Free printable resources · Original practice
Use angle rays and concentric circles to plot radius and direction. Choose degree or radian labels, then compare three points with their Cartesian coordinates.
Use angle rays and concentric circles to plot radius and direction. Choose degree or radian labels, then compare three points with their Cartesian coordinates.
Choose US Letter or A4, then download the PDF for printing or the SVG for editing. Keep the student worksheet separate from its answer or reference completion. All files, including the answers, are free without signup.
Original blank printable. Choose Letter or A4; print at actual size.
Original blank printable. Choose Letter or A4; print at actual size.
Original blank printable. Choose Letter or A4; print at actual size.
Use the starter sheet to plot A = (3,30°), B = (2,150°), C = (4,270°), then draw r = 2. A radius is a distance from the pole; an angle tells you its ray. Positive angles turn counterclockwise from the rightward zero ray.
| Point | Exact Cartesian (x, y) | Approximate check |
|---|---|---|
| A | (3√3/2, 3/2) | (2.598, 1.5) |
| B | (−√3, 1) | (−1.732, 1) |
| C | (0, −4) | On the downward ray |
Use x = r cos θ and y = r sin θ. The circle r = 2 becomes x² + y² = 4. For radian labels, 30° = π/6, 90° = π/2 and 180° = π. Calculator angle mode must match the input units.
Original practice with a separate reference answer. Choose Letter or A4.
A point can have more than one polar coordinate: (2,30°) and (2,390°) occupy the same position. A negative radius reverses the ray, so (−2,30°) is the same position as (2,210°). The starter asks only positive-radius problems; the blank grids remain usable for negative-radius equations.
Use the full degree grid for rays every 15°, or the full radian grid for rays every π/6 (30°). The four-grid sheet puts four 30° degree plots on one page for short separate problems. All have a radial range from 0 to 5 units. The full grid has a printed radius of 70 mm, or 14 mm per radial unit. Each small grid has a radius of 32 mm, or 6.4 mm per unit. Those are radii, not diameters. Choose a full grid when labels or a longer curve would crowd a small one.
Confirm the orientation before plotting to ensure accuracy. The zero angle is positioned to the right, and positive angles increase in a counterclockwise direction. Radian labels progress from zero through pi over six, pi over three, pi over two, up to eleven pi over six. Ensure your calculator or device is set to the angle mode that matches the stated units on the paper. This alignment prevents misinterpretation of coordinates, especially when converting between degrees and radians for points like A, B, and C on the polar plane.
Begin by plotting point A at three units and thirty degrees, point B at two units and one hundred fifty degrees, and point C at four units and two hundred seventy degrees. The starter problems use positive radii, measured outward along each named ray. Draw the circle with radius two centered at the pole. Verify these against Cartesian equivalents: A is three square root three over two by one point five, B is negative square root three by one, and C is zero by negative four. Use the radial lines to guide your placement.
| Watch for | Check instead |
|---|---|
| Assuming the small grid has a different mathematical range than the full grid. | Both grids cover the same radial range from zero to five units, differing only in printed size and unit scale. |
| Treating a negative radius as an invalid polar coordinate. | A negative radius is plotted in the direction opposite the stated angle: (−2, 30°) shares a point with (2, 210°). This starter activity uses positive radii so that you can learn orientation first. |
| Misaligning the zero angle position. | The zero angle is strictly to the right, with positive angles moving counterclockwise, not clockwise or upward. |
Use the PDF that matches the paper loaded in your printer. Choose Actual size / 100% and turn off Fit, Shrink and browser headers. The SVG keeps millimeter dimensions, but an editor or browser can rescale it; the PDF is the simpler print path. Measure any calibration bars with a separate physical ruler when the exercise depends on printed size. A screen preview does not establish physical scale.
Packs combine the same individual pages; they are not extra different exercises. Print a student copy first and keep references nearby for review. Letter and A4 preserve the stated grid spacing or model definition while their surrounding white space can differ. Color is optional: the references also use line position, text or relative lightness to explain the result.
The resource is available in PDF and SVG formats, compatible with both Letter and A4 paper sizes.
The small-grid sheet contains four separate degree grids with 30° rays. Each spans r = 0 to 5, the same numerical radial range as the full-page grids.
Yes, the circle with radius two is centered at the pole, which is the origin of the polar coordinate system.
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