How to Plot Polar Coordinates
Set up the polar plane., Understand polar coordinates., Review the unit circle., Construct a circle with radius r{\displaystyle r}., Measure an angle of θ{\displaystyle \theta } from the polar axis., Draw a line based on the sign of r{\displaystyle...
Step-by-Step Guide
-
Step 1: Set up the polar plane.
You've probably graphed points with Cartesian coordinates before, using (x,y){\displaystyle (x,y)} notation to mark locations on a rectangular grid.
Polar coordinates use a different kind of graph instead, based on circles:
The center point of the graph (or "origin" in a rectangular grid) is the pole.
You can label 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 have special polar graph paper, it will include many circles of different sizes, all centered on the pole.
You do not have to draw these yourself if using blank paper. -
Step 2: Understand polar coordinates.
On the polar plane, a point is represented by a coordinate in the form (r,θ){\displaystyle (r,\theta )}:
The first variable, r{\displaystyle r}, stands for radius.
The point is located on a circle with radius r{\displaystyle r}, centered on the pole (origin).
The second variable, θ{\displaystyle \theta }, represents an angle.
The point is located along a line that passes through the pole and forms an angle θ{\displaystyle \theta } with the polar axis. , In polar coordinates, the angle is usually measured in radians instead of degrees.
In this system, one full rotation (360º or a full circle) covers an angle of 2π{\displaystyle \pi } radians. (This value is chosen because a circle with radius 1 has a circumference of 2π{\displaystyle \pi }.) Familiarizing yourself with the unit circle will make working with polar coordinates much easier.
If your textbook uses degrees, you don't need to worry about this for now.
It is possible to plot polar points using degree values for θ{\displaystyle \theta }. , Any point P{\displaystyle P} has polar coordinates in the form (r,θ){\displaystyle (r,\theta )}.
Begin by drawing a circle with radius r{\displaystyle r}, centred on the pole.
The pole is the center point of the graph, where the origin is on the rectangular coordinate plane.
For example, to plot the point (5,π2){\displaystyle (5,{\frac {\pi }{2}})}, place your compass on the pole.
Extend the pencil end of the compass to 5 units along the polar axis.
Rotate the compass to draw a circle. , Place a protractor so the center is on the pole, and the edge runs along the polar axis.
Measure the angle θ{\displaystyle \theta } from this axis.
If the angle is in radians and your protractor only shows degrees, you can convert the units or refer to the unit circle for help.
For the point (5,π2){\displaystyle (5,{\frac {\pi }{2}})}, the unit circle tells you that π2{\displaystyle {\frac {\pi }{2}}} is ¼ of the way around the circle, equivalent to 90 degrees from the polar axis.
Always measure positive angles counter-clockwise from the axis.
Measure negative angles clockwise from the axis. , The next step will be to draw a line along the angle you measured.
Before you can do this, however, you need to know which way to draw the line.
Refer back to the polar coordinates (r,θ){\displaystyle (r,\theta )} to find out:
If r{\displaystyle r} is positive, draw the line "forward"
from the pole straight through the angle marking you just made.
If r{\displaystyle r} is negative, draw the line "backward": from the angle marking back through the pole, to intersect the circle on the opposite side.
Don't be confused by rectangular coordinates: this does not correspond to positive or negative values on an x- or y- axis. , This is the point (r,θ){\displaystyle (r,\theta )}.
The point (5,π2){\displaystyle (5,{\frac {\pi }{2}})} is located on a circle with radius 5 centered on the pole, ¼ of the way along the circle's circumference in a counter-clockwise direction from the polar axis. (This point is equivalent to (0, 5) in rectangular coordinates.) , Use the pole as its centre. , Measure this angle from the polar axis (equivalent to the positive x-axis).
Since the angle −π3{\displaystyle {\frac {-\pi }{3}}} is negative, measure this angle in a clockwise direction. , Start at the pole (origin).
Since the radius is positive, move forward from the pole through the angle you measured.
The point where the line intersects the circle is (4,−π3){\displaystyle (4,{\frac {-\pi }{3}})}. , Use the pole as its centre.
Although the radius is actually
-2, the sign is not important for this step. , Since the angle 3π2{\displaystyle {\frac {3\pi }{2}}} is positive, you must go counter-clockwise from the polar axis. , Since the radius −2{\displaystyle
-2} is negative, you must go from the pole in the opposite direction of the given angle.
The point where the line intersects the circle is (−2,3π2){\displaystyle (-2,{\frac {3\pi }{2}})}. , Starting at the origin, draw a line segment 2 units along the positive x-axis.
