These 25 problems practice standard G-GPE.A.1: writing the equation of a circle from its center and radius, reading the center and radius from an equation, completing the square, and deciding whether points are inside, on or outside a circle.
Problems get harder as you go. The last six use maps, towers and signals. Keep radicals exact unless told to round.
Problem 1: Center at the origin
Write the equation of the circle with center \((0,0)\) and radius 5.
Problem 2: Center and radius given
Write the equation of the circle with center \((3,2)\) and radius 4.
Problem 3: Negative x-coordinate
Write the equation of the circle with center \((-2,5)\) and radius 3, as drawn.
Problem 4: Radius is a square root
Write the equation of the circle with center \((4,-1)\) and radius \(\sqrt{7}\).
Problem 5: Given the diameter length
A circle has center \((-6,-3)\) and a diameter of 20. Write its equation.
Problem 6: Read the center and radius
Find the center and the radius of the circle \((x-5)^{2}+(y+2)^{2}=49\).
Problem 7: Circle at the origin
Find the center and the radius of the circle \(x^{2}+y^{2}=36\).
Problem 8: Radius that is not a whole number
Find the center and the exact radius of the circle \((x+1)^{2}+(y-4)^{2}=20\).
Problem 9: Unit circle shifted
Find the center and the radius of the circle \((x+8)^{2}+(y+8)^{2}=1\).
Problem 10: Is the point on the circle?
Determine whether the point \((6,6)\) lies on the circle \((x-2)^{2}+(y-3)^{2}=25\).
Problem 11: Inside, on or outside
The circle \((x-1)^{2}+(y+2)^{2}=36\) is shown. Classify each point as inside, on, or outside the circle: A\((4,2)\), B\((7,-2)\), C\((6,5)\).
Problem 12: Center and a point on the circle
A circle has center \((1,2)\) and passes through \((4,6)\). Write its equation.
Problem 13: Endpoints of a diameter
The endpoints of a diameter of a circle are A\((-2,1)\) and B\((4,9)\). Write the equation of the circle.
Problem 14: Find a missing coordinate
The point \((5,k)\) lies on the circle with center \((2,-3)\) and radius 5. Find both possible values of \(k\).
Problem 15: Tangent to the x-axis
A circle with center \((3,4)\) is tangent to the x-axis. Write its equation.
Problem 16: Tangent to the y-axis
A circle with center \((-5,2)\) is tangent to the y-axis. Write its equation.
Problem 17: Complete the square, case 1
Rewrite \(x^{2}+y^{2}+8x-2y+8=0\) in standard form. Then state the center and the radius.
Problem 18: Complete the square, case 2
Rewrite \(x^{2}+y^{2}-10x-4y+4=0\) in standard form. Then state the center and the radius.
Problem 19: Complete the square, case 3
Rewrite \(x^{2}+y^{2}+2x+6y-15=0\) in standard form. Then state the center and the radius.
Problem 20: Only one variable shifted
Rewrite \(x^{2}+y^{2}-12y+20=0\) in standard form. Then state the center and the radius.
Problem 21: Points with a given x-coordinate
Find the y-coordinates of the points on the circle \((x-2)^{2}+(y+1)^{2}=100\) whose x-coordinate is 8.
Problem 22: Cell tower coverage
On a map marked in miles, a cell tower stands at (3, −4) and its signal reaches 10 miles. Write an equation for the edge of the coverage area, then decide whether Town A at (9, 4), Town B at (−3, 2) and Town C at (12, −10) get a signal.
Problem 23: Wi-Fi router
A router sits at (2, 1), measured in feet from a corner of a gym, and its signal reaches 25 ft. Write the equation of the signal’s boundary. Does a laptop at (20, 18) get a signal? Does a laptop at (20, 20)?
Problem 24: Pond on a map
A park map shows a circular pond with equation \(x^{2}+y^{2}-8x+6y=0\), in yards. Find the center and the radius of the pond. Is the origin, where the park entrance stands, on the edge of the pond?
Problem 25: Earthquake zone
An earthquake is felt up to 15 miles from its epicenter at (−6, 8) on a grid in miles. Write the equation of the boundary of the felt zone. Is the shaking felt at the point (6, −1) and at the origin?
See every chapter on the Geometry math page.