Geometry Equations of Circles and Parabolas Worksheet (PDF)

Geometry math worksheet: Equations of Circles and Parabolas – Maths PDF

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This circle equations worksheet for Geometry covers equations of circles and parabolas (Common Core G-GPE.A.1, G-GPE.A.2), with a free PDF to print.

These 20 problems practice equations of circles, completing the square and parabolas (Common Core G-GPE.A.1, G-GPE.A.2). The first problems are quick one-step checks of the definitions and formulas from the lesson. The middle problems ask for diagrams, equations and careful reasoning in the style of a high school Geometry class. The last six problems are applications with radio towers, fire towers and satellite dishes.

Geometry · Common Core State Standards: G-GPE.A.1, G-GPE.A.2

Circle equations worksheet: exercises 1 to 10

Exercise 1 – Circle at the origin

Write the equation of the circle with center at the origin and radius 7.

Exercise 2 – Center and radius given

Circle drawn around (3, -2) with radius 5.

Write an equation of the circle with center \((3, -2)\) and radius 5.

Exercise 3 – Read the equation

Find the center and radius of \((x + 4)^{2} + (y – 1)^{2} = 36\).

Exercise 4 – Radius from r squared

A circle has equation \((x – 2)^{2} + (y + 3)^{2} = 20\). Find the radius in simplest radical form.

Exercise 5 – Point on the circle?

Point A lies on the circle, B lies outside and D lies inside.

Is the point \(A(4, 6)\) on the circle \((x – 1)^{2} + (y – 2)^{2} = 25\)?

Exercise 6 – Outside or inside?

Decide whether \(B(7, 3)\) and \(D(2, 3)\) are inside, on, or outside the circle \((x – 1)^{2} + (y – 2)^{2} = 25\).

Exercise 7 – Circle through a point

A circle has center \((-1, 2)\) and passes through \((3, 5)\). Write its equation.

Exercise 8 – Circle from a diameter

The endpoints of a diameter are \((-2, 3)\) and \((6, 7)\). Write the equation of the circle.

Exercise 9 – Complete the square (1)

Rewrite \(x^{2} + y^{2} – 6x + 4y – 12 = 0\) in center-radius form. Find the center and radius.

Exercise 10 – Complete the square (2)

Rewrite \(x^{2} + y^{2} – 12x + 6y + 20 = 0\) in center-radius form. Find the center and radius.

Exercises 11 and up

Exercise 11 – Complete the square (3)

Rewrite \(x^{2} + y^{2} – 10x = 0\) in center-radius form.

Exercise 12 – Tangent to the x-axis

Circle tangent to the x-axis.

A circle with center \((4, 3)\) is tangent to the x-axis. Write its equation and the point of tangency.

Exercise 13 – Line and circle

Find the points where the line \(y = x + 1\) meets the circle \(x^{2} + y^{2} = 25\).

Exercise 14 – Area and circumference

A circle has equation \(x^{2} + y^{2} = 81\). Find its circumference and area.

Exercise 15 – Parabola equation

Each point of the curve is the same distance from F and from the line y = -2.

A parabola has focus \((0, 2)\) and directrix \(y = -2\). Show that its equation is \(x^{2} = 8y\).

Exercise 16 – Parabola with focus (0, 3)

Write the equation of the parabola with focus \((0, 3)\) and directrix \(y = -3\).

Exercise 17 – Check a point on a parabola

Show that \(P(4, 2)\) is on the parabola \(x^{2} = 8y\) by comparing its distances to the focus \((0, 2)\) and to the directrix \(y = -2\).

Exercise 18 – Find the focus

A parabola has equation \(y = \frac{x^{2}}{16}\). Find its focus and directrix.

Exercise 19 – Radio coverage

Radio tower range.

A radio tower at \((10, -5)\) (miles from a map origin) broadcasts 13 miles. Write the equation of its coverage boundary and decide whether a city at \((22, 0)\) is inside, on, or outside it.

Exercise 20 – Satellite dish

A satellite dish has a cross-section \(y = \frac{x^{2}}{24}\) (units in feet). Find the position of the receiver, which sits at the focus, and the depth of the dish where its width is 24 feet.

Key facts behind this circle equations worksheet

Key facts

Circle: (x – h)^2 + (y – k)^2 = r^2, center (h, k), radius r.
Read the center with opposite signs from the equation.
Inside, on, or outside: compare (x – h)^2 + (y – k)^2 with r^2.
Complete the square to convert x^2 + y^2 + Dx + Ey + F = 0.
Parabola with focus (0, p) and directrix y = -p: x^2 = 4py.

How to use this circle equations worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the circle equations worksheet as a PDF with the button below and keep it with your notes.

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