CalcHack Try it free

Circle equations: type it as given, read the centre and radius

An original SAT-style practice question on the equation of a circle, worked step by step in Desmos — the same graphing calculator that's built into Bluebook — with completing the square alongside.

Practice question

In the xy-plane, the graph of the equation x² − 8x + y² + 6y = 24 is a circle. What is the radius of the circle?

⚡ Desmos is the fastest way here Desmos draws the circle straight from the expanded equation, so you can skip completing the square and read the size off the grid.
CirclesCentre and radiusMedium⏱ ~40 sec

Step by step

1

Type the equation exactly as given

No need to rearrange or solve for y. Desmos graphs equations with x² and y² in them directly, expanded or not.

x^{2}-8x+y^{2}+6y=24A circle appears
−6−4−22681214−10−8−6−44 r = 7 (4, −3) (−3, −3) (11, −3)
What you see: a circle sitting mostly to the right of the y-axis and below the x-axis. Zoom out a step if part of it is off the screen.
2

Read the left and right edges off the grid

The widest points of a circle are level with its centre. Here the leftmost point lands on the gridlines at (−3, −3) and the rightmost at (11, −3). Zoom in on either one if you want to be sure it sits exactly on the lines.

11−(−3) = 14Width across = the diameter
What you see: the circle is 14 units wide, so the radius is half that, 7. The centre is halfway between the edges, at (4, −3).
3

Confirm with standard form

On a new line, type the circle you think it is. If it lands exactly on top of the first one, your centre and radius are right.

(x-4)^{2}+(y+3)^{2}=7^{2}Overlaps perfectly — radius 7
Answer
7

The circle has centre (4, −3) and radius 7. Check: the point (11, −3) is on it, since 11² − 8(11) + (−3)² + 6(−3) = 121 − 88 + 9 − 18 = 24 ✓.

⚠ Common trap

Answering 49 — that's r², the number on the right after completing the square — or 24, the number on the right before you do. Reading the width straight off the graph gives 14, which is the diameter, not the radius. And if a question asks for the centre, watch the signs: (x − 4)² + (y + 3)² means the centre is (4, −3), not (−4, 3).

The algebra way, if you prefer

Complete the square: take half of each linear coefficient and square it. For x: half of −8 is −4, and (−4)² = 16. For y: half of 6 is 3, and 3² = 9. Add both to each side: x² − 8x + 16 + y² + 6y + 9 = 24 + 16 + 9. That factors to (x − 4)² + (y + 3)² = 49.

Standard form is (x − h)² + (y − k)² = r², so the centre is (4, −3) and r² = 49, which makes the radius 7. The usual slip is forgetting to add the 16 and 9 to the right-hand side too.

More walkthroughs: quadratic roots · vertex and maximum value · systems of linear equations · sum of a quadratic's solutions · exponential growth · line of best fit · solving for x by graphing · linear inequalities · absolute value equations · Desmos cheat sheet · all walkthroughs

Stuck on a different practice question?

Snap a photo of any SAT-style math practice question and CalcHack walks you through the fastest route, in Desmos or by hand. Five free scans every day, no account needed to start.

Try it on your own question