Worksheets · Precalculus

Polar form of conic sections worksheet

One equation covers all three conics in polar form, and the eccentricity decides which one you have. Getting it into the standard shape is a matter of dividing until the constant term in the bottom is 1.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Polar Forms of Conic Sections

Date Period

Identify each conic, and find what is asked for.

  1. Identify the conic

    Identify the conic:
    r=182−cos⁡θ
  2. Identify the conic:
    r=213−sin⁡θ
  3. Identify the conic:
    r=273−5sin⁡θ
  4. Identify the conic:
    r=162−2cos⁡θ
  5. Identify the conic:
    r=91+5sin⁡θ

  1. Find the eccentricity

    Find the eccentricity:
    r=162+4cos⁡θ
  2. Find the eccentricity:
    r=91+3sin⁡θ
  3. Find the eccentricity:
    r=213−7cos⁡θ
  4. Find the eccentricity:
    r=213−3sin⁡θ
  5. Find the eccentricity:
    r=182+sin⁡θ

  1. Write the polar equation from ee and the directrix

    A conic has a focus at the pole, e=12, and directrix x=1. Write its polar equation.
  2. A conic has a focus at the pole, e=12, and directrix x=−4. Write its polar equation.
  3. A conic has a focus at the pole, e=2, and directrix y=−6. Write its polar equation.
  4. A conic has a focus at the pole, e=2, and directrix x=1. Write its polar equation.
  5. A conic has a focus at the pole, e=2, and directrix x=5. Write its polar equation.

  1. Find the vertices and name the conic

    Find the vertex of r=55+5cos⁡θ and name the conic.
  2. Find the vertices of r=23−sin⁡θ and name the conic.
  3. Find the vertices of r=92+6sin⁡θ and name the conic.
  4. Find the vertices of r=94−2cos⁡θ and name the conic.
  5. Find the vertices of r=112−cos⁡θ and name the conic.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Polar Forms of Conic SectionsAnswer keyVersion 1

Date Period

Identify each conic, and find what is asked for.

  1. Identify the conic

    Identify the conic:
    r=182−cos⁡θ    Ellipse
  2. Identify the conic:
    r=213−sin⁡θ    Ellipse
  3. Identify the conic:
    r=273−5sin⁡θ    Hyperbola
  4. Identify the conic:
    r=162−2cos⁡θ    Parabola
  5. Identify the conic:
    r=91+5sin⁡θ    Hyperbola

  1. Find the eccentricity

    Find the eccentricity:
    r=162+4cos⁡θ    2
  2. Find the eccentricity:
    r=91+3sin⁡θ    3
  3. Find the eccentricity:
    r=213−7cos⁡θ    73
  4. Find the eccentricity:
    r=213−3sin⁡θ    1
  5. Find the eccentricity:
    r=182+sin⁡θ    12

  1. Write the polar equation from ee and the directrix

    A conic has a focus at the pole, e=12, and directrix x=1. Write its polar equation.r=12+cos⁡θ
  2. A conic has a focus at the pole, e=12, and directrix x=−4. Write its polar equation.r=42−cos⁡θ
  3. A conic has a focus at the pole, e=2, and directrix y=−6. Write its polar equation.r=121−2sin⁡θ
  4. A conic has a focus at the pole, e=2, and directrix x=1. Write its polar equation.r=21+2cos⁡θ
  5. A conic has a focus at the pole, e=2, and directrix x=5. Write its polar equation.r=101+2cos⁡θ

  1. Find the vertices and name the conic

    Find the vertex of r=55+5cos⁡θ and name the conic.(12,0); a parabola
  2. Find the vertices of r=23−sin⁡θ and name the conic.(1,π2) and (12,3π2); an ellipse
  3. Find the vertices of r=92+6sin⁡θ and name the conic.(98,π2) and (−94,3π2); a hyperbola
  4. Find the vertices of r=94−2cos⁡θ and name the conic.(92,0) and (32,π); an ellipse
  5. Find the vertices of r=112−cos⁡θ and name the conic.(11,0) and (113,π); an ellipse

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

r=123+2cos⁡θ

  1. Divide top and bottom by 3.
  2. That gives 4 over 1 plus two thirds cos theta.
  3. The eccentricity is two thirds, which is less than 1.

The answer is Ellipse, e=23.

Where students go wrong

Reading the eccentricity straight off the printed equation. In 12 over 3 plus 2 cos theta the eccentricity is not 2; you have to divide through first so the constant in the denominator is 1.

Also called conics in polar form, directrix or focus directrix form.

What you can put on this worksheet

Name the conic
Identify the conic:
r=82+sin⁡θ → Ellipse
The eccentricity
Find the eccentricity:
r=82+sin⁡θ → 12
The directrix
Find the directrix:
r=82+sin⁡θ → y=8
Write it from e and the directrix
A conic has a focus at the pole, e=12, and directrix y=−1. Write its polar equation. → r=12−sin⁡θ
The vertices
Find the vertex of r=104+4cos⁡θ and name the conic. → (54,0); a parabola
Decimal coefficients
Identify the conic, and give its eccentricity and directrix:
r=244+4cos⁡θ → Parabola; e=1; x=6
From e and the vertices
A conic with a focus at the pole has e=1 and vertex (3,0). Find the directrix and write its polar equation. → x=6; r=61+cos⁡θ
Graph a polar conic
Graph r=62−sin⁡θ. Give its type, eccentricity and directrix. — a blank polar grid → Ellipse; e=12; y=−6

Questions about these worksheets

Yes. Every polar form of conic sections sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the polar form of conic sections questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 8: name the conic, the eccentricity, the directrix, write it from e and the directrix, the vertices, decimal coefficients, from e and the vertices and graph a polar conic. Tick as many as you want and set how many of each, or let it spread them evenly.

Reading the eccentricity straight off the printed equation. In 12 over 3 plus 2 cos theta the eccentricity is not 2; you have to divide through first so the constant in the denominator is 1.