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Use a compass and straightedge, but treat these as two different constructions. An equilateral Gothic arch is an open profile made from two equal circular arcs; a Reuleaux triangle is a closed, constant-width curve made from three arcs. Start with the arch if you need a pointed opening, then use the triangle procedure when you need a precisely defined curved shape.
Tools and setup
- Compass capable of holding a fixed radius
- Straightedge for the span, triangle, and construction lines
- Sharp pencil and a clean, flat drawing surface
Keep the compass point firmly on each centre and do not change the radius between matching arcs. Draw light construction lines first, then darken only the finished curves.
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Construct an equilateral Gothic arch
1. Draw the springing line
Draw a horizontal line segment AB. Its length is the opening span of the arch. Points A and B are the springing points where the curves begin.
2. Transfer the span as the radius
Set the compass point on A and open it to exactly B. This makes the compass radius equal to the full span AB.
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3. Draw the two sides
- With A as centre, draw an arc upward from B.
- Without changing the radius, place the compass point on B and draw a matching arc upward from A.
- Mark the arcs’ intersection as the apex.
The two arcs meet at the apex and form the pointed opening. A, B, and the apex are the vertices of an equilateral triangle, so this particular profile is an equilateral pointed arch.
What the drawing does—and does not—show
This is a geometric profile, not a structural design. A built masonry arch must be assessed for its geometry, thrust, support conditions, and possible movement; a pointed outline alone does not establish that it is safe to construct.
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Construct a Gothic arch from a specified span and rise
If the opening span and the required rise are both fixed, the circle centres may not be the two springing points. Draw the span AB and mark the desired apex. Join one springing point to the apex, then construct the perpendicular bisector of that line. The intersection of this bisector with the springing line locates the centre for that side’s arc. Mirror the construction from the other springing point to locate the second centre.
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The centre locations determine the profile:
| Centre locations | Resulting profile |
|---|---|
| At the two span endpoints | Equilateral arch |
| Outside the span | Acute or lancet arch |
| Inside the span | Drop arch |
“Gothic arch” is therefore a family of pointed profiles, not one universal curve. Confirm the intended type before setting out work at full scale.
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Construct a Reuleaux triangle
1. Draw the generating triangle
Draw an equilateral triangle and label its vertices A, B, and C. The side length AB will define both the compass radius and the finished curve’s constant width.
2. Draw the three arcs
- Set the compass radius to AB. With A as centre, draw the arc from B to C.
- Keep the same radius. With B as centre, draw the arc from C to A.
- With C as centre, draw the arc from A to B.
Those three arcs meet to form the closed Reuleaux triangle. Its boundary is entirely curved; it is not an ordinary triangle whose straight sides have been rounded with arbitrary fillets.
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Why its width is constant
Each boundary arc has the same radius as a side of the generating equilateral triangle, and each arc is centred on the opposite vertex. In every direction, the distance between parallel supporting lines is therefore the side length of the original equilateral triangle. That distance is the Reuleaux triangle’s constant width.
Arch or Reuleaux triangle: which construction fits?
| Feature | Gothic equilateral arch | Reuleaux triangle |
|---|---|---|
| Starting geometry | Horizontal span AB | Equilateral triangle ABC |
| Radius | The full opening span | One side of the equilateral triangle |
| Boundary | Two arcs meeting at an apex | Three arcs forming a closed curve |
| Typical use | Pointed architectural opening or profile | Constant-width geometric design |
| Profile options | Equilateral, lancet, or drop, depending on centre locations | Fixed by the equal-radius three-arc construction |
Accuracy checks before you finish
- For the equilateral arch, verify that both arcs use the unchanged span AB as their radius.
- Check that the two arch arcs meet at one clearly marked apex.
- For a specified-rise arch, confirm each centre lies on the perpendicular-bisector construction for its side.
- For the Reuleaux triangle, confirm all three arcs have the same radius and are centred at A, B, and C respectively.
- Use a straightedge only for construction lines; the finished boundaries should be compass-drawn arcs.
Materials note
A basic geometry compass is sufficient for these paper or layout exercises because both methods rely on transferring a radius and drawing circular arcs. The construction does not establish a particular brand, precision rating, or suitability for structural site work.
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