Draw a second line segment from that point 1 unit in the positive y direction.
You are now at point (2, 1), so label this point P. , Draw a line between O and P.
This line has length r{\displaystyle r} in polar coordinates.
It is also the hypotenuse of a right triangle, so you can find the hypotenuse's length using geometry.
For example:
The legs of this right triangle have values of 2 and
1.
With the Pythagorean theorem, calculate that the hypotenuse's length is 22+12=4+1=5≈2.236{\displaystyle {\sqrt {2^{2}+1^{2}}}={\sqrt {4+1}}={\sqrt {5}}\approx
2.236}.
The general formula to find r{\displaystyle r} from Cartesian coordinates is r=x2+y2{\displaystyle r={\sqrt {x^{2}+y^{2}}}}, where x{\displaystyle x} is the Cartesian x-coordinate and y{\displaystyle y} the Cartesian y-coordinate. , Use trigonometry to find this value: tan(θ)=oppositeadjacent=12{\displaystyle \tan(\theta )={\frac {opposite}{adjacent}}={\frac {1}{2}}}tan−1(12)=θ=26.56∘{\displaystyle \tan ^{-1}({\frac {1}{2}})=\theta =26.56^{\circ }} The general formula to find θ{\displaystyle \theta } is θ=tan−1(yx){\displaystyle \theta =\tan ^{-1}({\frac {y}{x}})}, where y{\displaystyle y} is the Cartesian y-coordinate and x{\displaystyle x} the Cartesian x-coordinate. , You now have the values of r{\displaystyle r} and θ{\displaystyle \theta }.
The rectangular coordinates (2, 1) convert to approximate polar coordinates of (2.24,
26.6º), or exact coordinates of (5,tan−1(12)){\displaystyle ({\sqrt {5}},\tan ^{-1}({\frac {1}{2}}))}. -
Step 3: Review the unit circle.
-
Step 4: Construct a circle with radius r{\displaystyle r}.
-
Step 5: Measure an angle of θ{\displaystyle \theta } from the polar axis.
-
Step 6: Draw a line based on the sign of r{\displaystyle r}.
-
Step 7: Label the point where the line and circle meet.
-
Step 8: Construct a circle with radius r=4{\displaystyle r=4}.
-
Step 9: Measure the angle −π3{\displaystyle {\frac {-\pi }{3}}} radians.
-
Step 10: Draw a line at this angle.
-
Step 11: Construct a circle with radius r=2{\displaystyle r=2}.
-
Step 12: Measure the angle 3π2{\displaystyle {\frac {3\pi }{2}}} radians.
-
Step 13: Construct a line opposite that angle.
-
Step 14: Consider the point P(2
-
Step 15: 1){\displaystyle P(2
-
Step 16: 1)} in the Cartesian plane.
-
Step 17: Find the distance between the origin O{\displaystyle O} and P{\displaystyle P}.
-
Step 18: Find the angle between OP{\displaystyle OP} and the positive x-axis.
-
Step 19: Write down the polar coordinates.
Detailed Guide
You've probably graphed points with Cartesian coordinates before, using (x,y){\displaystyle (x,y)} notation to mark locations on a rectangular grid.
Polar coordinates use a different kind of graph instead, based on circles:
The center point of the graph (or "origin" in a rectangular grid) is the pole.
You can label 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 have special polar graph paper, it will include many circles of different sizes, all centered on the pole.
You do not have to draw these yourself if using blank paper.
On the polar plane, a point is represented by a coordinate in the form (r,θ){\displaystyle (r,\theta )}:
The first variable, r{\displaystyle r}, stands for radius.
The point is located on a circle with radius r{\displaystyle r}, centered on the pole (origin).
The second variable, θ{\displaystyle \theta }, represents an angle.
The point is located along a line that passes through the pole and forms an angle θ{\displaystyle \theta } with the polar axis. , In polar coordinates, the angle is usually measured in radians instead of degrees.
In this system, one full rotation (360º or a full circle) covers an angle of 2π{\displaystyle \pi } radians. (This value is chosen because a circle with radius 1 has a circumference of 2π{\displaystyle \pi }.) Familiarizing yourself with the unit circle will make working with polar coordinates much easier.
If your textbook uses degrees, you don't need to worry about this for now.
It is possible to plot polar points using degree values for θ{\displaystyle \theta }. , Any point P{\displaystyle P} has polar coordinates in the form (r,θ){\displaystyle (r,\theta )}.
Begin by drawing a circle with radius r{\displaystyle r}, centred on the pole.
The pole is the center point of the graph, where the origin is on the rectangular coordinate plane.
