Angles can be drawn facing any direction you want, but when you put them in standard position, it's easier to estimate the angle measure and work with them using graphs and coordinates. In this lesson, we'll explore drawing angles in standard position.
You can draw angles in any position without affecting their measure. You can turn them upside down or have them look like a greater-than sign or less-than sign, any position you like. However, in angular geometry there is one position that's considered the standard position for angles. You see, angular geometry is all about rotation around a circle, and a number of degrees (or radians) will tell you how far you've rotated from a starting point.
Say that you're out in the woods wandering around and someone says 'go east.' You need to know which way east is, so you grab your compass, which tells you which way north is. From there, you know to turn 45 degrees to the right to go straight east. Angles are the same way. You start at one point, the origin, then measure from there. That's what standard position for angles is all about: measuring from a pre-defined starting point.
Standard Position for Angles
So what exactly does the standard position for angles look like? On the Cartesian Coordinate Plane, standard position for angles is when the vertex of the angle sits at the origin of the x-axis and y-axis, and one side of the angle starts at the origin and extends to the right along the x-axis. There's a special term for that side of the angle along the x-axis. The initial side is the side of the angle that starts at the origin and extends to the right along the x-axis between Quadrant I and Quadrant III. To complete the angle, we need a terminal side, which also starts at the origin and extends in a direction counter-clockwise from the initial side.
Practice Using Degrees
Let's try our hand at drawing an angle in standard position measured in degrees. Remember, there are 360° in a full circle. You start with the x- and y-axes, which cross at a right (90°) angle. The initial side of your angle will start at the origin, the point where the axes cross, and then go straight to the right.
Now comes the fun part. You have to figure out how far to rotate around the origin to reach the measure that you need. An excellent way to do this accurately is to use a protractor. Let's say your angle is 30°. Line up the baseline of your protractor with the initial line of your angle, making sure that the origin is centered on the protractor. Now follow the curve on the protractor until you get to the 30° point. Once you connect that mark to the origin, you've drawn a 30° angle in standard position.
Practice Using Radians
Let's try another example, this time where the angle is measured in radians. Remember, a radian is a unit of measurement based on the fact that π is the ratio between the diameter and the circumference of a circle. That means that the distance all the way around the origin is π times the diameter. Since angles in standard form are all based on a unit circle, where a hypotenuse length equals 1, then the radius of the circle you're working with is 1. This makes trig functions a lot easier to do.
A radius of 1 means that the diameter of our circle rotation will be 2 (twice the radius), and so the distance around the circle is 2π. Angle measurement in radians is based on that 2π number. And so, π radians is halfway around the circle, π/2 is a fourth of the way around the circle, and so forth.
Protractors usually mark angles in degrees, so we'll do a conversion. Let's draw an angle in standard position that measures 2π/3 radians. First, let's convert 2π/3 to degrees. Remember, 360° = 2π radians, and converting means multiplying by 360°/2π or 2π/360°, depending on which measure we want. The unit we already have will be on the bottom; the unit we want will be on top. So, it looks like that devious 2π/3 they gave us is merely a 120° angle.
Let's draw it! Using the initial side you drew before, we'll put the protractor back and mark the 120° position. Now we can connect that mark to our origin, completing the 120° angle.
Standard position for angles starts on the line between Quadrants I and IV in the Cartesian Coordinate Plane, then rotates counterclockwise as the angle increases. The initial side of the angle starts at the origin, where the x- and y-axes cross, and goes straight to the right. The terminal side starts at the origin and extends out at some angle from the initial side. You can use a protractor to measure angles.
To draw an angle in standard position:
- Draw one line extending from the origin along the 0° position (the line between Quadrants I and IV)
- Use that line for a reference on your protractor and mark a point at the angle you want
- Draw a second line by connecting the origin to the point you made.
Remember, you can convert from radians to degrees by multiplying by 360°/2π, and from degrees to radians by multiplying by 2π/360°. With practice, drawing angles in standard position should be easy for you!