Question Video: Converting Radians to Degrees | Nagwa Question Video: Converting Radians to Degrees | Nagwa

Question Video: Converting Radians to Degrees Mathematics • First Year of Secondary School

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True or False: To convert from radians to degrees, you should multiply by πœ‹/180Β°.

03:17

Video Transcript

True or False: To convert from radians to degrees, you should multiply by πœ‹ over 180 degrees.

Radians and degrees are both units of measurements of angles. And so let’s begin by recalling what we mean by radians. We say that one radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. What we most commonly need to do, however, is work out a conversion between an angle in radians to one in degrees, or vice versa. To do this, we begin by recalling that the circumference is the distance around the outside of a circle. It can be calculated by working out πœ‹ times the diameter or alternatively as two times πœ‹ times the radius.

If we look at the sector in the circle with an arc length of π‘Ÿ, then we might wonder how many arcs of length π‘Ÿ can be drawn on the circumference. As we define the length of the arc to be π‘Ÿ, which is the same as the radius π‘Ÿ, then that means that the number of arcs that can fit on the circumference would be two πœ‹π‘Ÿ over π‘Ÿ, which simplifies to two πœ‹. We can even verify this by sketching on the circle. We know that πœ‹ is approximately equal to 3.14 to two decimal places. Multiplying that by two would give us 6.28 to two decimal places. We can see on the diagram that there are six arcs of length π‘Ÿ and a little piece of the circumference left over.

So let’s consider how all of this helps us to convert between radians and degrees. Because we know that two πœ‹ arcs make up the full circumference of the circle, then there must be two πœ‹ sectors making up the whole circle. And since the angle of each sector is one radian, then we can say that the sum of these angles is two πœ‹ radians. But of course, we also know that the sum of the angles in a circle in degrees is 360 degrees. And this gives us our first useful conversion that two πœ‹ radians is equal to 360 degrees. We can also divide both sides of this equation to give us that πœ‹ radians is equal to 180 degrees.

It’s useful to remember either one or both of these conversions to help us convert angle measures. We can do this using the second conversion to find that one radian must be equal to 180 degrees over πœ‹. And then, if we had any unknown angle of πœƒ radians, then we would take πœƒ and multiply it by 180 degrees over πœ‹. Notice that the statement in the question says that to convert from radians to degrees, you should multiply by πœ‹ over 180 degrees. This is false. In fact, alongside the two other conversions, we should also memorize the fact that to change any angle in radian measure to one in degrees, we multiply the given angle by 180 degrees over πœ‹.

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