How To Make A Half Circle In Desmos

Creating a half circle in Desmos is a fundamental skill that unlocks a surprising amount of creative and mathematical potential, from designing logos and artwork to modeling physical phenomena like projectile motion or architectural arches. Whether you are a student visualizing domain and range restrictions, a teacher building interactive activities, or an artist crafting detailed graphs, understanding the syntax for semicircles is essential. This guide walks through every method—from the explicit function approach to the more strong implicit and parametric forms—ensuring you have the right tool for every specific scenario Still holds up..

No fluff here — just what actually works.

The Explicit Function Method: Solving for y

The most intuitive way to draw a half circle in Desmos is by solving the standard circle equation for y. The standard equation for a circle centered at $(h, k)$ with radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$. To graph this as a function $y = f(x)$, you must isolate $y$, which introduces a plus-or-minus ($\pm$) symbol. This symbol represents the two halves: the top half (positive square root) and the bottom half (negative square root) Most people skip this — try not to..

To graph the upper half (a "smile" shape), type the following into the expression list:

$y = k + \sqrt{r^2 - (x-h)^2}$

To graph the lower half (a "frown" shape), use the negative root:

$y = k - \sqrt{r^2 - (x-h)^2}$

Breaking down the syntax:

• h and k: These represent the coordinates of the center. If you want the center at the origin $(0,0)$, simply omit them or type 0.
• r: This is the radius. Remember to type the radius value, not the diameter.
• sqrt(...): Desmos requires the function notation sqrt() for the square root. Do not use the radical symbol $\sqrt{}$ directly from a keyboard palette unless it converts to sqrt().
• Domain Restrictions: Crucially, Desmos automatically handles the domain restriction because the expression inside the square root, $r^2 - (x-h)^2$, must be non-negative. The graph will naturally stop exactly where the circle ends, at $x = h \pm r$.

Example: To draw a semicircle with radius $3$ centered at $(2, -1)$ opening upward: y = -1 + sqrt(9 - (x-2)^2)

This method is perfect for quick plotting and calculus operations (like finding derivatives or integrals), but it has a limitation: it fails completely for vertical half circles (left or right halves) because a vertical semicircle fails the vertical line test and cannot be written as a single function $y = f(x)$ Surprisingly effective..

The Implicit Equation Method: Domain and Range Restrictions

For maximum flexibility—including vertical half circles or partial arcs—the implicit form combined with restriction brackets { } is the professional standard. You start with the full circle equation and restrict either the domain (x-values) or the range (y-values) to carve out the half you need Turns out it matters..

The base equation for a circle centered at $(h,k)$ with radius $r$: $(x-h)^2 + (y-k)^2 = r^2$

In Desmos, type this exactly, using = (not := or y=) And that's really what it comes down to..

Creating Horizontal Halves (Top/Bottom)

Restrict the range (y-values) using curly braces at the end of the equation.

Top Half: (x-h)^2 + (y-k)^2 = r^2 {y >= k}

Bottom Half: (x-h)^2 + (y-k)^2 = r^2 {y <= k}

Creating Vertical Halves (Left/Right)

Restrict the domain (x-values). This is impossible with the explicit $y=$ method Not complicated — just consistent..

Right Half: (x-h)^2 + (y-k)^2 = r^2 {x >= h}

Left Half: (x-h)^2 + (y-k)^2 = r^2 {x <= h}

Why this is superior:

1. Orientation Agnostic: Works for any rotation of the half circle.
2. Clean Syntax: The geometry remains a single relation, not two separate functions.
3. Easy Shading: You can instantly turn the boundary into a shaded region (inequality) by changing = to <= or >=. For a solid half-disk: (x-h)^2 + (y-k)^2 <= r^2 {y >= k}.

The Parametric Method: Precision and Animation

Parametric equations define both $x$ and $y$ in terms of a third parameter, usually $t$ (often representing angle or time). This is the most powerful method for animation, oriented arcs, and dynamic labeling. In Desmos, parametric curves are entered as an ordered pair: (x(t), y(t)).

For a circle centered at $(h,k)$ with radius $r$, the standard parametrization is: $(h + r\cos(t),; k + r\sin(t))$

To make it a half circle, you simply restrict the domain of the parameter $t$. Think about it: by default, Desmos graphs $t$ from $0$ to $2\pi$ (or sometimes $-2\pi$ to $2\pi$). You restrict this by adding a condition in brackets at the end.

