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Date: 03/19/2002 at 23:09:20
From: Mandy

How do you find the degree measure for an angle from pi/60 rad?  I
have a chart that I made of conversions, but I don't know how to do it

If the radian measure of an angle is doubled, is the degree measure of
the same angle also doubled?

Thank you.
```

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Date: 03/20/2002 at 09:08:00
From: Doctor Peterson

Hi, Mandy.

Radians and degrees are just different units for measuring angles,
like feet and meters; so they are proportional, and when you double an
angle, its measure doubles whether in degrees or radians.

Imagine drawing an angle, and then drawing a circle (the size doesn't
matter) with its center at the vertex. We measure an angle by
measuring the length of arc subtended by the angle - that is, by
finding how long the part of the circle between the two sides of the
angle is. (That's what a protractor does, for example; it's just a
ruler curved around a circle.)

To measure in degrees, we might wrap a flexible measuring tape around
the whole circle and mark off the circumference; then lay it out
straight and divide that into 360 equal parts, so that each degree on
the tape is 1/360 of the whole circumference. Now we can wrap it back
around the circle, and measure the angle by counting degrees.

To measure in radians, we lay the tape straight and mark off the
radius of the circle. That itself becomes our unit - do you see how
much more natural that is than degrees? We can then divide the radian
into whatever parts we choose, say hundredths. Then we wrap it around
the circle and measure how many radians, in hundredths, the angle
covers.

or about 6.28, radians, since the circumference is 2 pi times the
radius, which is the unit we used. Since a whole circle is also 360
degrees, just because that is the number of parts we chose to divide
the circle into, we see that

360 degrees = 2 pi radians

Now we can convert between the two units by using the fraction

360 degrees
------------ = 1

For example, given an angle of pi/60 rad, this is equal to

360 deg    pi    360
pi/60 rad * -------- = -- * ---- deg = 3 deg
2 pi rad   60   2 pi

because we can cancel pi, 60, and 2 from the product.

It is also important to know that the area of a sector is proportional
to the angle, so that for example a 90 degree sector is 90/360 = 1/4
of a circle, and has area 1/4 pi r^2.

Write back if you need more help.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Definitions
High School Euclidean/Plane Geometry
High School Geometry
Middle School Definitions
Middle School Geometry
Middle School Terms/Units of Measurement
Middle School Two-Dimensional Geometry

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