Table of Contents
1. Revision
-Circumference of a circle = Radius•2•π
-The diameter is double the radius, so you could also say: Diameter•π
Legend
• = Multiplication
π = Pi
≈ = Approximately equal to
2. Unit circle
A unit circle is a circle with a radius of 1.
3. Arc
The arc is a part of the circumference of a circle.
4. Radian
A radian is an arc that has the same length as the radius. We know that a unit circle has a radius length of 1. That means that for a unit circle 1 radian will have an arc with the length of 1.
5. Dividing a circle in radians
When you are dividing a circle in radians, about 3.14 radians will fit in each half of the circle. So a full circle fits about 6.28 radians. The exact amount of radians that fit in half a circle is π (which is about 3.14), so a full circle fits 2π radians (which is about 6.28).
6. Angle of π radians
An angle of 180° is equal to a π amount of radians. This means that the amount of radians of a full circle equals to 2π. 2π therefore is 2•180° = 360°.
Calculating angles of radians
Lets first try something simple. For example a quarter of a circle. This is 1/4 part of the circle. 1/4 part of a 360° circle gives us the formula 1/4•360°. This gives us the solution of 90°.
Now an angle of a radian. A circle consists of 2π radians (about 6.28 radians) which equals to 360°. What we want to know now is an 1/ 2π part of a circle (so actually about an 1/6.28 part). This gives us the formula 1/(2π)•360°. This gives us the solution of about 57.30°.
This can be done a bit more simple. Instead of calculating 1/2π part of a full circle, you could also calculate 1/π part of half a circle of 180°. Now when we calculate the angle of a radian we get the formula: 1/π•180°.
This works the same when you try to calculate the angle of 4/10π radians. 4/10π is actually 4/10 part of π. We now know that π equals to 180°. So what we want to calculate is 4/10 part of 180°. Then we get the formula: 4/10•180°. This gives us a solution of 72°.
7. In summary
The size of the angle of 1 radian is about 57.30°. In math we use this angle as measurement. We call this measurement a radian, instead of the angle. So when a radian is mentioned, it refers to an angle of about 57.30°. The abbreviation of a radian is “rad”. P.S. “π rad” isn’t pronounced “pirate”. 😉
The ratio, in a radian, between the arc and the radius is always the same, because the length of the arc in a radian is the same as the radius. This means that a circle always consists of 2π radians and that the angle of the radian will always be 57.30°. In every circle with whatever measurements.
In short a radian is an arc with a length that is equal to the length of the radius. And a radian always has an angle of about 57.30°.
8. Formulas
From angles to π rad:
45° = 45/180°•π = 1/4•π rad
From π rad to angles:
0.5π rad = 0.5•180° = 90°
From decimal rad to angles:
1.5 rad = 1.5•(180°/π) ≈ 86°
From angles to decimal rad:
86° = 86/180°•π ≈ 1.5 rad
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