expression `2π/360` (which is the same as `π/180`). So the answer is yes, that works, but it's entirely ridiculous.
• What exactly is a radian? Other than "unit of measure for angles".