How do you translate radian measure into measurements?

Answer 1

When you work with triangles you are using degrees to measure its angles, but when you are working with circles is more convenient to use the radian measure.

What is a radian?

Definition: A radian is the measure of an angle that, when drawn as a central angle of a circle, intercepts an arc whose length is equal to the length of a radius of the circle.

For example,

1. When a central angle of a circle with a radius 1cm intercepts an arc of the circle which is also 1cm, this angle has 1 radian measure.

2. If you'll make this central angle bigger in order to intercept an arc of the circle whose length will be 2cm (or 2 radius length long), this bigger angle has a measure of 2 radians.

3. If you'll make the second central angle of the circle bigger in order to intercept an arc whose length will be 3cm (or 3 radius length long), this angle has a measure of 3 radians.

These three cases illustrate the following relationship:

measure of an angle in radians = length of the intercepted arc/length of the radius

So, if you label the angle with x, the length of the intercepted arc with s, and the length of the radius with r, you find the general formula: x = s/r

If you look at the picture of this circle you will see that in a half of the circle you can find a little more than 3 radians, and you need to go a little more far to make a full half of the circle. So, what is the number of radians that you need to go in order to make a full half of the circle?

Let's look at this.

Let's make a circle. If AB is a diameter of circle O with radius of length r, the points A and B separate the circle into two semicircles. Let's label the angle AOB with x, and the semicircle length with s.

You will see that the angle AOB is a central angle of the circle, so the measure of this angle in radians is equal to the length of the intercepted arc AB, which is s, divided by the length of the radius which is r. (See the general formula above)

You know that the circumference of the circle is:

C = 2(pi)r, since s = C/2 we write:

s = [2(pi)r]/2

s = (pi)r, substitute this at the general formula

x = s/r

x = [(pi)r]/r

x = pi

Since x is in radians we can say that in a half of the circle there is pi radians.

But we know at the same time that in a half of circle there is 180 degrees, so we can say that the following relationship is true.

180 deg = pi rad

Now you can see why is convenient to work with radians in a circle, because there is exactly pi radians in a half of the circle, and from this relationship we can see how radians are relating to degree measures.

Now for a full circle how many degrees are?

There are 360 degrees. How many radians it that will be?

180 deg = pi rad

360 deg = 2pi rad

So there are 2pi rad in a full of a circle.

Now you start to see why the circumference of a circle is 2(pi)r. Because we know that there are 2pi radius length arcs on a circle.

How to convert degrees to radians?

We know that:

180 deg = pi rad Let's write this a little bit differently

180(1 deg) = pi rad

1 deg = (pi/180) rad So, what about 5 degrees? Just multiply both sides by 5

5(1 deg) = 5(pi/180) rad What about x deg?

So, let' go t the general formula:

x(1 deg) = x(pi/180) rad

so you have the general formula to use it when you need to convert degrees to radians

How to convert radians to degrees? pi rad = 180 deg pi(1 rad) = 180 deg 1 rad = 180/pi deg What about 5 radians? Just multiply both sides by 5 5(1 rad) = 5(180/pi) deg What about x radians? So let's go to the general formula: x(1 rad) = x(180/pi) deg

So you have the general formula to use it when you need to convert radians to degrees.