Project Results
Interactive Spiral Diagram
Use the mouse wheel to zoom. Drag the diagram to pan. Labels can be hidden for dense spirals.
Calculator and Project Settings
Step-by-Step Construction Table
| Step | Previous radius | Fixed leg | New radius | Exact radius | Added angle | Total angle | Area | X | Y | Copy |
|---|
Formula Used
The spiral begins with radius r0 = a. Every triangle adds a perpendicular leg of length b. The next radius follows the Pythagorean theorem.
For unit lengths, the radii become √1, √2, √3, and onward. Triangle step n ends at radius √(n + 1). This indexing keeps every construction step clear and consistent.
An = ½ × b × rn−1
xn = x0 + rn cos(θn)
yn = y0 + rn sin(θn)
The total displayed area sums every constructed triangle. Later triangles can overlap visually after many rotations. Treat that sum as construction area, not union area.
How to Use This Calculator
- Choose a preset or enter custom spiral dimensions.
- Set triangle count, starting point, and initial direction.
- Select units, precision, diagram labels, and paper settings.
- Press the calculate button to rebuild every result.
- Inspect the selected step and construction table carefully.
- Zoom, pan, print, or export the finished project.
Start with seventeen triangles for a traditional classroom construction. Use fewer labels when the diagram becomes visually crowded. Export measurements before drawing your final project by hand.
Worked Example
Set both leg lengths to one unit. Generate seventeen triangles from a zero-degree starting direction. The seventeenth new radius equals the square root of eighteen.
| Input | Example value | Purpose |
|---|---|---|
| Triangles | 17 | Creates seventeen right triangles. |
| Starting leg | 1 unit | Creates the initial radius. |
| Fixed leg | 1 unit | Adds one perpendicular side each step. |
| Final radius | √18 ≈ 4.2426 units | Shows the seventeenth constructed hypotenuse. |
Compass-and-Ruler Construction Guide
- Draw the starting radius using the selected length.
- Construct a perpendicular line at its outer endpoint.
- Mark the fixed leg length along that perpendicular.
- Connect the new point back to the original center.
- Repeat from each new endpoint using identical steps.
- Number points and verify lengths against the table.
Keep every perpendicular construction accurate before extending the spiral. Small angle errors grow across later triangles. Frequent radius checks protect the final geometry project.
Frequently Asked Questions
What is a Pythagorean spiral?
It is a chain of connected right triangles. Each new triangle uses the previous hypotenuse as one leg. The radii follow square-root values in sequence.
Is it also called the Spiral of Theodorus?
Yes, both names describe the same geometric construction. The historical attribution is commonly linked with Theodorus of Cyrene. Classroom projects often use seventeen triangles.
Why does the radius formula contain n?
Each triangle adds another squared fixed leg. Repeating that addition produces the term n times b squared. The square root gives the new hypotenuse.
Can the leg length differ from one?
Yes, any positive starting and fixed lengths are supported. Equal scaled lengths create a proportional version of the standard spiral. Different lengths create a generalized construction.
Why do later triangles overlap?
The cumulative turning angle eventually wraps around the center. New triangles can then cross earlier regions. The calculator still preserves every mathematical construction step.
How should I print the diagram?
Select paper dimensions and an appropriate drawing scale. Check the project warning before printing. Landscape orientation usually fits wider spirals better.
What does exact radical form mean?
It shows a square root before decimal approximation. Integer dimensions can sometimes produce simplified radicals. Decimal dimensions remain displayed through their full radicand expression.
Can I export coordinates for other software?
Yes, coordinates can be copied or downloaded. Available formats include Cartesian, polar, JSON, text, and SVG paths. CSV files contain every calculated step.
How many triangles should a school project use?
Seventeen triangles match the traditional classroom version. Larger models reveal more rotation but require better precision. Choose a count that fits your paper.