How to Draw a Circle in a Cartesian Plane

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The familiar rectangular grid is an easy organization to acquire, but information technology is non convenient in all situations. What if you want to plot the spokes on a wheel, or the movement of water down a drain? In these cases, a circular coordinate system is a more natural fit. In fact, yous've already used the basic idea of polar coordinates in everyday life.^[one] If you're locating the source of a siren, for example, you demand two piece of information: how far abroad it is, and which direction the sound is coming from. The polar coordinate system maps points the same way, describing the distance ${\displaystyle r}$ from a fixed point, and the bending ${\displaystyle \theta }$ from a fixed ray.

1. 1

   Set up the polar plane. You've probably graphed points with Cartesian coordinates before, using ${\displaystyle (x,y)}$ notation to mark locations on a rectangular grid. Polar coordinates use a different kind of graph instead, based on circles:^[2]

   • The centre bespeak of the graph (or "origin" in a rectangular grid) is the pole. You can characterization this with the letter O.
   • Starting from the pole, draw a horizontal line to the right. This is the polar axis. Label the axis with units as you would the positive x-axis on a rectangular grid.
   • If you lot have special polar graph paper, it volition include many circles of different sizes, all centered on the pole. Y'all do non have to draw these yourself if using blank paper.

2. two

   Empathise polar coordinates. On the polar plane, a point is represented by a coordinate in the form ${\displaystyle (r,\theta )}$ :

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3. 3

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1. ane

2. 2

   Measure an angle of $$ θ {\displaystyle \theta } from the polar axis. Place a protractor then the center is on the pole, and the edge runs forth the polar centrality. Measure the angle ${\displaystyle \theta }$ from this axis. If the angle is in radians and your protractor only shows degrees, you tin convert the units or refer to the unit of measurement circumvolve for aid.

3. three

   Draw a line based on the sign of $$ r {\displaystyle r} . The next step will exist to draw a line forth the angle you lot measured. Before you can do this, however, you need to know which manner to draw the line. Refer back to the polar coordinates ${\displaystyle (r,\theta )}$ to find out:

4. 4

   Label the bespeak where the line and circumvolve meet. This is the point ${\displaystyle (r,\theta )}$ .

   • The point ${\displaystyle (5,{\frac {\pi }{2}})}$ is located on a circle with radius v centered on the pole, ¼ of the way along the circle's circumference in a counter-clockwise direction from the polar centrality. (This point is equivalent to (0, 5) in rectangular coordinates.)

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Starting time Example Download Article

Plot the betoken P located at ${\displaystyle (4,{\frac {-\pi }{three}})}$ on the polar plane

1. 1

   Construct a circle with radius $$ r = 4 {\displaystyle r=iv} . Use the pole as its centre.

2. 2

   Mensurate the angle $$ − π 3 {\displaystyle {\frac {-\pi }{3}}} radians. Measure this bending from the polar centrality (equivalent to the positive 10-axis). Since the angle ${\displaystyle {\frac {-\pi }{three}}}$ is negative, measure this angle in a clockwise direction.

3. iii

   Draw a line at this angle. Outset at the pole (origin). Since the radius is positive, movement forward from the pole through the angle you measured. The betoken where the line intersects the circle is ${\displaystyle (4,{\frac {-\pi }{3}})}$ .

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Second Example Download Article

Plot the point Q located at ${\displaystyle (-2,{\frac {3\pi }{two}})}$ on the polar aeroplane.

1. 1

   Construct a circle with radius $$ r = ii {\displaystyle r=2} . Apply the pole every bit its center. Although the radius is actually -two, the sign is not important for this step.

2. ii

   Measure the angle $$ 3 π two {\displaystyle {\frac {three\pi }{2}}} radians. Since the angle ${\displaystyle {\frac {3\pi }{ii}}}$ is positive, you must go counter-clockwise from the polar axis.

3. 3

   Construct a line opposite that angle. Since the radius ${\displaystyle -2}$ is negative, you must become from the pole in the reverse direction of the given angle. The point where the line intersects the circle is ${\displaystyle (-2,{\frac {3\pi }{2}})}$ .

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1. 1

   Consider the point $$ P ( 2 , ane ) {\displaystyle P(ii,i)} in the Cartesian airplane. Starting at the origin, draw a line segment two units along the positive x-axis. Draw a second line segment from that point 1 unit in the positive y direction. You lot are at present at indicate (2, i), so label this bespeak P.

2. ii

3. three

   Observe the angle between $$ O P {\displaystyle OP} and the positive x-centrality. Use trigonometry to discover this value:

4. 4

   Write down the polar coordinates. You lot now have the values of ${\displaystyle r}$ and ${\displaystyle \theta }$ . The rectangular coordinates (2, i) convert to approximate polar coordinates of (ii.24, 26.6º), or exact coordinates of ${\displaystyle ({\sqrt {five}},\tan ^{-1}({\frac {one}{ii}}))}$ .

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• Question

  How tin I calculate a vector in 3D?

  

  Vectors in 3D are represented using the x, y, and z axes. You tin can notice their intersections and lengths just similar y'all would in 2nd vectors.

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• Unlike the rectangular coordinate organisation, a point has infinite polar coordinates. For example, the indicate (1, 2π) is the same as the point (-1, π). It is likewise the aforementioned equally the points (1, 4π), (1, 6π), (1, 8π), and and so on. Each ane instructs you to "circle around" a dissimilar number of times, but they all finish up in the aforementioned identify.^[3]

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Things Yous'll Need

• Paper
• Pencil
• Drawing compass
• Protractor

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Article Summary Ten

To plot polar coordinates, set up upwardly the polar plane by drawing a dot labeled "O" on your graph at your point of origin. Draw a horizontal line to the right to ready the polar axis. When y'all expect at the polar coordinate, the first number is the radius of a circumvolve. To plot the coordinate, draw a circle centered on point O with that radius. The second coordinate is an angle. Utilise a protractor to describe a line that intersects point O at that angle. The point where the circle and the angled line meet is the polar coordinate. To acquire what management to draw your line, keep reading!

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Source: https://www.wikihow.com/Plot-Polar-Coordinates

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