Arch Designs

Trefoil Arch

Arches in buildings often combine several curved parts into one outline. Before the outline is drawn on paper or marked on a surface, its supports and arc centres must be planned. A trefoil arch has three rounded lobes: a central lobe and matching lobes on the sides.

Begin with a base segment AD. Construct equal inward-facing angles at A and D, using the angle-copying method. On the two resulting rays mark B and C so that AB = CD. The matching base angles and equal side lengths give a symmetric supporting framework. BC forms the support for the central lobe.

The source invites you to adjust the radii to make an arch look pleasing. However, the two side changes must match. Changing only one radius while keeping the other unchanged would generally break the mirror symmetry. Keep the framework visible until you have checked both sides.

Equal Support Lines and Angles

The framework can be constructed exactly, and then used to produce a particular arch. One straightforward version uses each of AB, BC, and CD as a diameter of a semicircle. A diameter passes through a circle’s centre, so finding its midpoint locates the required arc centre.

1. Draw AD and construct matching inward angles at A and D.
2. Choose one length and use the compass to mark AB = CD on the two rays. Join BC.
3. Construct the midpoint M of AB, N of BC, and P of CD using perpendicular bisectors.
4. With centre M and radius MA, draw the outer semicircle from A to B. With centre N and radius NB, draw the upper semicircle from B to C.
5. With centre P and radius PC, draw the matching outer semicircle from C to D. Trace the three lobes as the arch outline.

This is a specified construction, rather than the only possible trefoil. Here the radius of a semicircle is half its diameter. Since AB = CD, the two side radii are equal. The midpoint of BC lies on the symmetry line of the framework, so its top arc is symmetric too. The support segments are not part of the finished outline.

A curved outline may be made from several exact arcs without being one circle. Each part has its own centre and radius. When drawing it, stop at the intended joining point; otherwise an arc extension can become an unwanted extra curve.

A Pointed Arch

Some arches meet in a point rather than a rounded central lobe. In the source framework, two segments of equal length meet at the top, and their midpoints help locate the change between the upper and lower curved parts. The previous idea of repeated, matching geometry is still useful.

Draw equal segments from the top point to the two feet of the arch. Construct their midpoints rather than estimating them. These split the support lengths into matching halves. Build a chosen circular arc on each half and copy the corresponding construction to the other side. The matching pairs create an outline with the same changes in curvature on both sides.

To locate a circular arc through two chosen endpoints, use the perpendicular bisector of their joining segment: every possible centre on that line is equally far from those endpoints. Choose a centre and radius that reproduce the intended curve, and reflect or copy that choice for the matching side. Centres at different distances from a chord produce different curvature. Keep any shared joining points fixed while investigating those choices.

Here is a fully specified version of that idea. Let T be the top, L and R the feet, and M and N the midpoints of TL and TR. Use a symmetric sloping framework so the construction below has distinct intersections. A horizontal line through T will fix a centre for the upper arc; reflection through M then fixes a matching lower centre.

1. Draw the symmetry line through T and the midpoint of LR, then construct a perpendicular to that line through T.
2. Construct the perpendicular bisector of TM. Let C be its intersection with the new perpendicular through T on the left side. C is equally far from T and M.
3. With centre C and radius CT, draw the short arc from T to M.
4. Extend CM past M and mark C′ with MC′ = MC. Because M is also the midpoint of TL, reflection through M sends T to L and C to C′.
5. With centre C′ and the same radius, draw the short arc from M to L that continues the outline.
6. Copy the two centres and matching arcs symmetrically on the right arm, using N. Trace the four curved parts and retain the support lines faintly.

Changing the Radii

An arch can become flatter or more strongly curved when its arc centres move. The endpoints and symmetry conditions guide those changes. A radius cannot be changed independently of the centre if the same two endpoints must remain on the arc.

Making Your Own Arch

Designing your own arch gives you a reason to combine several earlier constructions. Start from the relationships you want to preserve: a base midpoint, equal arms, matching angles, and pairs of arcs. Then choose details that create your preferred outline.

Make two versions of an arch on the same base. Keep a labelled first version with all supporting lines and centres, and a clean second version with only the outline. Compare the versions by explaining which compass distances remain equal. This provides a clearer justification than saying that the arch looks balanced.

Architectural forms are useful context, but this lesson concerns how their planar outlines can be constructed. A drawing of an arch is not by itself a calculation of its structural strength. Concentrate on the geometry that determines the visible shape.

Check Your Understanding

Use these questions to check the conditions behind each method, as well as the result. For a diagram-based question, follow the labels and given information rather than judging by appearance.