For example, to plot the point (5,π2){\displaystyle (5,{\frac {\pi }{2}})}, place your compass on the pole.
Extend the pencil end of the compass to 5 units along the polar axis.
Rotate the compass to draw a circle. , Place a protractor so the center is on the pole, and the edge runs along the polar axis.
Measure the angle θ{\displaystyle \theta } from this axis.
If the angle is in radians and your protractor only shows degrees, you can convert the units or refer to the unit circle for help.
For the point (5,π2){\displaystyle (5,{\frac {\pi }{2}})}, the unit circle tells you that π2{\displaystyle {\frac {\pi }{2}}} is ¼ of the way around the circle, equivalent to 90 degrees from the polar axis.
Always measure positive angles counter-clockwise from the axis.
Measure negative angles clockwise from the axis. , The next step will be to draw a line along the angle you measured.
Before you can do this, however, you need to know which way to draw the line.
Refer back to the polar coordinates (r,θ){\displaystyle (r,\theta )} to find out:
If r{\displaystyle r} is positive, draw the line "forward"
from the pole straight through the angle marking you just made.
If r{\displaystyle r} is negative, draw the line "backward": from the angle marking back through the pole, to intersect the circle on the opposite side.
Don't be confused by rectangular coordinates: this does not correspond to positive or negative values on an x- or y- axis. , This is the point (r,θ){\displaystyle (r,\theta )}.
The point (5,π2){\displaystyle (5,{\frac {\pi }{2}})} is located on a circle with radius 5 centered on the pole, ¼ of the way along the circle's circumference in a counter-clockwise direction from the polar axis. (This point is equivalent to (0, 5) in rectangular coordinates.) , Use the pole as its centre. , Measure this angle from the polar axis (equivalent to the positive x-axis).
Since the angle −π3{\displaystyle {\frac {-\pi }{3}}} is negative, measure this angle in a clockwise direction. , Start at the pole (origin).
Since the radius is positive, move forward from the pole through the angle you measured.
The point where the line intersects the circle is (4,−π3){\displaystyle (4,{\frac {-\pi }{3}})}. , Use the pole as its centre.
Although the radius is actually
-2, the sign is not important for this step. , Since the angle 3π2{\displaystyle {\frac {3\pi }{2}}} is positive, you must go counter-clockwise from the polar axis. , Since the radius −2{\displaystyle
-2} is negative, you must go from the pole in the opposite direction of the given angle.
The point where the line intersects the circle is (−2,3π2){\displaystyle (-2,{\frac {3\pi }{2}})}. , Starting at the origin, draw a line segment 2 units along the positive x-axis.
Draw a second line segment from that point 1 unit in the positive y direction.
You are now at point (2, 1), so label this point P. , Draw a line between O and P.
This line has length r{\displaystyle r} in polar coordinates.
It is also the hypotenuse of a right triangle, so you can find the hypotenuse's length using geometry.
For example:
The legs of this right triangle have values of 2 and
1.
With the Pythagorean theorem, calculate that the hypotenuse's length is 22+12=4+1=5≈2.236{\displaystyle {\sqrt {2^{2}+1^{2}}}={\sqrt {4+1}}={\sqrt {5}}\approx
2.236}.
The general formula to find r{\displaystyle r} from Cartesian coordinates is r=x2+y2{\displaystyle r={\sqrt {x^{2}+y^{2}}}}, where x{\displaystyle x} is the Cartesian x-coordinate and y{\displaystyle y} the Cartesian y-coordinate. , Use trigonometry to find this value: tan(θ)=oppositeadjacent=12{\displaystyle \tan(\theta )={\frac {opposite}{adjacent}}={\frac {1}{2}}}tan−1(12)=θ=26.56∘{\displaystyle \tan ^{-1}({\frac {1}{2}})=\theta =26.56^{\circ }} The general formula to find θ{\displaystyle \theta } is θ=tan−1(yx){\displaystyle \theta =\tan ^{-1}({\frac {y}{x}})}, where y{\displaystyle y} is the Cartesian y-coordinate and x{\displaystyle x} the Cartesian x-coordinate. , You now have the values of r{\displaystyle r} and θ{\displaystyle \theta }.
The rectangular coordinates (2, 1) convert to approximate polar coordinates of (2.24,
26.6º), or exact coordinates of (5,tan−1(12)){\displaystyle ({\sqrt {5}},\tan ^{-1}({\frac {1}{2}}))}.
About the Author
Judith Gibson
Professional writer focused on creating easy-to-follow creative arts tutorials.
Rate This Guide
How helpful was this guide? Click to rate: