In the realm of mathematical graphing, the almighty circle reigns supreme as a symbol of perfection and endless possibilities. Its smooth, symmetrical form encapsulates countless applications, from celestial bodies to engineering marvels. With the advent of digital graphing tools like Desmos, creating circles has become as effortless as tracing a finger in the sand. Step into the captivating world of Desmos, where we embark on an enlightening journey to unveil the secrets of crafting circles with the utmost precision.
At the heart of Desmos lies a user-friendly interface that empowers you to effortlessly summon circles onto your virtual canvas. With just a few simple commands, you can conjure circles of any size, centered at any point on the coordinate plane. By specifying the coordinates of the circle’s center and its radius, you gain complete control over its position and dimensions. Desmos’ intuitive syntax makes this process as smooth as gliding on ice, ensuring that even novice graphers can produce stunning circular masterpieces.
However, the true magic of Desmos lies in its versatility. Not content with mere static circles, Desmos empowers you to unleash your creativity by creating circles that dance and transform before your eyes. By incorporating animation effects, you can watch circles expand, shrink, and slide effortlessly across the screen. Moreover, the ability to define circles parametrically opens up a whole new world of possibilities, allowing you to generate circles with intricate patterns and awe-inspiring movements. Desmos becomes your playground, where circles are not just mathematical objects but dynamic works of art.
Creating a Circle Using the Equation
A circle in Desmos can be defined using its equation. The general equation of a circle is x^2 + y^2 = r^2, where (x, y) are the coordinates of any point on the circle and r is the radius. To create a circle using this equation, follow these steps:
- Enter the equation in the input field: Click on the “New Graph” button in the top toolbar. A new graph will appear in the workspace. In the input field below the graph, type in the equation of the circle. For example, to create a circle with radius 5 centered at the origin, type in the equation x^2 + y^2 = 25.
- Adjust the equation as needed: Once you have entered the equation, you can adjust the values of r and (x, y) to change the size and position of the circle. For example, to change the radius to 10, you would change the equation to x^2 + y^2 = 100.
- Press enter: After adjusting the equation, press the enter key to create the circle. The circle will appear in the graph.
- Open Desmos and click on the “Graph” tab.
- Click on the “Add Function” button and enter the following equation:
- Replace
hwith the x-coordinate of the circle’s center,kwith the y-coordinate of the circle’s center, andrwith the radius of the circle. - Click on the “Enter” button.
- Open Desmos in your web browser.
- Click on the “Graph” tab.
- In the “Function” field, enter the following equation: `(x – h)^2 + (y – k)^2 = r^2`
- Replace `h` with the x-coordinate of the center of the circle, `k` with the y-coordinate of the center of the circle, and `r` with the radius of the circle.
- Click on the “Graph” button.
By using the equation, you can create circles of any size and position. This method is particularly useful when you want to precisely control the dimensions of the circle.
Defining the Radius and Center
The radius of a circle is the distance from the center of the circle to any point on the circle. The center of a circle is the point equidistant from all points on the circle.
Further Detail on Defining the Center
To define the center of a circle in Desmos, you can use the following syntax:
| Syntax | Description |
|---|---|
| (x1, y1) | The center of the circle is located at the point (x1, y1). |
For example, to define a circle with center at the point (2, 3), you would use the following syntax:
(x - 2)^2 + (y - 3)^2 = r^2
Where r is the radius of the circle.
Using Parameters and Sliders
Desmos provides a variety of tools to help you create circles. One such tool is the parameter slider. Parameter sliders allow you to dynamically change the values of parameters in your equations. This can be incredibly useful for exploring different shapes and graphs.
To create a parameter slider, simply click on the “Sliders” button in the Desmos toolbar. This will open a menu where you can choose the parameters you want to control with sliders. Once you have selected your parameters, click on the “Create” button.
Your parameter slider will appear in the upper-right corner of your Desmos graph. You can use the slider to adjust the values of your parameters in real-time. This allows you to explore different shapes and graphs without having to re-enter your equations.
Here are some examples of how you can use parameter sliders to create circles:
1. Create a slider for the radius of a circle:
“`
radius = slider(0, 10)
circle(0, 0, radius)
“`
2. Create a slider for the center of a circle:
“`
x = slider(-10, 10)
y = slider(-10, 10)
circle(x, y, 5)
“`
3. Create a slider for the color of a circle:
“`
color = slider(0, 360)
circle(0, 0, 5, {color: “hsl(” + color + “, 100%, 50%)”})
“`
Drawing a Circle with a Given Radius
To draw a circle with a given radius in Desmos, follow these steps:
“`
(x – h)^2 + (y – k)^2 = r^2
“`
The circle will be drawn on the graph. You can use the “Slider” tool to adjust the value of r and see how the circle changes.
Example:
To draw a circle with a radius of 5 centered at the origin, enter the following equation into the “Add Function” box:
“`
(x – 0)^2 + (y – 0)^2 = 5^2
“`
Click on the “Enter” button and the circle will be drawn on the graph.
| Expression | Description |
|---|---|
| (x – h)^2 | The horizontal distance from the point (x, y) to the center of the circle, (h, k) |
| (y – k)^2 | The vertical distance from the point (x, y) to the center of the circle, (h, k) |
| r^2 | The square of the radius of the circle |
Centering the Circle on the Origin
To center the circle on the origin, you need to specify the coordinates of the center as (0,0). This will place the circle at the intersection of the x-axis and y-axis.
Step 5: Fine-tuning the Circle
Once you have the basic circle equation, you can fine-tune it to adjust the appearance and behavior of the circle.
Here is a table summarizing the parameters you can adjust and their effects:
| Parameter | Effect |
|---|---|
| a | Scales the circle horizontally |
| b | Scales the circle vertically |
| c | Shifts the circle horizontally |
| d | Shifts the circle vertically |
| f(x) | Changes the orientation of the circle |
By experimenting with these parameters, you can create circles of various sizes, positions, and orientations. For example, to create an ellipse, you would adjust the values of a and b to different values.
Shifting the Circle with Transformations
To shift the circle either vertically or horizontally, we need to use the transformation equations for shifting a point. For example, to shift a circle with radius r and center (h,k) by a units to the right, we use the equation x → x + a.
Similarly, to shift the circle by b units upward, we use the equation y → y + b.
The following table summarizes the transformations for shifting a circle:
| Transformation | Equation |
|---|---|
| Shift a units to the right | x → x + a |
| Shift b units upward | y → y + b |
Example:
Shift the circle (x – 3)^2 + (y + 1)^2 = 4 by 2 units to the right and 3 units downward.
Using the transformation equations, we have:
(x – 3) → (x – 3) + 2 = x – 1
(y + 1) → (y + 1) – 3 = y – 2
Therefore, the equation of the transformed circle is: (x – 1)^2 + (y – 2)^2 = 4
Creating an Equation for a Circle
To represent a circle using an equation in Desmos, you’ll need the general form of a circle’s equation: (x – h)² + (y – k)² = r². In this equation, (h, k) represents the center of the circle and ‘r’ represents its radius.
For example, to graph a circle with its center at (3, 4) and radius of 5, you would input the equation (x – 3)² + (y – 4)² = 25 into Desmos.
Customizing Line Style and Color
Once you have the basic circle equation entered, you can customize the appearance of the graph by modifying the line style and color.
Line Style
To change the line style, click on the Style tab on the right-hand panel. Here, you can choose from various line styles, including solid, dashed, dotted, and hidden.
Line Thickness
Adjust the Weight slider to modify the thickness of the line. A higher weight value results in a thicker line.
Line Color
To change the line color, click on the Color tab on the right-hand panel. A color palette will appear, allowing you to select the desired color for your circle.
Custom Color
If you want to use a specific color that is not available in the palette, you can input its hexadecimal code in the Custom field.
Color Translucency
Use the Opacity slider to adjust the translucency of the line. A lower opacity value makes the line more transparent.
| Property | Description |
|---|---|
| Line Style | Determines the appearance of the line (solid, dashed, dotted) |
| Line Thickness | Adjusts the width of the circle’s outline |
| Line Color | Sets the color of the circle’s outline |
| Custom Color | Allows you to input specific color codes for the outline |
| Color Translucency | Controls the transparency of the circle’s outline |
Animating the Circle
To animate the circle, you can use the sliders to control the values of the parameters a and b. As you move the sliders, the circle will change its size, position, and color. You can also use the sliders to create animations, such as making the circle move around the screen or change color over time.
Creating an Animation
To create an animation, you can use the “Animate” button on the Desmos toolbar. This button will open a dialog box where you can choose the parameters you want to animate, the duration of the animation, and the number of frames per second. Once you have chosen your settings, click the “Start” button to start the animation.
Example
In the following example, we have created an animation that makes the circle move around the screen in a circular path. We have used the “a” and “b” parameters to control the size and position of the circle, and we have used the “color” parameter to control the color of the circle. The animation lasts for 10 seconds and has 30 frames per second.
| Parameter | Value |
|---|---|
| a | sin(t) + 2 |
| b | cos(t) + 2 |
| color | blue |
Using Properties to Measure the Circle
Once you have created a circle in Desmos, you can use its properties to measure its radius, circumference, and area. To do this, click on the circle to select it and then click on the “Properties” tab in the right-hand panel.
The Properties tab will display the following information about the circle:
Radius
The radius of a circle is the distance from the center of the circle to any point on the circle. In Desmos, the radius is displayed in the Properties tab as “r”.
Center
The center of a circle is the point that is equidistant from all points on the circle. In Desmos, the center is displayed in the Properties tab as “(h, k)”, where h is the x-coordinate of the center and k is the y-coordinate of the center.
Circumference
The circumference of a circle is the distance around the circle. In Desmos, the circumference is displayed in the Properties tab as “2πr”, where r is the radius of the circle.
Area
The area of a circle is the amount of space inside the circle. In Desmos, the area is displayed in the Properties tab as “πr²”, where r is the radius of the circle.
Exploring Advanced Circle Functions
### The Equation of a Circle
The equation of a circle is given by:
“`
(x – h)^2 + (y – k)^2 = r^2
“`
where:
* (h, k) is the center of the circle
* r is the radius of the circle
### Intersecting Circles
Two circles intersect if the distance between their centers is less than the sum of their radii. The points of intersection can be found by solving the system of equations:
“`
(x – h1)^2 + (y – k1)^2 = r1^2
(x – h2)^2 + (y – k2)^2 = r2^2
“`
where:
* (h1, k1), r1 are the center and radius of the first circle
* (h2, k2), r2 are the center and radius of the second circle
### Tangent Lines to Circles
A tangent line to a circle is a line that touches the circle at exactly one point. The equation of a tangent line to a circle at the point (x0, y0) is given by:
“`
y – y0 = m(x – x0)
“`
where:
* m is the slope of the tangent line
* (x0, y0) is the point of tangency
### Advanced Circle Functions
#### Circumference and Area
The circumference of a circle is given by:
“`
C = 2πr
“`
where:
* r is the radius of the circle
The area of a circle is given by:
“`
A = πr^2
“`
#### Sector Area
The area of a sector of a circle is given by:
“`
A = (θ/360°)πr^2
“`
where:
* θ is the central angle of the sector in degrees
* r is the radius of the circle
#### Arc Length
The length of an arc of a circle is given by:
“`
L = (θ/360°)2πr
“`
where:
* θ is the central angle of the arc in degrees
* r is the radius of the circle
How To Make A Circle In Desmos
Desmos is a free online graphing calculator that can be used to create a variety of graphs, including circles. To make a circle in Desmos, you can use the following steps:
Your circle will now be displayed in the graph window.
People Also Ask About How To Make A Circle In Desmos
How do I make a circle with a specific radius?
To make a circle with a specific radius, simply replace the `r` in the equation with the desired radius.
How do I make a circle that is not centered at the origin?
To make a circle that is not centered at the origin, simply replace the `h` and `k` in the equation with the desired x- and y-coordinates of the center of the circle.
How do I make a filled circle?
To make a filled circle, click on the “Style” tab and select the “Fill” option.