Standard Orientations (Counter-clockwise from 3 o'clock)

• Right Half: (h + r*cos(t), k + r*sin(t)) {0 <= t <= pi} Wait, standard param: t=0 is rightmost. t=pi is leftmost. So 0 to pi is the upper half? Let's check.
  • $t=0 \rightarrow (r, 0)$ (Rightmost)
  • $t=\pi/2 \rightarrow (0, r)$ (Top)
  • $t=\pi \rightarrow (-r, 0)$ (Leftmost)
  • So $0 \le t \le \pi$ draws the UPPER half.
  • $\pi \le t \le 2\pi$ draws the LOWER half.
• Upper Half: (h + r*cos(t), k + r*sin(t)) {0 <= t <= pi}
• Lower Half: (h + r*cos(t), k + r*sin(t)) {pi <= t <= 2pi}

Vertical Halves (Left/Right)

• Right Half: (h + r*cos(t), k + r*sin(t)) {-pi/2 <= t <= pi/2}
• Left Half: (h + r*cos(t), k + r*sin(t)) {pi/2 <= t <= 3pi/2}

Pro Tip for Animation: Replace the static bounds with a slider variable.

1. Create a slider a (set bounds $0$ to $2\pi$).
2. Type: (h + r*cos(t), k + r*sin(t)) {0 <= t <= a}
3. Press play on the slider a to watch the half circle draw itself in real-time. This is invaluable for presentations or understanding the parameter's role.

Using Sliders for Dynamic Exploration

Static graphs are useful, but Desmos shines when variables become interactive. Turn $h$, $k$, and $r$ into sliders to explore transformations in real-time.

1. Type h = 0, k = 0, r = 2 in separate lines.

2. Click the "add slider" prompt that appears for each

3. Click the “add slider” prompt that appears for each variable and configure it to suit your exploration. Desmos will automatically create a slider labeled h, k, and r with default bounds of 0 to 10. You can fine‑tune each one by clicking the gear icon and specifying:

• Minimum / Maximum – set the range of possible values (e.g., h: -5 … 5, k: -5 … 5, r: 0.5 … 10).
• Step – control the granularity of movement (useful for fine‑tuning radius or center positions).
• Initial value – the starting point when the graph first loads.
• Show value – toggle whether the current number appears next to the slider.

With these adjustments you can instantly see how the half‑circle shifts, stretches, or flips as you drag the controls It's one of those things that adds up. Less friction, more output..

Combining Sliders with the Parametric Half‑Circle

Now that h, k, and r are interactive, pair them with the parametric half‑circle to create a dynamic half‑disk that updates in real time:

(h + r*cos(t), k + r*sin(t)) {0 <= t <= pi}          // upper half
(h + r*cos(t), k + r*sin(t)) {pi <= t <= 2pi}        // lower half
(x - h)^2 + (y - k)^2 <= r^2 {y >= k}               // shaded half‑disk

• The first two lines draw the two semicircular arcs.
• The third line uses the same h, k, and r sliders to shade the interior of the half‑disk (the inequality {y >= k} restricts it to the upper side).

Because all three objects reference the same sliders, moving any of them instantly redraws the arcs and the shaded region, giving you an intuitive feel for how the center and radius affect the shape.

Animating the Parameter with a Time‑Based Slider

If you want to animate the drawing process itself, introduce a second slider that acts as a “time” variable:

a = 0                     // slider a: 0 → 2π, step 0.01
(h + r*cos(t), k + r*sin(t)) {0 <= t <= a}           // half‑circle draws up to angle a

Press the play button on slider a and watch the half‑circle trace out from its starting point (rightmost) to its endpoint (leftmost) in a smooth, continuous motion. This technique is especially handy for:

• Classroom demonstrations – illustrate how the parameter t maps to points on the curve.
• Presentations – create a live‑drawing effect that highlights the relationship between angle and position.
• Concept videos – use the animation as a visual aid to explain parametric equations.

Advanced Tricks with Sliders

People argue about this. Here's where I land on it.

Putting It All Together – A Complete Example

Below is a ready‑to‑paste Desmos script that demonstrates a fully interactive half‑disk with animation, shading, and dynamic labeling:

// 1. Sliders for center and radius
h = -2
k = 0
r = 3

// 2. Time slider for animation
a = 0   // bounds: 0 → 2π, step: 0.01

// 3. Upper