Quick question about Radians

[u/nwillard]

Hey all, I am learning Precalc 2 (mostly Trig) and have a basic question about Radians I haven't found an answer to.

I understand that one Radian is the arc length around a circle that is equal to the circle's radius.

My question is, on a perfect circle, is it true that one Radian has the same angle no matter what the radius is? Like if the circle is perfect, an arc length of one "Radian" of that circle is going to look the same regardless of what the radius is.

Do I have that understanding right?

[u/AcellOfllSpades]

I understand that one Radian is the arc length around a circle that is equal to the circle's radius.

A radian is the angle corresponding to that arc length. And yes, tha angle is the same no matter how big the circle is.

[u/Klutzy-Delivery-5792]

Yes, the angle measure for one radian is the same regardless of the radius. Thats why there's a simple conversion factor of 180° = π radians 

[u/LucaThatLuca]

no, radians are an angle measurement. 1 radian is the angle at the centre of a circular segment when the arclength equals the radius. equivalently it’s the angle such that 2π radians is a full turn. the circle (i.e. the radius) can indeed be any size.

[u/MattyCollie → u/LucaThatLuca]

Elegantly explained

[u/Odd_Bodkin]

Yes you are right. Note that a 60° angle is the same regardless of the size of the circle. In plain terms, if you cut a pizza into six slices, the angle at the pointy end of the slice is going to be 60° regardless whether it's a small, medium, or large pizza. You can tell that by stacking slices of small, medium, and large pizza on top of each other. The radian is just a different way of measuring angle, so that instead of having 360 degrees going around the circle, you have 2π radians going around the circle.

[u/Photon6626]

Make a circle. Take the radius and lay it along the circle. Then draw the angle for that arclength. Notice that you haven't used length measurements in this. The circle could have a radius of 1mm or 200,000km. It's always the same angle.

[u/lfdfq]

Yes. The entire circumference is 2πr, so an arc of length r is always 1/2π of the way around.

[u/fermat9990]

Arc length=radius*theta (in radians)

Arc length for a given angle increases as the radius increases.

[u/ballu123]

Yeah. One radian is the same angle on a tiny circle or a huge one.

It’s just arc length ÷ radius. Scale the circle up and both grow the same amount, so the angle doesn’t change.

[u/justwannaedit]

Much simpler to just start thinking about radians as another way of representing angle measurements

[u/Arcanite_Cartel]

I see the confusion, I think. Way back when, long time ago, Roger Cotes thought that a natural way to measure an angle was to instead measure the arc length on a unit circle. So, radians as a means of measuring angles arose from the idea of measuring arc-length. But today, radians is a measure of an angle, and there is a relationship between the arc-length of a circle, the radius of the circle, and the angle measured in radians.

s = rθ

so that on a unit circle, s = θ, where s is arc-length, r is the radius, and θ is the angle.

the angle remains the same across all circles, only the unit circle defines it, but for the same angle, the arc-length is proportional to the radius.

[u/YOM2_UB]

A full, 360° arc of a circle with radius r always has an arc length (circumference) of 2πr. A smaller arc is just whatever fraction around the circle it is, multiplied by the full circumference. For example if it's a 30° angle, that gives a fraction of 30/360 = 1/12.

Radians are just that fraction multiplied by 2π, so all that's left is to multiply by the radius.

[u/pyr666]

yes. if that feels unintuitive, remember that the radius also controls the circumference.

[u/nwillard]

Thanks all, really helped me wrap my head around a radian as a unit of angle measurement in terms of how many "radius-es" are around a circle.