What a Gubby Is and Why People Draw Them

A gubby is a straightforward cartoon face made entirely from equations and shapes in Desmos Graphing Calculator. It typically has a circular head, two eyes, a nose, and a mouth — all created by graphing circles, lines, and other mathematical functions. People draw gubbies in Desmos because it combines math practice with creative play: you learn how equations control the position and size of shapes while making something that looks intentional rather than random.

The gubby has become a common project in math classes and online communities because it requires no special skills beyond basic algebra and geometry. You do not need to understand calculus or advanced functions. A straightforward gubby takes about 10 to 15 minutes to build once you know the steps.

Key Takeaways

  • A gubby is a cartoon face drawn using circles, lines, and straightforward equations in Desmos, and you can build one in under 20 minutes.
  • The head is a circle equation in the form (x − h)² + (y − k)² = r², where you choose the center point and radius.
  • Eyes, nose, and mouth are smaller circles and line segments positioned using the same equation format with different center points.
  • You can change colors, sizes, and positions by editing the numbers in each equation without starting over.
  • Desmos saves your work automatically, so you can close and reopen the same gubby later.

Setting Up Desmos and Starting Your First Circle

Open Desmos Graphing Calculator by going to desmos.com/calculator in any web browser. You do not need to create an account, though signing in lets you save your work to your account. The calculator opens with a blank grid and an input line at the top left where you type equations.

Click the input line and type the equation for your head: (x − 0)² + (y − 0)² = 25. This creates a circle centered at the point (0, 0) with a radius of 5. You should see a circle appear on the grid. The numbers inside the parentheses control where the circle sits (the center), and the number on the right side (25) controls the size. Since radius 5 squared equals 25, the radius is 5 units.

If the circle is too small or too large, edit the equation. Change the 25 to a different number: try 36 for a bigger head (radius 6) or 16 for a smaller one (radius 4). Press Enter after each change to see the result.

Adding Eyes Using Smaller Circles

Eyes are smaller circles positioned on either side of the head. Click a new input line below your head equation and type: (x − 2)² + (y − 2)² = 1. This creates a small circle (radius 1) positioned 2 units to the right and 2 units up from the center. You should see a dot appear on the upper right of your head.

Add a second eye on the left side by typing: (x + 2)² + (y − 2)² = 1. The plus sign before the 2 in the first parenthesis moves this circle 2 units to the left instead of right. Now you have two eyes symmetrically placed.

If the eyes are too close together or too far apart, change the 2 values. If they are too high or too low, change the second 2 (the y-coordinate). If they are too big or small, change the 1 to a different number — try 0.5 for tiny eyes or 2 for larger ones.

Creating a Nose and Mouth with Lines

A nose can be a short vertical line. Click a new input line and type: x = 0 for a vertical line down the middle, then add a restriction to make it shorter: x = 0 {−1 < y < 0}. The part in curly braces limits the line to only show between y = −1 and y = 0, creating a short line segment below the eyes.

For a mouth, you can use a semicircle or a straightforward curved line. Type: y = −√(4 − x²) − 2. This creates a downward-facing curve (like a smile) centered below the nose. If the smile is too wide, change the 4 to a smaller number like 2. If it is too low, change the −2 to a different number like −1.5.

Alternatively, draw a straight mouth using: y = −2 {−2 < x < 2}. This creates a horizontal line segment that looks like a neutral expression.

Adjusting Colors and Making Your Gubby Unique

Each equation in Desmos has a colored dot next to it on the left side. Click that dot to open a color menu and choose a different color for that shape. Make your head one color, your eyes another, and your mouth a third color to make the gubby stand out.

You can also change line thickness. Click the three dots at the right end of any equation line and select "Line thickness" to make the mouth or nose bolder. Experiment with different combinations until your gubby looks the way you want.

Saving and Sharing Your Gubby

If you are signed into Desmos with an account, your work saves automatically. You can close the browser and come back later to find your gubby still there. If you are not signed in, your work stays in the browser tab but disappears if you close it completely.

To share your gubby, click the "Share" button in the top right corner. Desmos generates a link you can send to friends or classmates. They can open the link and see your gubby, and they can also click "Edit a Copy" to modify it and make their own version.

Common Mistakes and How to Fix Them

If a circle does not appear, check that you typed the equation exactly: parentheses around each part, a minus sign or plus sign before the numbers, and an equals sign before the radius squared. Desmos shows an error message if the syntax is wrong — read it and look for missing symbols.

If shapes overlap in the wrong way, you may have positioned them incorrectly. Remember that the first number in parentheses (x) moves shapes left and right, and the second number (y) moves them up and down. Negative numbers move left or down; positive numbers move right or up.

If your mouth or nose is not visible, it might be hidden behind the head or positioned outside the viewing area. Try changing the y-coordinate to move it, or zoom out by scrolling your mouse wheel to see more of the grid.

Frequently Asked Questions

Can I make a gubby with just circles, no lines?

Yes. You can draw the mouth as a small circle positioned low on the head, and the nose as an even smaller circle in the middle. This creates a simpler gubby that uses only circle equations. Some people prefer this style because it is easier to remember and modify.

What if I want to move my entire gubby to a different spot on the grid?

Edit each equation and change the center coordinates. If your head is at (0, 0) and you want to move it to (5, 5), change the head equation to (x − 5)² + (y − 5)² = 25, then add 5 to the x-coordinate and 5 to the y-coordinate in every other shape. This shifts everything together.

Can I make the eyes blink or move?

Desmos does not animate equations, so shapes stay still. However, you can create two versions of your gubby — one with eyes open and one with eyes closed — by typing both sets of equations and toggling them on and off by clicking the colored dots.

How do I make a gubby with a hat or other features?

Add more circles or lines above the head for a hat, or add circles to the sides for ears. Use the same equation format: decide where you want the shape, pick a center point, choose a radius, and type the equation. Each new shape is just another equation in the list.

Is there a way to make curved eyebrows or other curved features?

Yes, using functions like parabolas or parts of circles. For example, y = −(x − 2)² + 3 creates a downward-facing curve that could work as an eyebrow. Experiment with different functions and restrictions to create the curves you want.