How to graph circles on Desmos

Circle questions on the digital SAT look intimidating in general form — all those x², y² and stray linear terms. Desmos graphs them straight from the page, so you can see the center and radius instead of wrestling the algebra first.

Type a circle in standard form — (x−h)² + (y−k)² = r² — and Desmos graphs it: the center is (h, k) and the radius is r. Given general form like x²+y²+Dx+Ey+F=0, just type it as-is — Desmos still draws the circle, so you can read the center and radius off the picture (or complete the square for exact values). You can also add a line and click where it crosses the circle.

Practice this on SAT-style Math All Desmos techniques

Desmos circle: Make the mathematical object in this brief the hero: a Desmos panel: a circle centered below and right of the origin with a marked center dot and a radius segment drawn to the edge, a straight line cutting across it with one intersection ringed, and an expression list showing a circle equation

Type it rawBoth standard and general form graph without rearranging.

(h, k)Standard form shows the center directly: opposite the signs.

r = √(r²)The radius is the square root of the right-hand side.

Add a lineGraph a line too and click the crossing points.

Step by step

From equation to center and radius

Four steps whether the question gives you tidy standard form or a messy general-form equation full of linear terms.

Step 1

Type the equation

Enter it on one expression line exactly as printed. Standard form (x-3)^2+(y+2)^2=25 or general form x^2+y^2-6x+4y-12=0 — Desmos graphs either as a circle.

Step 2

Find the center

In standard form the center is (h, k) — flip the signs inside the parentheses. From general form, read the middle of the circle on the grid, or hover to confirm the coordinates.

Step 3

Read the radius

It's √(right side) in standard form. Otherwise measure from the center to a clear edge point — click the top or right of the circle and compare to the center.

Step 4

Add a line if needed

If the question involves a line, type it on a second line. Both graph together; click an intersection dot to read where the line meets the circle.

Lock in the setup on real items.

Circle questions repeat the same handful of moves. Practice them on SAT-style Math with the built-in calculator until reading the center and radius is instant.

Start free practice

Worked example

A general-form circle, decoded

Center and radius from a messy equation

SAT-style problem. A circle in the xy-plane is given by x² + y² − 6x + 4y − 12 = 0. What are the coordinates of its center, and what is its radius?

Fastest path — graph it. Type x^2+y^2-6x+4y-12=0 on one line in Desmos. A circle appears, sitting to the right of the y-axis and below the x-axis. Hover the middle and you'll read a center near (3, −2); the circle reaches from x = −2 to x = 8, a diameter of 10, so the radius is 5.

Exact path — complete the square. Group and balance: (x² − 6x) + (y² + 4y) = 12 → (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4 → (x − 3)² + (y + 2)² = 25. Now it's standard form: center (3, −2), radius √25 = 5.

Both routes agree. Typing the standard form (x-3)^2+(y+2)^2=25 as a second line in Desmos draws the identical circle — a clean visual confirmation.

The completing-the-square half-steps

The numbers you add come from half the linear coefficient, squared. For −6x: half of −6 is −3, squared is 9. For +4y: half of 4 is 2, squared is 4. Add both to each side, then the right side becomes 12 + 9 + 4 = 25 = r².

Answer: center (3, −2), radius 5. Note the signs flip — (x − 3)² means h = +3, and (y + 2)² means k = −2. Mixing those up is the classic circle mistake; the graph catches it for you.

When to use / pitfalls

Graph the circle — or compute it

Use it

For general form, every time.

When the equation has loose x and y terms, typing it in beats rearranging by hand. You see the circle, estimate center and radius, and decide if you even need exact values.

Use it

For line-and-circle questions.

Tangent, chord, or no intersection? Graph both and look. Clicking a crossing point gives exact coordinates without solving a quadratic system by hand.

Watch out

For exact vs. eyeballed values.

If the radius is something like √20, the grid won't read cleanly — complete the square for the exact number, then let Desmos confirm. And remember the sign flip: (x − h) means the center coordinate is +h.

FAQ

Desmos circles on the SAT, answered

How do I graph a circle on the Desmos SAT calculator?

Type the circle's equation straight into an expression line. From standard form, enter (x-h)^2+(y-k)^2=r^2 and Desmos draws the circle. The center is (h, k) and the radius is r — the square root of the right-hand side.

What if the circle is in general form, like x^2+y^2+Dx+Ey+F=0?

Type it in exactly as written. Desmos graphs general form without any rearranging, so you can read the center and radius from the picture. To get exact values, click the graph or complete the square to convert it to standard form.

How do I find the center and radius from the graph?

The center is the middle of the circle — hover or click to read its coordinates, or note where the circle is symmetric. The radius is the distance from that center to any point on the circle; click the topmost or rightmost point and compare its coordinates to the center.

Can Desmos find where a line crosses a circle?

Yes. Enter the circle on one line and the line on another. Both graph together, and clicking a point where they cross shows the intersection coordinates. This is handy for questions about whether a line is a chord, tangent, or misses the circle.

Do I still need to complete the square if Desmos graphs it?

Not to get the picture, but sometimes for an exact answer. If a question wants the center or radius as an exact number and the graph reading is ambiguous, completing the square converts general form to standard form so the values are exact. Desmos confirms your result visually.

Turn Desmos into points.

Start free and practice SAT-style Math with the built-in calculator, until graphing circles is second nature.

Start free practice Desmos guide